Learning Survival Distributions with Individually Calibrated Asymmetric Laplace Distribution
Authors: Deming Sheng
Ricardo Henao
Venue: ICLR 2026 Poster (2026)
Keywords: Machine Learning; Probabilistic methods; Survival Analysis; Asymmetric Laplace Distribution; Calibration
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We propose ICALD, a unified framework for individually calibrated survival prediction using the Asymmetric Laplace Distribution, supporting both pre- and post-calibration, and achieving competitive performance across 21 datasets.

Abstract: Survival analysis plays a critical role in modeling time-to-event outcomes across various domains. 
Although recent advances have focused on improving _predictive accuracy_ and _concordance_, fine-grained _calibration_ remains comparatively underexplored. 
In this paper, we propose a survival modeling framework based on the Individually Calibrated Asymmetric Laplace Distribution (ICALD), which unifies _parametric_ and _nonparametric_ approaches based on the ALD.
We begin by revisiting the probabilistic foundation of the widely used _pinball_ loss in _quantile regression_ and its reparameterization as the _asymmetry form_ of the ALD. 
This reparameterization enables a principled shift to _parametric_ modeling while preserving the flexibility of _nonparametric_ methods.
Furthermore, we show theoretically that ICALD, with the _quantile regression_ loss is probably approximately individually calibrated.
Then we design an extended ICALD framework that supports both _pre-calibration_ and _post-calibration_ strategies. 
Extensive experiments on 14 synthetic and 7 real-world datasets demonstrate that our method achieves competitive performance in terms of _predictive accuracy_, _concordance_, and _calibration_, while outperforming 12 existing baselines including recent _pre-calibration_  and _post-calibration_ methods.

Published as a conference paper at ICLR 2026
LEARNING SURVIVAL DISTRIBUTIONS WITH INDIVIDUALLY CALIBRATED ASYMMETRIC LAPLACE DISTRIBUTION
Deming Sheng, Ricardo Henao*
{ (deming. sheng, ricardo. henao duke edu
ABSTRACT
Survival analysis plays critical role in modeling time-to-event outcomes across various domains. Although recent advances have focused on improving predic- tive accuracy and concordance, fine-grained calibration remains comparatively underexplored. In this paper, we propose a survival modeling framework based on the Individually Calibrated Asymmetric Laplace Distribution (ICALD), which unifies parametric and ronparametric approaches based on the ALD. We begin by revisiting the probabilistic foundation of the widely used pinball loss in quantile regression and its reparameterization reparameterization enables principled shift to parametric modeling while preserv- ing the flexibility of nomparametric methods. Furthermore, we show theoretically that ICALD, with the quantile regression loss is probably approximately individ- ually calibrated. Then design extended ICALD framework that supports both rre-calibration and cost-calibration strategies. Extensive experiments on 14 synthetic and real-word wallw datasets demonstrate that our method achieves competi- tive performance in terms of predictive accuracy, concordance, and calibration, while outperforming 12 existing baselines including recent pre-calibration and post-calibration methods.
the asymmetry form of the ALD. This
INTRODUCTION
Survival analysis, modeling the distribution of time-to-event outcomes, has gained popularity in recent years (Lánczky & Györffy 2021, Emmerson & Brown 2021). Survival models can be broadly categorized into parametric, remi-parametric and nonparametric approaches, depending on the strength of the distributional assumptions they make about the underlying time-to-event distribution. Parametric models make strong assumptions by specifying a particular probability distribution for event times, such the exponential (Feigl & Zelen| 1965). Weibull (Scholz & Works 1996). log- normal RRoston 2001), asymmetric Laplace (Kotz et al. 2012), or their mixtures (Nagpal et al. 2021) In contrast, remi-parameets model, such as the Cox proportional hazards (CPH) model (Cox 1972). relax these assumptions by modeling the hazard multiplicatively without specifying the baseline hazard. Nomparametric models, such as Gradient Boosting Machines (GBM) (Dembek et al. 2014) and Random Survival Forests (RSF) (Ishwaran et al., 2008). do not rely on predefined forms for the survival distribution or the hazard function, and instead estimate survival quantities directly from the data. Alternatively, neural networks have significantly advanced survival modeling in both (semi--parametric and nonparametric paradigms. For example, the parametric LogNorm- (Hoseini et al.] (017)) improves parameter estimation for survival distributions under the log-normal assumption (Royston| 2001). The semi-parametric DeepSurv model (Katzman et al. 2018) leverages neural networks to capture complex nonlinear covariate effects while preserving the multiplicative structure of Cox models (Cox) 1972). Nonparametrie models such as DeepHit (Lee et al. (018)) and CQRNN (Pearce et al. (222)) utilize neural architectures to directly estimate individualized survival distributions, offering greater expressiveness and scalability than non-neural models like GBM (Dembek al. 2014) and RSF (Ishwaran et al. 2008
model
"Corresponding author, Duke University.

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To assess the performance of survival models, evaluation metrics are typically grouped into three major categories: predictive accuracy (Graf et al., 1999). concordance (Harrell et al., 1982 Uno et al., 20TI), and calibration (Haider al.] 2020). Predictive accuracy measures how well the estimated survival probabilities or times align with the observed outcomes, and is particularly useful in scenarios where precise event time estimates are critical, such as predicting patient prognosis, forecasting treatment duration, or scheduling follow-up assessments. Concordance assesses the model's ability to correctly rank individuals by risk, making it valuable for pairwise comparisons, such as prioritizing patients for treatment. Calibration reflects the reliability of the predicted survival probabilities, i.e., if predicted risks are consistent with empirical observations. More specifically, calibration can be further assessed at the average, group, individual level (Gneiting et al. 2007). Individual calibration (Villani 2009) is important for high-stakes decisions at the patient level, such as eligibility for high-risk interventions, while average and group calibration (Naeini et al.) 2015) are more relevant for population-leve -leeel decisions, such as allocating clinical resources or designing screening strategies.
all,
While most recent works (Katzman et al. 2018 Lee et al. 2018 Pearce et al. 2022) primarily
focus on predictive accuracy, concordance, and coarse- coarse-graine grained notions of calibration, our objective is to conduct a more comprehensive evaluation of model performance, with particular emphasis on fine-grained calibration (Gneiting al.) 2007: Villani 2009 Naeini et al., 2015). In addition, we propose the Individually Calibrated Asymmetric Lapaae Distribution (ICALD) model, which significantly improves the calibration performance. Our contributions are summarized below.
We propose the ICALD model that synthesizes the complementary advantages of parametric and nonparametric ALD-based approaches. Our model not only enhances calibration and mitigates issues associated with distribution mismatch in parametric ALD approach (Sheng
& Henao 2025), but also effectively addresses the issues of discretization and quantile crossing commonly encountered in nonparametric methods (Pearce et al. 2022).
Specifically, ICALD admits two theoretically equivalent loss functions, each of which is provably capable of rendering the model Probably Approximately Individually Calibrated (PAIC: /Zhoo al. (020)). More importantly, the model supports both pre-calibration and post-calibration with either loss, providing a unified and flexible framework where the calibration strategy and loss function can be independently selected based on the specific application requirements.
We comprehensively evaluate method on 14 synthetic and 7 real-world datasets using 7 performance metrics that span predictive accuracy, concordance and calibration. Our method is compared against 9 strong baselines covering a wide spectrum of survival models, including both gemii-)paaametri and nonparametric approaches, as well as both neural and non-neural architectures. Furthermore, we compare with 1 pre-calibration method (X-CAL: Goldstein et al. 2020) and 2 post-calibration methods (CSD; Qi et al. 20244 and CiPOT: Qi et al. 2024bb that target average calibration of survival distributions. Overall, our method consistently outperforms these baselines, achieving superior performance in terms of predictive, concordance and calibration.
2 BACKGROUND
We use capital letters X, Y, Q to denote random variables, lowercase letters y, q to denote fixed values, bold lowercase letters x to denote vectors, and X, y, Q to denote the sets of all possible values they can take.
Survival Data A survival dataset D is composed of a set of triplets {(xn: Un. 8,,) ) n= each contain- ing covariates x, E IRdd, an observed time yv E TR,, and an event indicator 5, E {0, Moreover, the observed time is defined as the minimum between the true event time e, and the censoring time ca. i.e., Vn- min(en, Cn), and the event indicator is defined as 8, = I(e., I Cx). denoting whether the event was observed (6,, = 1) or censored (6,, = 0). In this work, we adopt the common assumption that the event and censoring distributions are conditionally independent given the covariates, i.e., e I c x. Moreover, although we focus on right-censored data; less common types of censoring (e.g., e, left and interval) can also be readily accommodated (Klein & Moeschherger 2006).
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Asymmetric Laplace Distribution The Asymmetrie Laplace Distribution (ALD) Kotz al. 2012
has two common parameterizations. In its quantile form, let the random variable Y ~ A o. q).
where 0 R is the location (distribution mode), is the scale, and g (0. 1) is the target quantile. This form is quite useful in quantile regression (Koenker & Bassett Jr following probability density function (PDF):
1978). and has the
9(1 9) exp g y))
y 2 8
f(y:0,0,9)
(I)
o
exp (- (y 0)) y 0.
Alternatively, in its asymmetry form, the ALD is reparameterized as AC(O,, Ø. x), where x > 0
denotes the asymmetry parameter, and the latter is related to through g It Its PDF is:
x²
fly:0, V2
K
exp
/2x (0 v) (y 0))
0,
(2)
1
R2
exp
<< 0.
Quantile Value and Percentage Given a random variable Y with its conditional cumulative distribu- tion function (CDF) (y)x), the quantile value Yg corresponding a quantile percentage q E [0 is defined as (Garthwaite et al. 2002):
3/ Fr (gx) - inf e R Fr 2 g).
where V4 represents the threshold below which proportion q of observations y ) lie. The quantile percentage q indicates the probability mass accumulated up to y/.. i.e., the fraction of the distribution that falls below this threshold.
Average Calibration CDF predictive model F& (y/x)) parameterized by & is said to be perfectly averagely calibrated, if its predicted distribution aligns with the empirical distribution of the target population (represented by test set). Formally, the model should satisfy (Gneiting et al. 2007):
Pr (y S F5 (glx)xx E X)
or Pr Pr x) S q|x E 2) = Ai E [0, 1), Ay EV. (4)
Group Calibration A CDF predictive model Fe x) parameterized by & is said to be perfectly
group calibrated with respect to a collection = { C X predefined subsets if, for every group Sg, the predicted distribution is consistent with the empirical outcome distribution within that group. Formally, the model should satisfy (Gneiting et al. 2007p:
Pr (y S (glx) x E Sx) q or Pr (F & g/x E Sx) = 9, Ag E [0, 1]. Vy E y. S. (5)
Individual Calibration A CDF predictive model parameterized by &b is said to be perfectly individu- ally calibrated if its predicted conditional cumulative distribution function F& (Y|x) satisfies, for any given input x E X, the following condition (Gneiting et al., 2007):
Pr (Y S FF X x) = qq or Pr (Fo (Vlx) S g = x) = q. 4g E [0, 1), WE EV. (6) Definition (Probably Approximately Individually Calibrated (PAIC) (Zao et al. 2020p. A model with predictive CDF (Y(((x) (x) is said to be (e, -PAAI for all X, Y V, and q [0, 1], the following holds:
Pr V.
9f dg s or Pr V/ [Pr [P < al dg S
2 1 6.
We slightly extended the original definition for PAIC introduced by Zhao al. 2020 to also allow an equivalent expression based on the inverse CDF function. Definition is connected to earlier notions of calibration for regression models, including those in Gneiting al. (2007) and Kuleshoy
al. 2018), which formalize approximate individual calibration consistent with Eq. (6)
Note that (y x) represents the model that directly outputs the conditional CDF value at y E Y. However, popular models such as CQRNN (Pearce et al. 2022) based on Eq. ① produce Un =
Fo (q|x) for specific values of g, and DeepHit (Lee et al. 2018 produces Fx (2/(x), but assuming that it is piecewise constant. In contrast, parametric models such as the accelerated failure time model (AFT) (Weil 1992) return distributional parameters which then can be used to obtain Fg (v(x) for y E y. Specifically. a parametric model for Fg (3/x) based on the ALD (Sheng & Henao 2025
in Eq. 2 is denoted here as {0, a, k) m.).).
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3 SURVIVAL MODELING WITH THE ICALD
3.1 PARAMETRIC AND NONPARAMETRIC ALD APPROACHES
Quantile regression methods, such as CQRNN (Pearee et al. 2022), utilize the widely adopted pinball (or check) loss (Koenker& & Bassett Jr 1978) to estimate conditional quantiles of the response variable. The pinball loss for target value y and predicted quantile value Üg = F (g/xx) mm.g (x)) from a model for g is defined as:
Cpinbau +) = (y - m4., T[ma, (xx) v)),
(7)
which optimizes a weighted absolute deviation objective that asymmetrically penalizes the under- and over-estimation of y. This formulation yields a predictive function that statistically separates the q-th and (1 g)-th quantiles in a consistent manner. Moreover, the loss for CQRNN accounting for censoring leveraging the Portnoy estimator (Portnoy (003)) is defined as:
Legr(y;; (, 2, 9) = Σ neDo
Cpinban (v; L, 4) + Σ nEDe
Wn Wn 2,9) + (1 - wwn Spinhan (v"; 22,4) q), (8)
where Do and Dc are the subsets of the dataset D = Do Dc, corresponding to observed (5 = 1),
pseudo target set be sufficiently larger than all observed values of in the dataset, Wx, E 1) is weight that balances the contribution of the censored and imputed targets and the loss is optimized for a model m defined for a specific value of q. In fact, the pinball loss in Eq. 7 can be interpreted as the negative logllikelhood of the quantile form of the ALD, parameterized as AL(O = Ney a = 1, 1,9), up to an additive constant (see Lemma ] in Appendix A. for technical details). This connection provides a probabilistic interpretation of quantile regression and forms the basis for likelihood seeeeeood-bbaed extensions. Building on this foundation, it is possible to extend the nonparametric quantile regression into parametric modeling framework by adopting an alternative parameterization of the asymmetry form of the ALD, which explicitly models the location, scale, and asymmetry parameters of AL R.) (Sheng & Henao 2025). This reformulation enables the transition from pointwise quantile estimation to full conditional distribution estimation, offering a relatively more flexible modeling framework beyond fixed quantile levels. Formally, the loss of the parametric model can be summarized as follows:
censored (6 0) instances, y"
LALD b) = Σ DEDo
log fALD mx())) - Σ log SALD(y,: (x)),
(9)
nEDe
where {6, x) = m is a parametric model that maps covariates to the parameters of the ALD, and (.LD(.) and (ALD(.) denote the PDF and survival function (1 - F-) of AL a, k),, respectively (see Lemma 2 in Appendix for technical details).
It should be noted that both the nonparametric and parametric ALD-based modeling approaches in Eq. (8) and Eq. (9) respectively, come with their own limitations. Nonparametric approaches are inherently restrictive, each model (or head) can only capture single quantile value y4 specified by the quantile percentage q. This form of discretization, observed many other methods, can cause approximation errors when quantile grids sparse. Although inereasing the number of quantiles reduces this error, it requires training multiple models, leading to a fragmented formulation that behaves as a collection of independent ALD models. While this increases the density of estimated quantiles, it also introduces substantial computational overhead and fails to capture global coherence across quantile levels, which in turn gives rise to the crossing quantiles issue where higher quantile estimates may fall below lower ones, violating Ny, 2 Voz for all qi > Q2 (Bondell et al. 2010). A case study illustrating this behavior can be found Appendix
Alternatively, parametric approaches based on the ALD offer greater flexibility in computing various distributional summaries, such as mean, median, mode and quantiles, which result from having closed-form expressions for f-LD(-) and (ALD(-). It is also computationally efficient and maintains smoothness throughout the estimated distribution, thereby avoiding issues like discretization and crossing quantiles that commonly arise in nomparametric approaches. However, relying on a single ALD to model the entire conditional distribution is also restrictive. Although parametric ALD models typically perform well for central quantiles, their approximation error tends to increase the distribution tails, leading to degraded performance for extreme-quanttic estimation. More critically, distribution mismatch can occur in some cases, where the estimated ALD distribution significantly deviates from the ground truth, often manifesting as consistent over- or under estimation in regions of y. A case study illustrating this behavior can be found in Appendix C.4
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3.2 SURVIVAL MODELING WITH THE INDIVIDUALLY CALIBRATED ALD (ICALD)
Given the strengths and limitations of both non- parametric and parametric ALD-based survival models, combining them within a unified frame- work seems like a natural and effective strategy. Specifically, begin by adopting parametric ALD-based survival model as the backbone, which ensures global continuity and smoothness through- out the estimated distribution. Then, we include an adapter module in the backbone which takes q as input to produce refined ALD parameters through {0, a. k} = m- (x, q) The resulting model illus- trated in is optimized using the negative log- leelikelihood loss in Eq. ( and the quantile regression loss (with censoring) in Eq. (8) for a specific value of q used both as input to the model and the
FCLU
FC, A6LU
FC
Shared BC, RAU
FC, ReLU
PCDDO
Figure 1: Architecture of the proposed ICALD sur- vival model. Here, @ denotes concatenation oper- ation, and FC refers to a fully connected layer.
quantile regression loss as follows:
LALD-Cq(VV Q) = CALD(y; 2) + >CCar(y: 2)
(10)
where À is hyperparameter that balances the contributions of the two losses (see Appendix C.4 for ablation studies). In practice, to encourage individualized quantile calibration, we sample a quantile percentage random from q ~ 24(0, 1) for each instance during training, which in turn is used to
refine the prediction of model m (x. q) to minimize Ley (y; T. q) in Eq. (10) for a specific value of q, while also maximizing the likelihood of y over the full ALD distribution implied by the model
{0, k,k = ma (x, q) using EuI((y;)).
Conceptually, this model can be seen as continuous mixture of ALDs specified as I dq f(q) (y: m. (x q)), for p(q) = 24 1), in which the backbone captures the general shape of the conditional distribution, while the adapter module drives local adjustments improve the calibration of the model. Note that marginalizing over does not yield an ALD distribution and that, in practice, we approximate this mixture at test time by averaging predictions over a finite set of 2,000 samples drawn from p(g) ((00) 1). During training, we iterate for up to 2000 epochs with early stopping, which is necessary to prevent overfitting, particularly on datasets with high variance or limited sample size (see Appendix C.4. The properties of the continuous mixture of ALDs are discussed in Appendix In particular, the CDF of the mixed ALDs admits a closed-form expression, allowing for interpretation via standard scalar summaries (e.g.. the mean survival time of the mixed ALDs). Thus, off-the-shelf explanation tools such as SHAP (Lundberg & Lee 2017) can be applied to these scalar quantities to obtain factor-level attributions for individual predictions; the same analysis can also be aggregated across subgroups to provide cohort-level insights (e.g., how covariate contributions vary across risk strata or time horizons, see representative visualization and case study in Appendix C.4). In the end, the theoretical foundation supporting the individual calibration ability of the proposed ICALD model discussed below.
3.3 THE ICALD MODEL IS PAIC
By Definition 1 quantile regression model 0y - =F mx.((x) trained to minimize the loss Ly 0b ?) in Eq. D is in principle (c, 6)-PAI provided it satisfies the individual calibration condition for each outcome y E y and its corresponding covariate x X. However, as also noted in Definition [ verifying this condition empirically is challenging provided that for each (input) x, the model produces predictions By for a fixes set of q. This limitation makes it difficult to flexibly estimate the probability Pr y I F5 (g]x) across all quantile percentages g E [0, 1]. To overcome this issue, we instead consider evaluating the modified probability Pr [y < F (q x, q)]. where the model myp (x, q) trained via Eq. (IO) takes as inputs both the covariates x and quantile percentage q (0, 1). This change allows us to assess calibration across quantile percentages using each observed y with mo mx 4), under the assumption that is monotonic in q. Note that any continuous but non-monotonic function can be transformed into monotonic one by sorting its outputs over quantile levels (see Appendix [A.5] for more details on monotonicity). This idea draws inspiration from the reparameterization trick introduced by Kingma et al. (2013). Under this construction, we extend the notion of PAIC to the monotonic setting, resulting in the following definition.
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Definition 2 (Monotonic Probably Approximately Individually Calibrated (MPAIC) (Zhao et al. 2020)). A predictive CDF model Fz ( q) is said to be (e,8)-MPAIC if for all X E X. Y E V. and q [0, 1}, the following holds:
Pr [Y s F (g(x..)) - al dq S c) or Pr [1o [P S 9 dg s
Pr
2 1 - 8.
Thus, by Definition 2 we can conclude that the ICALD model m. (x,(x,0) q). trained with the loss CALDDCar (y; 4) in Eq. (1)). is (e, OMPPACC Moreover, with Theorem below (proof is provided in Appendix A.4), this further implies that the ICALD model is also PAIC.
Theorem (MPAIC is a sufficient (but not necessary) condition for PAIC (Zhao al. 2020)). Ifa
predictive CDF model F& is (e,A)-MPAIC. then for any é' >> E, it is also
-PAIC.
In addition to using the quantile regression LAIDDCOU to improve the individual calibration, we note that an equivalent alternative is to use calibration loss Lcat- defined directly over the predicted cumulative distribution. Specifically, this calibration loss measures the discrepancy between the predicted cumulative probability F4 (3/xx, q) and the target quantile percentage q, and is defined as:
Leal o 9) Fo (vlx. - -gl-
(11)
This loss enforces the calibration condition that, for given input x, the predicted CDF of true event time distribution aligns with the queried quantile percentage q, which essentially equivalent to the MPAIC loss in Zhao et al. (2020). Importantly, Lca is evaluated over the joint distribution of all (x. y. Q) pairs, where y is treated as a realization of the latent true event time random variable. Thus, whether an observation is censored or uncensored does not affect the validity of the assessment, since the modeling target remains the conditional distribution of true event time.
In the end, Leg satisfies the first condition in Definition 2 while Lea enforces the second condition, making them equivalent under the assumption of mæ (x,,x, 4). Hence, the overall training objective for the ICALD model in Eq. (10) can alternatively be formulated as:
CALD+Ca $ q. = LALD (y: m. (x,9)) + ARca(v;; Φ,9).
(12)
Furthermore, by Definition 2 and Theorem ! we can conclude that the ICALD model m4 q). trained with the loss LALD-Cal, is (e, -MPPAIC and also ( -PAIC. -PAIC.
3.4 PRE-CALIBRATION AND POST- DOST-CALIBRTTIN
potential issue of asynchronous convergence may arise in pre-calibration models trained with Eq. (10) or Eq. (12). This happens when the likeliidl and calibration losses converge at different speeds, which although not observed in most datasets, is an issue in heavier-tailed ones. To address this, we introduce warm-up calibration strategy (see Appendix C.4). where training initially focuses solely on the negative ligllikel loss before incorporating the calibration loss at a later stage. Alternatively, post-caliibee calsbraaibrttion offers even simpler and more effective approach for handling with this issue. As discussed above, the thcoretical guarantees of calibration arise from the properties of the quantile regression loss L (or calibration loss Lca) itself. This enables post-calibration to be applied as a lightweight post-processing step, without retraining or modifications to the original model architecture. By decoupling the additional loss (i.e., Legr or Lca) from the training dynamics, post-calibration also avoids noisy or conflicting gradient signals during early training stages, leading more stable and reliable calibration.
Referring back to the pre-calibration model architecture (denoted as m))) illustrated in Fig. 1 we can infer that the parameters of the Individually Calibrated Asymmetric Laplace distribution (ICALD) are conditioned on both the input x and the quantile percentage q ~ (0, 1). Therefore, we can utilize a simple adapter module (denoted as mn Fost), like in the top part of Fig. that takes x and g as input and outputs the post-calibration adjustment factors y E R3 for the ALD parameters estimated by the base model (denoted as m Base), Effectively, the pre-calibration model can be decomposed into a post-calibration model and base model with the quantile regression loss or calibration loss as:
m& (xx q) = (6), "
K = m Post (x, g) m& (x) 7 C (0g. og. Kig)
(13)
where @ denotes element- wise multiplication, and 0;- o
K are the ICALD parameters produced by the pre-calibration model. In practice, since both the quantile regression loss and the calibration loss

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Monte Carlo approximations of their respective theoretical expectations, it is crucial to sample as many quantile percentages ~ 24(0, 1) as possible during training. As shown in Theorem 2 (with proof provided in Appendix A.4). increasing the number of quantile samples improves the
approximation quality and enhances the model's individual calibration performance.
Theorem 2 (Concentration (Zhao al. 2020)). Let Fo be any (e, 8) -mPPIC predictive CDF model, and let {x,, 3a
lid.
FxY, 19
Ltd.
U(0, Then, with probability at least 1 % have:
a
log
I[Fo(yi 9:) - qil 5 8+
2n
To this end, we apply several practical strategies to improve both pre-calibration and post-calibration models. Deepening the culibration anchor: We first concatenate the quantile percentage q at each network layer to ensure its influence propagates deeply through the architecture. Then we increase the number of training epochs to allow the model more opportunity to align predictions with the target quantiles. Widening the calibration anchor: Rather than using a scalar as in ZZhao et al.) 2020). empirically found that expanding it into small vector (e-g., 4-dimensional when Titidden 32) enables richer interactions with learned features and improves the expressiveness of the quantile- conditioned outputs (see Appendix C.4). In general, both pre-calibration and post-calibration can be implemented with either LALD+Cqr or AAL+-Cal- offering a unified and flexible framework where the calibration strategy and loss function can be independently selected based on the use case.
General-purpose post-hoc calibrator Beyond calibrating our ALD instantiation, the proposed post-calibration module could be used general-purpose calibrator for other survival models (e.g.. RSF (Ishwaran et al., 2008), DeepSurv (Katzman et al., 2018), and DeepHit (Lee et al. 2018)). This primarily because: (i) The adapter module (i.e., the q-branch in Section (.2) is agnostic to the choice of backbone feature extractor. In principle, it can be attached to MLPs, tree-based models, CNNs for image input, and BERT-style encoders for text - all without architectural restrictions. (ii) The calibration losses introduce (i.e., Lcy: and Lca) only require access the estimated survival CDF (i.e,, Fg (3/ x, 9)), making them widely applicable as long as such a function can be computed.
However, we do note some practical considerations when adapting our calibrator to other models, such as RSF, DeepSurv, and DeepHit: (4) As discussed in Section and Appendix C.4 these models nonparametric, and their predicted distributions continuous. This poses a challenge when applying our calibration objectives, which assume access to well-defined and continuous CDF. In such cases, post- processocssing step (e.g., linear interpolation between adjacent CDF points (Haider et al. (0200) may be necessary to transform the discrete outputs into a usable continuous CDF. Also, nonparametric models have the crossing quantiles issue (See Section B.1), thus additional constraints or penalties (e.g., non-crossing loss (Bondell et al.) 2010)) might required to enforce monotonicity. (ii) Models like DeepHit already incorporate multiple training objectives (e.g.. the likelihood and ranking loss), so directly adding a calibration term may introduce training instability and exacerbate trade-offs between competing losses.
often discretized rather than smooth and
In summary, our framework is theoretically general and readily applies to many parametric survival models (e.g., (e.g.. |oseini et al. 2017;; Nagpal et al. 2021 Sheng & Henao 2025), where a smooth CDF is directly available. For nomparumelric models (e.g.. Ishwaran et al. 20.8 Dembek et al. 2014 Lee et al. 2018 Katzman et al. 2018 Pearce et al. 2022). ). extra implementation care may be needed because the output CDF is typically discrete piecewise-constant.
4 EXPERIMENTS
Datasets We evaluate our methods on a broad suite of datasets introduced by Pearce et al. 2022) These include two types: synthetic event data with synthetic censoring and real event data with real censoring. For synthetic datasets, inputs x are sampled uniformly from (1 2)d, where d is the number of features, with event times e and censoring times c generated from distinct, parameterized distributions to simulate diverse scenarios. For real-world datasets, we consider survival datasets from domains such as healthcare and oncology. Full descriptions for all datasets can be found in Appendix B.I We follow standard practice by running each experiment with 5 random train/test splits. The source code required to reproduce the experiments presented in this paper is available at: https: //g com/ dem ngwhhngg TCALD
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Table 1: Performance comparison of our pre-calibrated ICALD across 21 datasets, showing number of cases where it performs significantly better, worse, or equal, respectively. The last two rows report total counts and proportions across all 56 pairwise comparisons.
Metric
Paiseca or FauD
Kissea V. Fog
Aaovyy os FLLD
Ainco us
for
Cip-c v. Chipsty
Aupsca ue
ALDARCAL
Average Culibration
13
5
0
16
4
4
13
3
16
13
0
7
0
14
Group Callbration
13
11
10
7
2
12
0
18
14
0
9
0
12
Individual Calibration
8
7
6
5
9
3
2
Total Propoction
23
12
25
28
60.7
41.
21.4
54
50.0
Table 2: Performance comparison of 4Abb-ca against three post-calibrated ALD baselines (i.e,, L CCue LPe
ALD-CSDD and (Pact D.D-PiOT as well as its pre-calibrated counterpart LiD+Ca. across 21 datasets. Each triplet reports the number of datasets where Clae ALb-cca performs significantly better, worse, or the same, respectively, for each calibration metric. The final two rows show total counts and proportions across 56 pairwise comparisons.
/Pos
Metric
AlTiaa Vs.
PPo
Liutt ALDsESD
AlTiCa VX.
Coet Allbaaaa
Average Calibration
16
14
5
2
11
3
7
1
19
Group Calibration
16
1
4
14
6
1
13
0
8
6
2
13
Individual Calibration
11
12
0
2
7
0
7
1
12
Total
43
40
11
5
31
3
22
4
44
Proportion (%)
76.8
3.6
19.6
71.4
19.6
8.9
55.4
5.4
39.3
14.3
7.1
78.6
Metries We evaluate each model using three categories of metrics: Predictive Accuracy, Concordance, and Calibration. For predictive accuracy, we report the Mean Absolute Error (MAE) and the Integrated Brier Score (IBS) GGrf et al. 1999). which quantify the accuracy of survival time predictions over time. For concordance, we use Harrell's C-Index (Harrell et al.. 1982) and Uno's C-Index (Uno et al. 2011 to evaluate the model's ability to correctly rank survival times while accounting for censored observations. For calibration, we assess the reliability of survival probability estimates using Expected Calibration Error (ECE) (Naeini et al. (015) for both average and group calibration, and the average Wasserstein Distance (Villani] 2009) between predicted and empirical survival distributions to evaluate individual calibration. Details can be found in Appendix B.2
Baselines We compare the proposed method against 12 baselines: ALD (Sheng & Henao 2025), Log-
Norm (Hoseini et al. [0177]. DSM (Nagpal et al. [202] (log normal and Weibull), DeepSurv (Katzman cct al. 2018), CQRNN (Pearce et 2022), DeepHit (Lee et al. 2018), GBM (Dembek et al. 2014)) and RSF (Ishwaran et al.. 2008), covering broad spectrum of survival models, including (semi- Iparametric and nonparametric approaches, as well as both neural and non non-neural architectures. Furthermore, we compare to 1 pre-calibration method X-CAL (Goldstein al. 2020) and 2 recent post-calibration methods, CSD (Qi et al. 2024a) and CiPOT (Qi et al.) (024b). ). These baselines represent diverse modeling strategies and provide comprehensive and principled benchmark for
evaluation. A complementary discussion of Related Work is provided in Appendix A.6 while detailed
descriptions and implementation notes for each bascline are presented in Appendix
Pre-Calinn Fre-Calibration Comparison We first evaluate the impact of the proposed calibration strategy in the pre- pre-calibration setting. Table evaluates the impact of our proposed calibration strategies across three calibration dimensions: average, group, and individual calibration. When assessing the statistical significance of the different metrics (i.e., Tables [ 2
and 3) we use a Student's I test with p < 0.05 considered significant after correction for false discovery rate using Benjamini- Hochberg Benjammini & Hochberg 1995). We observe that both ICALD methods (i.e., CPrc CREDD-C and CPre ( consistently outperform the single ALD baselines (i.e., CALD and LCqr) and CPie A in the majority of cases. Among these, CALD+Cal demonstrates a clear advantage over Cee ALD+CCor (64.3% wins vs. 0% losses), confirming the efffectiveness of the calibration loss Lcal as a principled and consistent training objective. As discussed in Section 3.3 both ICALD models trained with Lhe
and Che C. are (e, 6) -MPAIC and therefore also (e', e-c) -PICC. Although both are expected to improve calibration, their empirical effectiveness depends on the performance of the additional loss function (i.e., Lea and fcz). The superior performance of Lea over Lay is likely due to issues of the latter when handling censored data effectively. We provide detailed discussion and analysis of this issue in Appendix C.4
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LALD-Cal with nine baselines across 21 datasets. Each group of three columns reports the number of datasets where our method performs significantly better, worse, or the same, respectively. The final two rows summarize total counts and proportions across 140 pairwise comparisons.
Table General comparison of Poee
Maric
EQWAN
LegNarn
Decdiarr
DRM (Webeeo
DSM (Lapheer
Dupels
GBM
EBS
16
14
2
13
Harwelss C-tades
15
14
6
71
11
lerregy Cilliration
14
11
1
16
15
1
5
15
3
13
3
13
Indiidaal Calioration
13
0
Propariia (9)
2566
40.7
121
47.
31.9
13.5
13.
19.6
114
5/
210
31.6
4100
47.9
13.
38,0
38.1
711
3111
Fig. 2 further illustrates the individual calibration results by highlighting the best and worst improve- ment cases of L ALD+Cal compared to CALD on the Norm Linear dataset. Full comparison for the pre-calibration results across all datasets are provided in Appendix As discussed in Section 3.1 both the original objectives LALD and Lcur exhibit limited performance due to their inherent limitations. In contrast, both CPr ALD.CCa and C significantly improve calibration perfor-
Norm linear: Individuel Ce alloo Cases
Bast impravement 325)
Vorst improvement (D:: 228)
CGr
Ideal D3-
Ideal CDD
04
0.6
0.2
0.2
110
20
18ml
118C
Figure 2: Illustration of the best (Left) and worst (Right) individual calibration improvement cases PPee ALD+caa vs. LALD) achieved by hybrid ALD-based survival models.
mance in most cases, seen in Fig. 2 (Left. These improvements lead the estimated CDFs to better align with the ideal CDFs, which is the essence of individual calibration. Notably, the performance of LPe ALD+Cal and CPr DgCC is comparable in this example, further suggesting that when their loss formulations (i.e., Eq. (TO) and Eq. (12) behave similarly, their calibration outcomes are expected to be similar as well. This theoretically consistent with the shared MPAIC property. However, there are cases where calibration yields only marginal benefits or even slight degradation, like in Fig. 2 (Right), possibly due to the inherent variability across individual samples.
Post- cast-Caibbraiinn Comparison We then evaluate the impact of the proposed calibration strategy in the post-calibration setting. Table T summarizes the comparison between the post-calibrated ICALD model Post
ALD+Cai and three strong castt-aalibaation baselines: CPost ALD+CCU ALD-CSDD and Post ALD-CiPor We also include a comparison with the original pre-calibrated [Pre L' to assess the relative benefit of applying calibration as post-processing step. Full post-calibration results across all datasets provided in Appendix C.2 Results clearly show that CPuss
Post
ALDDCaa outperforms all other post-calibration AL.D+Cyr 71.4% over [Post D and
baselines in most cases (i.e., significantly better: 76.8% over [Post
55.4% over Post AALDCRPOO). especially compared to the two recent novel post-calibration strategies CSD QQi et al. 2024a) and CiPOT (Qi et al. 2024b). Furthermore, although Poot slightly better pertormance than its pre-caaiiratee counterpart, this marginal gain may be attributed to the issue of asynchronous convergence in pre-calibration discussed in Section 3.4 In summary, post-calibration results reaffirm the robustness of the hybrid ALD model, particulariy when using Lcal.
AALDCCa shows only
General Comparison We also evaluate and compare our post-calibrated ICALD model Ca against 9 baseline methods across 7 evaluation metrics on 21 datasets. Complete results for all datasets are provided in Appendix C33 We further provide an interpretability and robustness case study in Appendix C.4 including stratified analyses across time horizons and quantile levels. Table 3 shows that our method consistently delivers strong performance, achieving gains over all baselines in substantial subset of comparisons.

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ICALD ALD & CQRNN: Our method achieves significant gains in individual calibration (9 significantly better and worse over ALD, and significantly better and O worse over CQRNN), and also improves average and group calibration on over half of the datasets. This indicates that simply relying on quantile regression Lcg or maximum likelihood LALD may lead to under-calibration, and that the proposed calibration strategy via Post ALD+Cal encourages individual-leve alignment with
ground-truth distributions.
ICALD vs. (Semi- Parametric & Mixture Models: Our model consistently improves both calibration and accuracy metrics. Specifically, P ALD-Caa outperforms the semi-parametric model SeepSury in metrics such as IBS (21 significantly better and O worse), average calibration (15 and 2), and group calibration (14 and 2). Moreover, compared to the parametric mixture DSM (Weibull) model, our method leads in 73.6% of all comparisons (103 out of 140), especially excelling in MAE, concordance, and all forms of calibration. This highlights that ICALD provides better generalization than fixed-form parametric assumptions or mixture modeling approaches.
ICALD vs. Nonparametric Models: Our model demonstrates superior calibration, especially in group and individual calibration. ICALD outperforms DeepHit on group calibration in 14 of 21 datasets, and achieves significantly better individual calibration on all 13 datasets. Similar trends are observed for GBM (12/1/8 for group calibration) and RSF (15/1/5), confirming that our model delivers better calibrated estimates, despite the strong expressiveness of ensemble or neural nonparametric baselines.
5 CONCLUSION
In this paper, we introduced a novel survival modeling framework ICALD that unifies the strengths of parameiric and nonparametric ALD approaches. By supporting two theoretically equivalent loss functions (i.e., CALD-Cal and LALD+Cor) with formal guarantees for Probably Approximately Individually Calibrated (PAIC) learning, ICALD offers a flcxible and principled framework for improving calibration in both pre- and costtcalibbation settings. Through extensive experiments on 21 benchmark datasets, we demonstrate that ICALD consistently outperforms a wide range of strong baselines, including both traditional and neural survival models, as well as recent pre-calibration and post-calibration techniques. These results highlight the effectiveness and generalizability of our approach in achieving accurate, concordant, and calibrated survival predictions.
Limitations While increasing the number of quantile percentage samples improves the Monte Carlo approximation and calibration quality (Theorem 2). it also prolongs training and may lead to overfitting, particularly on heavily skewed datasets (e.g.. LogNorm). Although we apply early stopping to maintain a proper trade-off between calibration and generalization, more principled solutions remain to be explored. In addition, for the joint loss formulations ALL+cau and Pie ALD+Car we observe the synchronous convergence issue in some cases. To mitigate this, we adopt a warm- up calibration strategy that delays the introduction of the additional loss (i.e., Leal and Ccgr) to encourage alignment between optimization objectives. However, this approach does not fully resolve the convergence mismatch during training. Although both LALD+CaI and LALD+Cor are theoretically grounded in improving individual calibration (Theorem D. the latter appears more sensitive to censoring. Its reliance on the Portnoy estimator may limit its ability to capture reliable calibration signals under censored conditions (see Appendix C.4). In contrast, LALD+CaI offers more robust and stable calibration objective for censored survival data. Thus, improving the performance of CALD-CC in the presence of censoring remains an open challenge for future work.
Lastly, our empirical evaluation is conducted on a benchmark suite chosen to ensure fair and consistent comparisons with prior work, following the datasets used in Pearce al. (2022) and [Sheng & Henao (2025). While this suite spans diverse distribution families and clinical censoring regimes, it is not exhaustive, and performance may vary across domains and dataset characteristics. Evaluating ICALD on larger, independent, and more diverse benchmarks is therefore an important direction for future work. To partially address concerns about external validity and scalability, we additionally evaluate ICALD on two highh-imensionall gene-expresse expressioe datasets (DBCD; Rosenwald et al. 2002 and DLBCL; Van Houwelingen et al. 2006), as well as four finance and engineering datasets from the PySurvival p In the end, we provide a dedicated scalability case study in Appendix C.4 where ICALD demonstrates competitive performance, particularly on calibration metrics.
https://gi thw. com www../////uuaeeey/r/rriio//
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A ANALYTICAL RESULTS
This section presents the theoretical foundations and formal guarantees of the proposed ICALD mod- els and their calibration properties. In Appendix A.1 we show that the widely used pinball loss can be interpreted as special case of the negative log-likelihood of the Asymmetric Laplace Distribution (ALD), providing a probabilistic justification for quantile- the full loss formulation for parumetric ALD model in the presence of censoring. Appendix AA.3] discusses the properties of continuous mixtures of ALDs. Appendix [A.4] establishes formal cal- ibration guarantees (Theorem 1) for the ICALD models, demonstrating that training with either CALD+Cal (Equation 12) or LALD+Cqr (Equation 10) yields models that satisfy (e, 6-MMAI and,
objectives. Appendix(A.2 [A.2 derives
consequently. (é', + PPAC.. In addition, Appendix A.4 includes Theorem 2, which provides
nigh-probability generalization bound showing that increasing the number of sampled quantile levels improves the Monte Carlo approximation and strengthens empirical individual calibration. Appendix A.5 provides an extended discussion of the monotonicity constraints, clarifying why mono- tonicity is required, when it can or cannot be ignored, and how to address nonmonotonic predictions using monotone rearrangement. Also, a discussion of Related Work is provided in Appendix A.6
A.1 PINBALL LOss AS A SPECIAL CASE OF THE ALD LIKELIHOOD
Lemma 1. The pinball loss is equivalent to the negative log-likelihood of the Asymmetric Laplace Distribution (ALD) in its quantile parameterization AC(O Dg. a = 1, 4..). up to an additive constant.
Proof. Let a random variable Y AL((,, o, g). where 0 E R is the location (the q-th quantile, i.e., Ng). a >> is the scale, and g E (0, 1) is the target quantile. Then, the probability density function (PDF) of the ALD in its quantile form is given by:
fly: (y::0,0,)) 9(1 -2)
exp (2 y))
y 8 0
(14)
σ
exp ( (y 0)) y 0.
The negative log-likelihood becomes:
o 0),
y y20, 0, L-g y), y 0.
- log f(y:0,0,0))) = log
(15)
q(1
(
On the other hand, the pinball loss used in quantile regression is defined as:
Cginhaul (y: Hg) (y--y) -a I(0, > v)) = ( - Bq):
2 Fg.
(16)
(11 () y), y 3q
Now, if we let and ÿg be the predicted quantile value. Then, we can conclude:
log fy: Uq. 1,9) = Cpinball (y:: ).)) - log (g(1 9)) ())-
(17)
This shows that the pinball loss is proportional the negative log- logliikllhood of the ALD, up to an additive constant. This connection provides probabilistic interpretation of quantile regression and justifies likelihood-based parameteric modeling extensions using the ALD.
A.2 Loss FUNCTION FOR THE PARAMETRIC ALD MODEL
Lemma 2. Let Y ~ AL(B, or Hi), where AL denotes the Asymmetric Laplace Distribution with R. scale parameter
location parameter
O. and asymmetry parameter 0. Suppose parametric model ma (x) = (0. a, 13 maps input covariates x to the corresponding ALD parameters. Then, the total loss function over both observed Do and censored data De is given by:
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CALo(y; +)= =-
Σ
log fALD mx
Σ
log (
(18)
nEDo
nEDe
where JALD and SALD = -- of the ALD, respectively.
FALD denote the probability density function (PDF) and survival function
Proof. The PDF and CDF are explicitly defined as:
V2
exp
y2 y2 (0 y)
VBR (y 8)
fA,
(19)
1 R2
exp
y 5<0.
ON
1
1 1tx²
exp
(-v2n (y 0) 2 0,
FALD(Y; 0n kK)
(20)
v2 (0 y)
Itx
exp
y <0
Accordingly, the negative log-likelihood for observed samples (6 = 1) is:
Σ
log (ALDDYNS mx(()))
(21)
DED
logo, on - llog
Kin
V2
Kn Knnyn 0n):
On 2 an
+ On
+
I (0n
Mn);
Un < On:
nEDo
Kn
For censored samples = 0), the loss is derived from the survival function:
Σ neDe
log (AID(ys,
(22)
log(n + 1) + v
kin (B -On)
Yn Pn
on
log( + 1) log I + Nit (1- exp (- (0, Mnn) vn)))]
Un On.
neDe
Jakn
These formulations enable efficient optimization of ALD-based models for both observed and censored survival data, by leveraging the closed-form expressions of the ALD's PDF and CDF.
A.3 THE PROPERTIES OF THE MIXTURE OF ALD
Given Y ~ AL(O, k), YMS f p(r) (p(r) (y; mo (x, r)) )r where ~ L(O, 1), and mx(xx1 r) (0,,0r- } are the ALD parameters predicted by the model for each quantile percentage r. Here, to avoid confusion with the quantile value J4 = Fy (g x), we use instead of q to denote the random quantile percentage. We have:
E| =
V2 K - Var[Y] = σ²
1 --x),
k)).
+ n22)
(23)
2 R2
M(MMi = E,-12(011)
L Y So (0, + 5 ( - Kir --xx)) dr ~ ΣN
[0: + 5
(1
] (24)
By the law of total variance, have:
Var[YMix] (ar + Var, [BY:]],
(25)
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where the variance is:
By-ee(o.1)[Var((]] = /
o- 2
ΣΣ
o.
+ +
+
mp
dr
~
(26)
k2
N
2
and the between- componet variance is:
1 N
Var, [[]]]
Σ
[E[Y;) - M[
(27)
Similar to the computation of the mixture mean Max, we can estimate the quantiles of the mixture ALD model by averaging quantile values y/ sampled over r ~ u(0, 1) from the predicted parameters {rr, or Kr Formally, the mixture quantile y/ estimated as:
0 oR
log
1+ x²
if E (0,
x²
V/2
1+x²
=
(28)
0
log
[1 + +x²)(1
4)]
if E
[4R2111
1 Σ yg.
3y 10, ~
(29)
A.4 THEORETICAL FOUNDATIONS OF INDIVIDUAL CALIBRATION
In this section, we will show the ICALD model {0, o, k} mo (x, 4). trained with the loss CALD-Cal or CALD+Cyr. is (c, 8)-MPAIC and also (',8- }=2 PAIC. Now, we begin by recalling the definitions of PAIC and MPAIC provided in Definition 1 and Definition [ These definitions are slight extensions of the original formulation in Zhao al. (2020). incorporating an equivalent expression based on the inverse CDF. Note that these extended definitions allow us to generalize the original proof in Zhao et al. (2020), which establishes that training model with the calibration loss CALD+Cal yields PAIC and MPAIC guarantees. In our case, we show that the same guarantees hold when the model is trained with the equivalent quantile- based loss LALDACQr-
Definition (Probably Approximately Individually Calibrated (PAIC: Zhao et al. 2020 ). A predictive CDF model Fo (Y xx) is said to be (c, -PPIC for all x X, Y V, and g 10 the following holds:
Pr V. Pr S x al da se Pr V." [Pr[F S al dg S > 1 1 6.
Pr
Definition (Monotonic Probably Approximately Individually Calibrated (MPAIC; Zhao et al. 20200). A predictive CDF model F (x),,,))))) 9) is said to be (c, -MMAIIC for all x E X, Y E V. and
q E [0 1}. the following holds:
Pr So Pr Y Fo (g|x, 4) - gf dg Se or Pr [co Pr [F S a dg 1-8.
Then, will show the proof for Theorem 1
Theorem (MPAIC is sufficient (but not necessary) condition for PAIC (Zhao et al. 2020)). Ifa
predictive CDF model Fo is (e, MMPPIIC, then for any é' 6, is also (e' 8 1= PAIC
Proof. Let Fylx and Q 44 1) be an independent random variable. Define the expected calibration error (ECE) using the 1- 1-Wasserstein distance as:
ECE(F&) = [ [P](Fo(V|X) s gl dg = dw. Fu)
(30)
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where F ()) denotes the true CDF of the predicted cumulative probabilities, and Fu denotes the CDF of 200, 1). Intuitively, dw, (r x). 1F) try to integrate the difference between the curve q Pr(F+(YiX) gl and the curve q > q- Then, we can define the calibration error
err(x, y) dw, (IF (v),,.)).) Fv) err(x) dw, (Y :F)
(31)
Case 1: Monotonic mapping q. If (yl increases monotonically Q, then we have:
e v) {-¹ (v(xx)))) gl dq = 1 (ul [I QI]
(32)
Let Z = ( Q), where Q ~ (4(0 1). Then the CDF of Z is given by:
F (2 Pr(2 S =) Pr(Fo Q) 2).
(33)
Now, F2(212 q) is monotonically nondecreasing continuous function of q, then the mapping q + F (v) is mecssrr--prssrrvigg This implies that:
(2 Pr(Q s F (y/ar, 2)),
(34)
and hence,
(2) Fu g). Vg E [0,
(35)
Let F denote the CDF of the uniform distribution 2/(0, 1). that is,
Fu(u) Pr(U - u) u, so Fú (q) q. Ag [0,
(36)
According to the property for the W-sssrrsn distance (Villani et al. 2008 between two distribu- tions μ and on the real line, if == and _F,,1 their respective quantile functions (i.e., the inverse
function of CDF), then:
dw, (H, J. | |= - F= dq.
(37)
Applying this identity to Z and Q, we can obtain:
1'' Fo g) dg,
dw: FF)
(38)
which is exactly Eq. 32
Case 2: General case without monotonicity. In general, even if monotonicity doesn't hold, the following inequality still applies:
err( x, 3) < EQ= [[ Q) - QI]-
(39)
is derived from the Kantorovich- duality for the I-Wasserstein distance Villlani et al. 2008
(f dh du dw; (1,, v) sup
ff dv
sup 11
[E,[vb] E. [w]],
(40)
where the supremum is taken over all 1-Lipschitz functions
R. i.e. functions that satisfy
(b(z) (v)) I |æ vl. Va, y R.
(41)
Applying this duality to Z and Q, and choosing the the 1-Lipschitz function (aaa) = a, we can obtain:
dw; (F Fu) < 1E [2) - Eq- Finally, applying Jensen's 's inequality (B[[]] < E[|A]] over Q ~ ((( 1)) yields:
= -IEQ-u(0.11
-QI
(42)
dw, (L Fu) (Q,)((,1)) [z Q]
(43)
(Q- 1 Q1 E Fo(u Q) Q1)
Importantly, inequality (39 holds even if F. (2)z, is not monotonic. This is because the dual form of the I-Wasserstein distance does not require any structural assumptions on the mapping from Q to 7 Fo Po(vla Q). Therefore, inequality (39) remains valid upper bound in the general case. On
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the role of inequality (39): Eq. (32 shows when inequality (39 becomes tight (I.e. q Folyla,9) is monotonically increasing).
Incquality (44) follows from the same reasoning as incquality (39). the application of Kan-
duality followed by Jensen's 's inequality, with an additional expectation over
Y ~ Fyla:
err(x) S By Q01((11)) (Y)x Q) - Q1]
(44)
Contradiction argument. Now suppose, for contradiction, that F not (''.'')-AICC. That is, by Definition | have:
Pr (r(e((x)) > < 8'.
(45)
If we define the set:
;, x By Q-2((.11) | Q) Q11 2 o).
(46)
and by the inequality (44 we can know that whenever err(x) 2 é' we have x E Sb. thus we can conclude:
Pr[X E S.) > 8'.
(47)
Then, for any < é' and x E Sb. by bounding the expectation, we have:
d $ Ey Q
QIl
(48)
S Er-Pr[| <= Q) Q]
(49)
where inequality (49 holds because the absolute deviation term [Y Q) - QI [0, Now,
letting p = Pr[| F x, Q) Q 2 e). we can solve:
S c11 p) +) 6-6
(50)
1-c
Combining this with the bound over x So (i.e., inequality (47) and applying the law of total probability, we can obtain:
Pr Fr - P
F
Q) - QI 2 c E Sp Pr S.)
(51)
> / -
+ Fr Q 2 c E S
Pr x E Sy]
8..
1 -
Violation of MPAIC. By Definition 2 MPPA requires: PY Q) - QI > d < 5. Thus,
equation (51) implies that F& is not (e, 8' - )MPACC.
Contrapositive and conclusion. We have shown:
Not (c'88))-PAIC = Not (c, 1- M-IIIMAAC, VE
(52)
Taking the contrapositive:
(e, 8)-MAII + (6',0. L VE'
(53)
In the end, we can conclude that if Fg is not (dAI')-PAIC, then for any e < é', is not (e.8' d'=e
T-c
mPAIC, which is equivalent to the Theorem ie., if Fo (c, (PPAII, then for any e' 6, is
also (e',0 6 |=c) -PAIC.
Then, we will show the proof for Theorem
Theorem 2 (Concentration (Zhao et al. 2020 Let Fa be any (c,8)-MPAIC predictive CDF model,
and let (x1; yi....., (Xn, Fn)
itd. FXY, and ....., In
lt U(0, 1). Then, with probability at least
1 7, we have:
I(Fo xx. - qul 2112)550 /
log
n
2n
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Proof. Define the sequence of Bernoulli random variables: b, [ (3t x.. 96l 2 {0, 1). By the definition of (c, 6M-PAIC, we know that:
Pr (b, 1) I or equivalently [[,,]] S d.
(54)
Now apply Hoeffding's s inequality for bounded i..d. Bernoulli variables b; E [0, 1|:
Pr
b, 2 8 +
S n
(55)
i=1
Set the RHS to 7, we solve for e':
26 -7 e'
log 7
(56)
2n
Therefore, with probability least 7, have:
((F xx 2.l (al2e) V
log
(57)
2n
In summary, Theorem 2 reveals key implication: increasing the number quantile samples improves the quality of the Monte Carlo approximation, thus enhancing the individual culibration performance of the model.
A.5 EXTENDED DISCUSSION ABOUT THE MONOTONICITY CONSTRAINTS.
We clarify the role of monotonicity in framework by answering the following four key questions.
(I) Why is monotonicity needed? Monotonicity is required for the first equality in Eq. 32) to hold, i.e., for err(x) and err(x. y) attain their maximal value, because the mapping q + F x, q) must be measure-presseeeeeee preserving. Without monotonicity, this mapping may fail to be maassurepreserving (leading, for example, to crossing quantiles), so the equality can break down. However, it is worth noting that inequality (39) and inequality 444 remain valid upper bounds (Full details can be found in Appendix(A.4)
(2) When can we ignore monotonicity? During model training, we do not explicitly enforce monotonicity. Inequality (39 and inequality (44) guarantee that the training loss always serves as valid upper bound on the true calibration error. For nonmonotone mappings, the actual calibration error is even smaller than this bound. Thus, ignoring monotonicity during optimization does not compromise the validity of the learning objective but instead leads to conservative estimate.
(3) When can we not ignore monotonicity? evaluating individual calibration (as in Definition I and Definition 2). This ensures that the calculated calibration error accurately reflects the theoretical definition, enabling a fair model comparison.
important to ensure monotonicity when directly
(4) What is the solution for nonmonotonic' Any continuous but nonmonotonic function can be transformed into a monotonic one by applying monotone rearrangement. In practice, we draw a finite number of quantile samples q1.-. ... .K (4(0, 1), compute fi F (y) x, qi). and sort these values into nondecreasing sequence fi,...,fx: which are then reassigned to QI...., (K- This transformation preserves the distributional meaning while ensuring monotonicity, enabling rigorous evaluation of calibration.
A.6 RELATED WORK
Survival Models Classical survival models can be categorized in general into parametric, semi- parametric, and nonparametric approaches, depending the assumptions they make about the underlying time-to-event distribution. Parametric models assume that event times follow a certain probability distribution, such exponential (Feigl & Zelen 1965). Weibull (Scholz & Works 1996). log normal (Royston) (001), Asymmetric Laplace (Kotz et al. 2012), or their mixtures (Nagpal al. 2021). These methods typically characterize event times using the conditional
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probability density function (t) and the corresponding cumulative distribution function F(t) x). Semi paraamerii models, most notably the Cox proportional hazards model (Cox 1972). decompose the hazard function into time-dependent baseline component and a covariate-depende
component, Le., h(()x) = (t) exp(x _ (3). The baseline hazard (t))) is left unspecified and it is estimated nonparametrically, while the covariate effects are modeled parametrically. More recently, neural extensions such as DeepSurv (Katzman et al., (2018) have improved the scalability and expressiveness of Cox models, particularly in digh-dimmnsional settings. Nonparametric models avoid explicit distributional assumptions, instead relying on data-driven estimators. Examples include Random Survival Forests (RSF) (Ishwaran al. 2008), Gradient Boosting Machines (GBM) (Dembek et al.,
2014), discrete-time models using categorical likelihoods (e.g., DeepHli (Lee et al. 2018)). quantile regression (e.g., CQRNN (Pearce et al.
2022 ), or generative modeling (Chapfuwa et al. 2018).
Calibration The notion of calibration has long history in statistics, with early definitions of average calibration (e.g., the Brier score) (Brier 1950, Murphy 1973] Dawid 1984). More recent interest in recalibrating classifiers has surged, especially for deep neural networks (Guo et al. 2017
[973]Dawid
Lakshminarayanan et al., 2017) Beyond average calibration, group calibration has been studied for both predefined (Kleinberg et al. 2017) and computationally defined groups (Kearns et al. 2018
Hébert-Johnson et al. 2018). In the context of survival analysis, recent post hoc calibration methods
(e.g.. CSD) (Qi et al. 2024a) and CiPOT (Qi al.] 2024b)) have been proposed to improve the average calibration of predicted survival functions by applying conformal prediction techniques and adjusting the estimated curves both in probability and in time. Individual calibration, which assesses the accuracy of predicted risks at the level of each instance, has been explored in fairness- learning "Sharifi-Malvajeee
al.) 2019). however, remains both computationally and statistically challenging. For example, (Foygel Barber et al., 2021) showed that achieving perfect individual calibration with tight confidence intervals subject to fundamental lower bounds. Moreover, recent work (Zhao et al., (020)) aimed to approximate individual calibration using randomized forecasting, and provided theoretical guarantees under certain assumptions on the model class and the data distribution.
B EXPERIMENTAL DETAILS
This section provides additional information about the experimental setup. All experiments were implemented using the PyTorch framework. Detailed descriptions of the datasets, evaluation metrics, method and baseline models, and implementation specifics are provided in Appendix BB.l Appendix B.2, and Appendix B.3 respectively.
Hardware. All experiments were conducted on a MacBook Pro equipped with Apple M3 Pro chip, featuring 12 cores (6 performance and 6 efficiency cores) and 18 GB of memory. All computations were performed the CPU, as the models predominantly utilized fully connected neural network architectures that did not require GPU acceleration.
B.1 DATASETS
Our datasets are designed following the settings outlined in Pearce et al. (2022). The first category consists of synthetic event data with synthetic censoring. In these datasets, the input features x are generated uniformly as x (0, 2)P, where D denotes the number of features. The event time p(e x) and censoring time p(c x) are drawn from distinct parameterized distributions, with the specific forms of these distributions varying across different dataset configurations. Table summarizes the distributional details of the event and censoring mechanisms.
The other type of dataset comprises real event data with real censoring, sourced from various domains and characterized by distinct features, sample sizes, and censoring proportions:
METABRIC (Molecular Taxonomy of Breast Cancer International Consortium): Contains genomic and clinical data for breast cancer patients. Includes features, 1523 training samples, and 381 testing samples, with a censoring proportion of 0.42. Retrieved from the DeepSurv Repository
WHAS (Worcester Heart Attack Study): Focuses predicting survival following acute myocardial infarction. Includes 6 features, 1310 training samples, and 328 testing samples, with a censoring proportion of 0.57. Retrieved from the DeepSurv Repository
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Table 4: Summary of dataset statistics, including the number of features (Feats), training and test set sizes, proportion of censored events (PropCens), and the distributions used for sampling event and censoring times. The coefficient vector [0.8, 0.6, 0.4,0.5, 0.3, 0.2, 0.0, -.77].
Dataset
Train
Test size
PropCens
Variables for time
Variables for censoring time
Type I: Synthetic event data with syntheric cemsoring
Norm linewr
500
10300
0..20
N 10, (x 1)")
[Vxx + 10, (0.8x 0.44")
nonlinear
1000
N (in + (0.5x 0.5)")
(2x 10.2")
Exponential
500
1000
0.30
Exp(2x +
Expl -3x + 15)
Weibull
500
1000
0.22
Weibelli ( 1 11) 10,5)
Weibull - 3x 20,5)
LogNom
500
1000
LogNorm 1)
24(0, 10)
Norm uniform
500
1000
0.62
N (2x)) + 13, (xc 0.5)%)
24(0, 18;
Norm heavy
2000
1000
0.80
(3x0 X5 x5 (sin(xaXs) 6x 0.5)P)
24 12)
Norn med.
2000
1000
0.49
2400, 12:
Norm light
2900
1000
(410, 20)
Norm same
2000
1000
0.50
Equal to target
LogNom heavy
40000
1000
0.75
LogNorm E, Box 13/10
14(0, 0.4)
Normorm med.
4000)
1000
2400 1.0)
LogNorm light
4300
1000
0.23
2/(0.3.5)
LogNorm same
4000
1000
0.50
Equal to target
Type Real event data with real
METABRIC WHAS
381
Real
1310
328
0.57
Real
Real
SUPPORT
14
7098
1775
0.32
Real
Real
GBSG
1785
447
01422
Real
Real
TMBImmuno
1328
BreasiMSK
1467
367
0.77
Real
Real
LGGBM
510
128
0.60
Real
Real
SUPPORT (Study to Understand Prognoses Preferences Outcomes and Risks of Treatment):
Provides survival data for critically ill hospitalized patients. Includes 14 features, 7098 training sam- ples, and 1775 testing samples, with a censoring proportion of 0.32. Covariates include demographic information and basic diagnostic data. Retrieved from the DeepSurv Repository
GBSG (German Breast Cancer Study Group): Tracks survival outcomes of breast cancer patients. Includes 7 features, 1785 training samples, and 447 testing samples, with a censoring proportion of 0.42. Retrieved from the DeepSurv Repository
TMBImmuno (Tumor Mutational Burden and Immunotherapy): Predicts survival time for patients with various cancer types using clinical data. Includes 3 features, 1328 training samples, and 332 testing samples, with a censoring proportion of 0.49. Covariates include age, sex, and mutation count. Retrieved from cBioPortal
Derived from the Memorial Sloan Kettering Cancer Center, this dataset focuses on survival prediction for breast cancer patients using tumor-related information. Includes 5 features, 1467 training samples, and 367 testing samples, with censoring proportion of 0.77. Retrieved from cBioPortal
BreastMSK:
Integrates survival data from low-grade glioma (LGG) and glioblastoma multiforme (GBM), often used for validating models in cancer genomics. Includes 5 features, 510 training samples, and 128 testing samples, with a censoring proportion of 0.60. Retrieved from cBioPortal
LGGGBM:
B.2 METRICS
We evaluate each model using three categories of metrics: Predictive Accuracy, Concorlance, and Calibration. For predictive accuracy, we report the Mean Absolute Error (MAE) and the Integrated Brier Score (IBS) (Graf et al. 1999). which quantify the accuracy of survival time predictions over time. For concordance, we use Harrell's C-Index (Harrell et al. 1982) and Uno's C-Index (Uno) et al. T) to evaluate the model's ability to correctly rank survival times while accounting for censored observations. For calibration, using Expected Calibration Error (ECE) (Nacini et al. 2015) for both average and group calibration, and the average Wasserstein Distance (Villani (009) between predicted and empirical survival distributions to evaluate individual calibration. These metrics provide a holistic evaluation framework that effectively captures the predictive accuracy, discriminative ability, and calibration quality of survival models.
assess the reliability of survival probability estimates
Mean Absolute Error (MAE):
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MAE
V/l
(58)
where yi is the ground-truth event time, Dr is the model's predicted survival time (e.g., the median value the estimated survival CDE, i.e., Fx (q 0.5|x)), and N is the number test samples.
Integrated Brier Score (IBS):
Σ N (1
1
N
Pa(1 xx) 2 =1,e=
Fa(1 x,) =1) = G(t)
BS(
(59)
G(y))
IBS t1 Jh
BS(() dl,
(60)
where (t x₂) denotes the predicted cumulative distribution function (CDF) time t, C(- the Kaplan- Laplan-Meier estimator (Kaplan & Meier [9588 of the censoring distribution, and 100 time points in the integration range [t, t2] are evenly selected from the 0.1 to 0.9 quantiles of the training set's y-distribution.
Harrell's C-Index:
Cи Eizj [L(o; > 65)+ + 1(0: = = I( < v,)6
(61)
Eits T(y y3 y,)6,
where Φ; Sys xx) - (ys xx) is the model's risk score. For implementation, function from the sksurv metrics module, as
utilize the www.ordddneeeeeeeee documented in the scikit-survival API
Uno's C-Index:
Cu EE
G(y)) [ide > 05)+ + T(Ø: = +5)] T(y < Uz. Vi < yr)
(62)
Σ E G())) I(y; リý D: v7)6;
where Vr is the cutoff value for the survival time. For implementation, we use the inw.orrdaneeeidexiiiie function from the met.iiss module, documented in the scikitt API
To evaluate average calibration, assess whether the model's predicted cumulative probabilities align with the ideal uniform distribution L((,, 1). Specifically, the Expected Calibration Error (ECE) of predictive CDF model Fs, is defined as the 1-Wasserstein distance between the empirical distribution of the predicted CDF values and the uniform distribution:
Average Calibration:
ECE(F4) = ( [Pr [P | XX x - gl = dw, (FFo(rI)))Fuu)
(63)
where Y Fy 9 (1, 1), IF (Y)x)) denotes the empirical CDF of the predicted cumulative probabilities, and FFz denotes the ideal uniform CDF In practice, compute (e xx) for each test instance (x; yi), where e; the true event time. Note that e; is only observable synthetic datasets, where the true generative distribution is known.
Predicted CDF values F (e; x;) are then sorted to form the empirical distribution, which is compared to uniformly spaced quantile targets (q; N u(0, 1). The calibration error is caleulated as:
i
ECE
Σ
Fa x;)
(64)
N
N
1=1
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For real-world datasets with censoring, replace e, with the observed time yss and compute ECE only on uncensored samples (ò, 1), resulting in:
Nit
ECE
(V (x))
x:)
(6.5)
Nobs
Nobs
where Noss is the number uncensored test samples.
Group Calibration: To evaluate group calibration, we partition the input space X' into structured subsets (i.e., S [
2) using combinations of feature dimensions. Let the input x E l.".
We define (2) feature pairs, and for each pair (gy, we form 4 subgroups based on median
thresholding:
(I) x4 (edian(x.) and z Median(x.)
(2) I2
Median(r) m.) and xy I Median(x.) (zy)
(3) 22 Median(r)) my) and z >> Median(r)) (xy)
(4) x; Median(x,) and 2 Median(x,)
This results in K = 4 x groups in total. Within each group, we compute the ECE in the average case.
/ x x St) S s) ol dg.
ECE, ECE,-
(66)
Also, to ensure statistical stability, each group must contain between 1 size and size of the full dataset. We then define the group calibration error as the worst (i.e., largest) ECE across all valid groups:
GroupECE max ECE
(67)
sES
Individual Calibration: evaluate individual calibration, we compare the predicted cumulative distribution Fy x) with the ground-trus truth CDF F*(y x) for each individual test input. The discrepancy between these two distributions is measured using the 1-Wasserstein distance:
12 Vinao
dw, (F,,
Fo x;) (* x;) dt,
(68)
where F. x;) is the estimated CDF produced by the model and F* x;) is the oracle CDF (ground truth) corresponding to the same input. In practice, we approximate this integral using discrete grid of 1000 evenly spaced time points (ty [ty]]on j=1 E [0, 1.2 x Umax]. where Ymax denotes the maximum observed event time in the test set. Since ground truth distributions only accessible for synthetic datasets, individual calibration can only be evaluated synthetic settings, where F (t x;) is analytically known for each test input.
1000
B.3 ICALD AND BASELINES
To comprehensively assess the performance of the proposed method (ICALD), we compare it to 11 strong baseline models, summarized in Table These baselines span a spectrum of survival modeling paradigms, including parametric, semi comii-arammtric, nonparametric, and post-calibration approaches:
Parametric models: ALD (Sheng & Henao 2025) and LogNorm (Royston 2001) assume fixed parametric distributions for event times. DSM (Nagpal et al., 2021) extends this to a mixture of parametric families such Weibull and Normol
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Demi-parametric model: DeepSurv (Kazzman et al. 2018) is neural extension of the Cox propor- tional hazards model (Fox & Weisberg, 2002). allowing for non-linear feature representations while maintaining proportional hazard assumptions.
Nonparametric models: CQRNN (Pearce et al. 2022) directly estimates quantiles using the pinball loss under censoring, while DeepHit (Lee et al. 2018) estimates the full discrete-time survival function via log-likelihood and ranking losses. Tree-based ensemble models such as GBM (Dembek) al. 2014) and RSF (Ishwaran al. 2008) also included, offering non-neural alternatives that model complex interactions.
Pre-calibration methods: X-CAL (Goldstein al. 2020) introduces explicit calibration objective for survival analysis by reformulating the distributional calibration ID-Calibration) metric (Haider ct al., 2020) into a differentiable loss, allowing calibration to be optimized jointly with predictive accuracy during model training.
Post-calibration methods: CSD (Qi al. 2024a) and CiPOT (Qi al. 2024b) are representative post-calibration strategies applied after model training to improve alignment between predicted and true distributions.
All neural baselines were trained using either the same network architecture our method or the default architecture provided by their official repositories, under consistent optimization protocol to ensure fair comparison. Specifically, the implementations for CQRNN and Log- Norm were adopted from the official CQRNN repository while DeepSurv and DeepHit were adapted from the pyсox. methods module" For the mixture-based baseline, we employed the Deep Survival Machines (DSM) model from the autonn auton surviva val. models dsmm DeepSurvivalMachine we used the official implementations from the sksurv library, namely RandomSur SandommSrviii va valFF malloree Forest and Gradi Botst inguurvi va Ana aimllssi s, both available in the ensembl b module For the pre-calibration baseline, we used the official implementation of X-CAL which introduces an explicit calibration loss for survival analysis. Finally, both the CSD and CiPOT post-calibration methods were re-implemented based on their official repository
[ library implemented via For ensemble-based baselines,
Table 5: Summary of baselines used for comparison.
Method
Type
Neural
Description
Shgus Henzo
Ponnsetric
Asuutes event times follww Asyymeeticc Lonlase disttbution
LagNNmm (Rayaon
DSM
permmnic diveitunisso,
Sdull Wirks
Nenral extession Uox propertinnal bazands moder
2002
CORNN
Non Nun-punameini:
Neural emnoral quatiic reprosiun msng the pinbal hos
Desstil
Non paraneiii
Poedies survival Fancions via fog diailbod and casting Josies
GRM
Non-panaaunii (Exuratiee)
Generalieed Heosird Model sdapted for surviaa tasks
RSF
Si 2e de 20081
Non patanetic (Exseuible
X
Randon Forest adapoed for survvaal usls
x CAL
Pre collivation
/
Fist coliiaation methed applied when surviaa model training
Porccollenttion Puticcallbestion
X
Prot-nliination mechaal spplisd afier saritca muiid inining Post-naiination mechad appliad after sarsicl model Inining
CIPOT Mreeie
Hyperparameter default settings. All experiments were repeated across 10 random seeds to ensure robust and reliable results. The hyperparameter settings were as follows:
Default Neural Network Architecture: Fully-connected network with two hidden layers, each consisting of 100 hidden nodes, using ReLU activations.
Default Epochs: 200
Default Batch Size: 128
Default Learning Rate: 0.01
Dropout Rate: 0.1
Optimizer: Adam
html
httpss//attbbb.coom///aeeesaaaap Cal i
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Batch Norm: FALSE
ALD and ICALD (Trained with Pte Pre PPost ALDACaP ALD+Cyrr ALD+Car and Post ALD++Cyr
The model architecture for our method (pre-calibrated with LPr ALD+Cal and LPe A is illustrated in Fig. [ Similar to our pre-calibration method, we employ residual connection between the shared feature extractor and the first hidden layer to improve gradient flow and training stability for the base ALD model. Each hidden layer consists of 32 neurons with ReLU activation. To enforce positivity constraints the ALD parameters, exponential activations corresponding to 8, a, and K. To mitigate overfitting, we randomly hold out 20% of the training set validation set and apply early stopping (default epochs for method is validation loss. The key distinction between our pre-calibration methods and the original ALD model lies in an additional adapter module that incorporates the quantile level q ((, 1) an input. This allows the model to learn crantilc-conditionnn ALD parameters {6) o Mig which are crucial for both quantile regression and calibration losses (see Equation equation TŪ and Equation equation [2].
applied the output heads
2000) based on
In the post-calibration setting (i.c., AL.Dacal and CPust D.Cp)) we first obtain the ALD parameters from pre-trained base ALD model. Then, we apply a lightweight post-calibration network that takes both the input x and quantile q using LPost ALD+Caa and L ALD-CCur to output adjustment factors E R This post-calib cost-calibration module is implemented as a compact MLP with two hidden layers of 16 units each and ReLU activation. An exponential activation is then applied the output to ensure positivity of the adjustment factors. These adjustment factors modulate the base parameters produce calibrated outputs via element-wise product (see Equation equation [3). This design allows the calibbalibatiion module to correct for miscalibration while preserving the base model's learned structure.
Finally, for prediction, we sample 2000 quantile percentages 4 U(0 1) to construct a mixture of ALD. Conceptually, this model can be interpreted as a continuous mixture over ceantile-speciffe ALD components, expressed as:
f(y [u]xx= x)= SALD (y: (x..)))))) dg.
(69)
where (ALD(: (x, g)) is the ALD parameterized by the post-calibrated model at quantile This formulation captures rich distributional information and enables fine- grn-ggraied calibration by averaging over wide spectrum of quantile- condttie--od predictions.
CQRNN. We followed the hyperparameter settings tuned the original paper (Pearce et al. 2022).
where three random splits used for validation (ensuring overlap with the random seeds used in the final test runs). The following settings were applied:
Weight Decay: 0.0001
Grid Size: 100
Pseudo Value: y² = 1.2 x max; 1/4
Dropout Rate: 0.333
The number of epochs and dropout usage were adjusted based on the dataset type:
Synthetic Datasets:
Norm linear, Norm non-linear, Exponential, Weibull, LogNorm, Norm uniform:
100 epochs with dropout disabled.
Norm heavy, Norm medium, Norm light, Norm same: 20 epochs with dropout disabled.
- -LogNorm heavy, LogNorm medium, LogNorm light, LogNorm same: 10 epochs with dropout disabled.
Real-World Datasets:
METABRIC: 20 epochs with dropout disabled.
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-WHAS: 100 epochs with dropout disabled.
SSUPPORT: 10 epochs with dropout disabled.
GBSG: 20 epochs with dropout enabled.
TMBImmuno: 50 epochs with dropout disabled.
BreastMSK: 100 epochs with dropout disabled.
LGGGBM: 50 epochs with dropout enabled.
LogNorm. The output dimensions of the default neural network architecture are 2, where the two outputs represent the mean and standard deviation of a Log-Normal distribution. To ensure the standard deviation prediction is always positive and differentiable, the output representing the standard deviation is passed through a SoftPlus activation function. We followed the hyperparameter settings tuned in the original paper (Pearce al. 2022). with dropout rate of 0.333, The number of epochs and the usage of dropouts were adjusted according to the type of dataset as follows:
Synthetic Datasets: The same settings described above for CQRNN.
Real-World Datasets:
METABRIC: epochs with dropout disabled.
WHAS:: 50 epochs with dropout disabled.
-SUPPORT: 20 epochs with dropout disabled.
GBSG: 10 epochs with dropout enabled.
TMBImmuno: 50 epochs with dropout disabled.
BreastMSK: 50 epochs with dropout disabled.
LGGGBM: 20 epochs with dropout enabled.
DeepSurv. We adhered to the official hyperparameter settings from the pyсox methods module (GitHub Link). Each of the two hidden layers contains 32 hidden nodes. A validation set was created by splitting 20% of the training set. Early stopping was used terminate training when validation performance stopped improving. Batch normalization was applied.
DeepHit. We adhered the official hyperparameter settings from the pyсох. methods module (ittHub Link). Fach the two hidden layers contains 32 hidden nodes. A validation set was created by splitting 20% of the training set. Early stopping used terminate training when validation performance stopped improving. Batch normalization was applied, with additional settings: nun.durati ions 100, alpha 0.2, and sigma 0.1.
DSM. We adopted the Deep Survival Machines (DSM) model from the auton- survival library B implemented via aut sutessuvviaal models dsmm SeepSurvivalMaeee Malhhcinnes. The model was configured with two hidden layers of 32 units each. For the LogNormal variant, the number of mixture components was set to l: 10, as increasing à led performance degradation. For the Weibull variant, we followed the default configuration with & = 100 to ensure sufficient capacity. The model was trained using observed event times and indicators, and the final prediction was constructed by evaluating the mixture distribution over fixed 1000-point time grid to obtain the cumulative distribution function (CDF).
GBM. We used the GradientBoost t ingsurvvvaalnaaaaaa.... sksurv. ensemble module The model was configured with n.estimators 100, learning rate 0.01, and max depth 3.
implementation from the
RSF. For the Random Survival Forest, we used the RandomStrviivalForess class from sksurv. entumvle We followed the standard configuration with neestimators 100.
e.html
http html
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C ADDITIONAL RESULTS
This section presents additional results to provide a comprehensive evaluation. The full results for pre- calibration, post-calibration, and general performance are provided in Appendix C.I Appendix C.2 and Appendix C,3 respectively. Case studies are provided in Appendix C.4
C.I PRE-CALIBRATION RESULTS
Table presents the full results for the pre-calibration setting. The best performance for each dataset and metric is highlighted in bold. Fig. iillustrats the best and worst individual calibration AA.D against CALD. achieved by the hybrid ALD-based survival model across all synthetic datasets.
improvement cases with the pre-calibration setting, comparing Lre
Table 6: Full results table pre-calibration for all datasets, methods, and metrics. represent the mean + standard error for the test set over runs.
The values
Dataset
Method
Average Calibration
Group Calibration
Individual Calibration
(CAD)
0.047 + 0.006
0.079 = 0.012
0.044 0.005
CQRNN (Lew)
0.035 + 0.006
0.054 = 0.005
0.018 0.002
Norm_linear
D
0.050 = 0.007
0.076 = 0.010
0.043 : 0.003
PPe
0.017 = 0.003
0.036 = 0.009
0.018 = 0.002
0.016 + 0.002
0.029 0.005
0.018 0.002
(CAL))
0,072 = 0.010
0.119 0.012
0.060 0.008
CQRNN (Cop)
0.034 = 0.011
0.078 = 0.011
0.029 : 0.002
Norm_nonlincal
Cere
0.070 + 0.007
0.112 = 0.009
0.056 0.005
CPic
0.034 = 0.003
0.045 = 0.006
0.018 0.001
ALDica
0.023 = 0.004
0.035 = 0.044
0.012 0.001
ALD (CALD)
0.095 + 0.009
0.159 = 0.020
0.098 0.020
CQRNN (Cow)
0.036 + 0.009
0,113 = 0.029
0.054 0.007
Norm uniform
DPre
0.078 + 0.013
0.144 0.022
0.085 0.020
ALD44G
CEre
0.102 0.004
0.141 0.009
0.092 0.004
ALDACal
0,027 = 0.004
0.038 = 0.006
0.018 = 0.002
(CALd)
0.018 + 0.011
0.030 = 0.014
0.016 + 0.003
CQRNN (Cew)
0.030 + 0.008
0.051 0.011
0.030 0.003
Exponential
PPw
0.017 = 0.009
0.029 0.011
0.013 0.004
ALDD
0.034 = 0.007
0.041 0.009
0.006
Phe ALisCCal
0,012 = 0.005
0.020 = 0.006
0.015 # 0.003
ALD (CAi))
0.048 = 0.009
0.067 = 0.006
0.042 + 0.003
CQRNN (Cop)
0.031 = 0.004
0.086 0.021
0.040 0.010
Weibull
0.045 0.008
0.061 0.005
0.039 0.003
ALDHOU
0.031 0.007
0.039 0.008
CPR AlLDACa
0.019 + 0.005
0.032 + 0.002
0.020 0.002
ALD ALD((AAD) (CALd)
0.020 + 0.006
0.031 = 0.012
0,128 + 0.006
CQRNN (Lop)
0.031 0.013
0.050 0.014
0.135 0.008
LogNorm
Pfre
+ 0.003
0.030
CEre
0.052 0.005
0.058 0.006
0.140 0.002
ALDiCa
Cls
0.015 = 0.003
0.021 0.002
0.131 0.003
ALD (LAL))
0.062 + 0.009
0.113 + 0.033
0.048 0.006
CQRNN (Ley)
0.020
0.157 = 0.023
Norm heavy
0.068 0.015
0.118
0.046 0.003
ALDDO
cFic
0.132 0.027
0.259 0.033
0.110 0.005
Cle ALD+Cal
0.063 = 0.010
0.109 = 0.017
0.003
ALD (LALd)
0.054 + 0.031
0.086 = 0.026
0.028 0.004
CQRNN (Cew)
0.050 0.015
0.093 0.013
0.001
Norm med.
PPn
0.044 0.012
0.079 0.009
0.025 0.003
LALDDOG
CEIc
0.085 0.003
0.111 0.006
0.071 0.004
LAIDaCal
CPi
0.044 = 0.004
0.076
0.020
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Dataset
Method
Average Calibration
Group Calibration
Individual Calibration
ALD (CAD) CQRNN (Lcx)
0.077 = 0.034
0.111 * 0.027
0.027 0.005
0.036 = 0.018
0.083 + 0.006
0.015 0.002
Norm light
CPee
0.055 0.013
0.097 0.003
0.023 0.002
0.048 0.004
0.068 0.007
0.037 0.002
the
ALDiCa
0.032 = 0.006
0,059 = 0.005
0.016 : 0.001
ALD (CALD)
0.065 = 0.012
0.090 0.017
0.044 : 0.008
CQRNN (Loq)
0.037 0.008
0.075 0.008
0.004
sarme
CPcc
0.062 = 0.008
0.088 0.012
0.043 0.005
Chic
0.029
0.067
CBee DeCa
0.025 = 0.003
0.053 + 0.009
0.023 * 0.002
ALD (CALD)
0.037 = 0.029
0.081 = 0.043
0,038 = 0.008
CORNN (Loq)
0.174 = 0.008
0,294 = 0.014
0.011
LogNorm heavy
LPe
0.027 0.013
0,070 0.022
0.002
DPf
0.024
0.061
0.040 0.003
Liee ALDiCa
0.025 0.005
0.072 0.009
ALD (CAL))
0.021 0.009
0.062 0.020
0.040 0.005
CQRNN (Log)
0.079
0.157
0.071 0.007
LogNorm med.
LPre
0.019 0.008
0,051 0.010
0.034 0.002
Chi
CLD
0.035 0.008
0.063
0.044 0.002
LPie DAli
0.050
ALD (CALd)
0.021 0.005
0,053 = 0.007
0.027 0.002
CORNN (Coo)
0.035 0.007
0.074 0.012
0.030 0.002
LogNorm light
[Pee
0.023 0.011
0.056 0.012
0.026 0.002
ALD-CCu
CPP
0.043 0.012
0.073 0.010
0.037 0.002
ALDiCA
Chee
0.017
0.050 0.009
0.025 0.001
(CALd)
0.006
0,052
CORNN (Co.))
0.029 0.008
0.068 0.004
0.014 0.001
LogNorm same
LXCCAL
0.030 0.018
0.059 0.019
0.012 0.002
ALD-CU
0.035 0.004
0.067 0.007
0.011 0.004
ALDICU
Cose
0.014 0.002
0.047 0.003
0.008 0.004
ALD (CALD)
0.136 0.013
0.265 0.020
CQRNN (Cog)
0.165 0.001
0.270 = 0.010
METABRIC
LXCAL
0.135 = 0.007
0.265 = 0.016
ALD-CCr
0.134 = 0.020
0.230
CPTe ALDiCa
0.100 0.016
0.222 2 0.018
ALD (CALD)
0.103 0.030
0.290 0.020
CORNN (Lo.)
0.144 = 0.021
0.348 = 0.021
WHAS
CXCAL
0.103 0.030
0,288 0.017
CLD
0.060 = 0.019
0.214
ALDICa
0.046 0.011
0.172 0.016
ALD (CALD)
0.263 0.008
0.307 0.015
CQRNN (Log)
0.174 = 0.005
0,218 + 0.004
SUPPORT
CXCALL
0.257 = 0.00S
0,301 + 0.011
Chfc CLL--Ca
0.175 = 0.016
0,214
ALD+Cal
0.164 + 0.015
0,205 + 0.014
ALD (ALD))
0.201 0.017
+
CQRNN (Lc.)
0,204 = 0.008
0,315
GBSG
AALD-OP
Clf
PPe
CAL
0.202 0.013
0.208 = 0.020
0,301 + 0.009
0.292 + 0.028
pls ALDiCal
0.161 0.099
ALD (LAD)
0.228 + 0.010
0.275 + 0.015
CQRNN (Log)
0.229 0.010
0
TMBImmuno
0.229
0
Chic ALD--CO
0.243 0.009
0.265
pbe ALDHCal
0,229
0,252
ALD (CALD)
0.272
0,286 0.031
CQRNN (Leg)
0.289 = 0.004
0.310 0.006
BreastMSK
0.276 0.020
0,292 0.024
Ctf CALD-CP
0.267 0.017
0.287 0.011
0.249
0.270 0.012
ALD (CALD)
0.173 0.050
0.348 0.034
CORNN (Log)
0.180
0.372
LGGGBM
CPe
0.165 0.046
0.334 0.033
ALD-CP
CPm
0.133 0.036
0.364 0.032
piic ALDHCa
0.135 = 0.023
0.360
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Published as a conference paper at ICLR 2026
Test Poroveee:: 1D, 1851
WS Ingroseeent 3D. 150
Deet inpoveeent 30, 5k0g
Wett improvere ne Ne
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Leghanne Indivdual Colbratto Casts
22
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Dest iwp.oveeeeetttee... 30. BB0)
Went improvere 756
Mil
Nom light Inchuidust Calibration Cabes
Nom Indhechan Callbretion CEBES
Cive
360 55
156
15 354
06
35
35
355
05
30
25
05
05
155
320
30
355
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Published as a conference paper at ICLR 2026
Lognarm neayy Individust Cansratznn Caoes
Loghtrm IREE INStviDu Contration CadES
0
Logitrm gic. indvitual Colbracion Capes
Log some indittual Colbration Cases
0
Figure 3: Illustraiion of the best and worst individual calibration improvement cases PCer CALD) achieved by the hybrid ALD-based survival model across all synthetic datasets.
ALLDaal VS.
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C.2 POST-CALIBRATION RESULTS
Table presents the full results for the post-calibration setting. The best performance for each dataset and metric is highlighted bold. Fig. iillutarut the best and worst individual calibration improvement cases with the post-calibration setting, comparing ALD+Cal against CALD. achieved by the hybrid ALD-based survival model across all synthetic datasets.
Table 7: Full results table on post-calibration for all datasets, methods, and metrics. The values represent the mean + standard error for the test set over 5 runs.
Dataset
Method
Average Calibration
Group Calibration
Individual Calibration
Pre
0.016 0.002
0.029 +: 0.005
0.018 + 0.002
Pest AAL
0.020 + 0.002
0.038 + 0.006
0.020 + 0.002
Nonear
ALDICSD
0.058 0.011
0.072 = 0.017
0.058 0.009
0.047 0.006
0.079 0.012
0.045 + 0.005
ALDeCal
0.018 0.003
0.027 0.003
0.017 0.001
Phe
ALDeCal
0,023 0.004
= 0.004
0.012 = 0.001
Chai
0.035 0.004
0.047 = 0.005
0.019 = 0.002
Nomn.nonlincar
PPo
0.104 + 0.006
0.138 # 0.017
0.082 # 0.008
ALDciPoo
0.074 + 0.009
0.120 0.011
0.061 + 0.008
0.021 0.002
0.027 0.003
0.012 + 0.001
ALDiCu
Phe
0.027 0.004
0.038 0.006
0.018 0.002
Plost
0.105 0.003
0.145 0.005
0.099 0.003
Norm uniform
LLDDCSD
Poot
0.145 = 0.039
0.176 = 0.028
0,127 = 0.018
clat
ALDyCIPOT
0.097 = 0.008
0.160 = 0.020
0.099 = 0.020
AlTicaal
0.027 = 0.002
0.031 = 0.001
0.016 = 0.002
Phe
0,012 0.005
0.020 0.006
0.015 0.003
ALDsC
0.027 0.008
0.038 0.006
0,020 0.005
Exponential
ALDDCSD
cPos
0.058 0.036
0.064 0.036
0.031 + 0.020
Cost
0.037 + 0.035
0.049 + 0.034
0.026 + 0.022
ALDrCa
Fos
0.012 0.005
0.019 + 0.006
0.011 + 0.004
ALDicU
0.019 0.005
0.032 0.002
0.020 = 0.002
ALDiCg
0.026 0.005
0.036 0.007
0.026 0.003
Weibull
ALDcSDD
0.119 0.029
0.153 = 0.045
0.054 = 0.009
Cket
ALD-CIPOT
0.022
0.029
0.048 = 0.006
Aa
Pid
0.021 0.004
0.030 = 0.005
0.020 = 0.003
Lhe
0.015 0.003
0.021 0.002
0,131 + 0.003
0.048 0.005
0.053 0.005
0,138 + 0.004
LogNorm
ALDDCSD
PPoot
0.070 0.015
0.081 0.014
0.141 0.008
Pott
ALDCCRPT
0.021 0.009
0.033 0.013
0.127 0.007
Al
plus
0.014 0.003
0.019 0.002
0.128 + 0.004
ALDsCs
Pie
cleat
ALDaCa
0.063 = 0.010
0.144 # 0.009
0.109 = 0.017
0.223 # 0.023
0.111 + 0.002
0.038 + 0.003
Norm heavy
Poo
ALDsCSD
0.155 0.047
0.217 0.030
0.144 0.01
ALD+CiPOT
0.063 0.010
0.134 0.036
0.048 0.006
Pid
ALD+Cal
0.033 0.006
0.067 0.003
0.030 0.002
Pw
0.044 + 0.004
0.076 = 0.006
0.020 + 0.001
0.081 + 0.004
0.107 = 0.008
0.072 + 0.003
Norm med.
LALDICSD
PPo
0.148 0.048
0.180 0.050
0.174 + 0.017
ALDiCIPOT
0.091 0.073
0.9 0.070
0.034 + 0.012
0.031 0.004
0.050 0.004
0.017 0.001
Pre
ALDiCa
0.032 0.006
0.005
0.016 0.001
ALDRIS
Reat
0.048 0.003
0.067 0.004
0.037 = 0.001
Norm light
DACSDD
0.113 + 0.020
0.133 = 0.027
0.194 + 0.017
A.DDCCP
0.074 + 0.035
0.107 0.029
0,027 + 0.005
ALD+Cal
0.028 0.002
0.043 0.002
0.014 = 0.001
AL
Pre
0.025 0.003
0.053 0.009
0.023 0.002
cloa
0.027 0.005
0.005
+ 0.002
Norm same
CPoz
ALDDICDD
0.201 + 0.027
0.231 + 0.023
0.055 + 0.018
ALDICIPOT
0.065 = 0.012
0.096 = 0.012
0.044 + 0.008
ALDiGa
0.021 # 0.004
0.043 & 0.003
0.017 + 0.001
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Dataset
Method
Average Calibration
Group Calibration
Individual Calibration
AaDaca
Lhe
0.025 0.005
0.072 0.009
0.039 0.005
Chest
0.032 0.001
0.056 0.003
0.040 0.002
LogNorm beavy
CALDICSD
CPost
0.239 = 0.006
0.256 = 0.011
0.153 = 0.015
Chal
ALD-CPOT
0.167 = 0.025
0.304 # 0.025
0.038 = 0.008
ALDiCal
0,020 = 0.003
0.058 # 0.008
0.041 = 0.003
Lh
0,018 0.006
0.050
0,035 0.003
LPaa
0,033 0.004
0,058 0.008
0,046 0.003
LogNorm med.
Lhot
0.080 0.005
0.090
0.073 = 0.012
Chat
0.015
= 0.019
0.040 = 0.005
0.014 0.003
0.046 # 0.003
0.040 + 0.002
ALDiCa
0.017 0.003
0.050 0.009
0.025 + 0.001
ALLDCy
0,043 0.004
0,072 0.007
0,035 + 0.001
LogNorm light
Cloat
AADDC
0.163 0.020
0.169 0.018
0.275 0.035
ALDACIPOT
Coat
0.021 0.007
0.060 0.010
0.027 0.002
Post
ALDrCal
0.001
0.003
0.030 0.001
ALL
Chon
Pre
ALDiCa
0.031 # 0.003
0.014 + 0.002
0.051 # 0.006
0.047 # 0.003
0.012 + 0.005
0.008 = 0.004
LogNorm same
CPout
0.291 0.009
0.309 0.014
0.562 0.075
CASD
ALD-CPPT
0,017 0.006
0.059 0.010
0,012 0.003
ALD+Cal
0.014 0.002
0,042 0.003
0.008 0.002
Pr
0,100 = 0.016
0.222 # 0.018
Cat
PPn
0.145 = 0.012
0.257 # 0.018
METABRIC
DACSD
0.095 0.006
0.105 0.006
chat ALDCCIPOT
0.108 0.009
0.229 0.047
ALDaca
0.120 # 0.015
0.242 0.008
APr
DaCa
0.011
0.016
Poo
ALDiCUr
0.017
0.014
WHAS
ALDDCCDD
Cleat
0.064 # 0.008
0.138 # 0.020
Cloat ALDACIPOT
0.222 + 0.020
0.387 + 0.011
ALDaCA
0.064 + 0.013
0.244 # 0.010
ALDiCal
Pie
0.164 0.015
0.205 0.014
CPost
ALD
0,215 0.005
0.251 0.012
SUPPORTT
DACDD
Clea
0.102 = 0.012
0.115 0.012
Clod
ALD-CPPT
0.139 + 0.011
0.324 0.032
ALD+Cal
CPost
0.240 0.003
0.283 = 0.007
0,161 0.009
0.272 # 0.018
0.217 0.008
0.304 0.011
GBSG
CPeat
LACD
0.002
0.021
PPs
ALD-CPOT
0.016
0.234 0.032
Ret
ALDiCa
0.003
0.009
LPe
0.229 + 0.009
0.252 + 0.010
LAIDD=Or
0.249 + 0.012
0.269 + 0.010
TMBImmuno
LALDiUSD
cPost
0.150 0.012
0.184 0.021
cloat
ALDACIPT
0.157 0.005
0.210 0.018
ALDicaa
0.214 0.010
0.242 0.014
CPr
ALD+Cal
0,249 + 0.016
0.270 + 0.012
ALD+CCr
0.270 # 0.014
0.287 # 0.016
BreastMSK
0.212 # 0.012
0.225 + 0.010
AADD
0.325 0.009
0.433 0.079
ALD+CIPT
ALD+Cal
0.247 0.011
0
ALD+Cal
Pm
0,135 0.023
0.360 =
0.152 0.021
0.335 0.044
LGGGBM
Cloat
0,351 + 0.027
0.431 + 0.022
CPoa
0.267 # 0.021
0.415 + 0.011
0.118 0.023
0.257 + 0.037
ALDiCa
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Published as a conference paper at ICLR 2026
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33

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Published as a conference paper at ICLR 2026
Logharm reayy individuel Cansritzon Caoes
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Figure 4: Illustraiion of the best and worst individual calibration improvement cases (CPod LALD) achieved by the hybrid ALD-based survival model across all synthetic datasets.
ALLDcaa VS.
34

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Published as a conference paper at ICLR 2026
C.3 OVERALL RESULTS
Table g summarizes the full results across 21 datasets, comparing our method with 9 baselines across 7 metrics. The best performance for each dataset and metric is highlighted in bold.
Table 8: Full results table for all datasets, methods, and metrics. The values represent the mean = 1 standard error for the test set over runs.
Dutused
Metluds
MAE
Harrel's C-index
Une's Cánseex
Avcruge Culibration
Group Culibrationn
Individual Cullbration
Du692 0..22
0.285
0.008
0.619 B1126
0u619 = 0.02
0.947
0.179 0.012
0.044 0.005
CORNN
2,877
0.268 0.087
0.656 0.011
0,650 0.000
0.055 00,006
0.054 0.005
0.015
Nomn. Ineor
DrppSrrv
LegNoen
0.251 000003
0.348 4 0.156
0.665 0.07
0:713400:00
0.677 + 0.000
0.577 e 0.000
0.611 00000
0.651 + 00008
0222 + 0,007
0010 + 0.007
0,51 0.023
B.000 00,015
0.014 0.082
0.016+ 0.006
(Wcialiaal))
1054 n 0.034
0.322 0.006
0.005
0663. 0.10
0.647 + 0.000
0.91 + 0.009
0.188 0.006
0.066 0.002
DSMI LogNoo)
1.688 a 0.031
0.323 0.00
0.611 = 0.005
3,666 = 0.208
0.090 0.012
0,200 : 0,010
0.057 4 0.005
Despilit
1.06 2 0.442
0 +-0022
0.665 01230
3.607 0.031
0,189 : 0.044
0.254 0.061
0.151 0.00
GBM
0..022
0.305 0.007
0.66 z 0.14
0,631 00012
IUI17 2
0.091 : 0.011
0.030 0.001
RSF
1.329 0.314
0.331 0.00
0,581 = 0.033
0.577 =
DJ23 = 0.002
B1053 0.006
0,102 a: 0.009
0.600 0.122
0.122
0.375
0.000
0.661 a IJI09
R.633 a 0J008
D018 + 0.003
IU127 a 0,013
0.017 + 0.001
ALI3
0.077 +..18
0.241 0000
0
= D0:0
0.57 = 0033
0.912 0,010
1. 0.012
0.000 0.006
CORNN
0.039
0003
0.994 = Bn12
0.560 0.013
= 0.011
E.078 0.011
0.002
Nomuneiizn
LogNorm
0.229 00000
00013
0.699 = 0106
0.643 = 0.014
0 2 0.005
0.016
0.005
SeepSure
0.850
0001
= Un0
0.649 0.015
0.023 = 0.007
0.065 0.012
0:460
0,627
0,097
0.002
LigNon)
0.083
0.003
0.915 0021
0,963 0.022
0.011
1.125 0.013
0.003
DecaHii
1027 0.346
0,042
0.600 0.49
= 0.041
0.114
0.024
GHM
0:314 a 0,090
0,327
0.055
0.65 a 0.211
0.688 = 0.19
0.26 0,003
0.006 0,006
0.000
RSP
0:508 0.061
0.341
0,004
0 0015
0.605 - 0014
0 0.001
0,50 0.001
0.04K 0.009
0.794 0,184
0,261 0.012
0.679 s tuIls
(.657 = 0.006
(U121 n 0.002
fU127 0,413
0,012 0.081
ALD
2 2 0,664
0,049
0.002
0.768 = 0.019
0.693 : 0.224
0.693 2 0,009
0.159 = 0,000
0 0.020
CORNN
0,000 0 0,328
0.154 0.077
0.767 + n022
0.80 - 0017
0.0 - 0.009
0.117 0.000
0.054 0.007
Noro stalfom
LeghNom
15876 3.017
0.387 0,04
0.5% + 0.175
0,74 - 0.000
0.218 + 0.003
n.234 0.001
0.336 0.013
DeepSurv
0.199
0.517 0.317 2000
6.784 00030
0.704 = 0.014
01009 = 0,014
0.102 0,006
0,064 = 0.003
LSSM Weihulll
1.30120.001
0,062 0.00
0.764 = 0000
0.079 = 0018
= 0.013
0.196 0.001
0, 0,106- 0.002
USMI Loghorm)
1.300 00011
0.01 0002
0:781 = 0013
01922 = 0021
0111 = 0.021
0.201 0 0.000
0,147 + 0.018
Desgffit
0.300
0.368 0.07
0.756 = 0.25
0,991 = 0.255
0.253 = 0:077
0.355 0-113
0.147: 0.040
GBM
0.112)
0000
0.746 = 0.09
0,77 = 0.122
0.032 = 0.013
0.156= 0.013
0,071 000004
0.086
00002
0.687 = 0022
= 0.011
0 = 0.010
BU772 0.012
0.075 0.000
Calbues
0.03
0 D006
0.690 = 0011
00277 = 0,002
[U131
0,016 0.092
ALD
0.554 0.153
0,242
0.044
0.564 BJN6
0.563 0.005
00018 0,001
0.030 0.014
0,016 0.003
CORNN
1682 0.316
0.009
0,553 0012
0.008
0,005
0,030 0.005
Expreziiall
DuspSurs
LegNorm
2.533 a 0.184
0.557
0
0.055
0.015
0.564 s IUIb4
0.566 0.41
0.563 = 0.004
0.518 0.040
0.019 = 0,203
0,005
0.056 0,005
0.037 0.016
0.018 0.006
0,054 0.006
Weiibll
1.915 0.237
0.237
0.293
0.004
0,563 z 0.00
0.562 = 0.004
0.616 2 0,005
Bu062 0.003
0,030 0.005
DSMd LngNom)
2 2 0.235
0,295 0.00
0,55 e n004
0.59 = 0.005
0 = 0,008
B.0K2 = 0,007
0,064 + 0.006
Dosplin
1.190 - 0.767
D469 0.011
0.5% + nm25
0,59 - 0023
D.88 - 0.040
0:121 0.054
0,000 0.040
GRM
2.190 0.236
0.296 0.005
0.577 + 0012
0.37 - 0010
D,21 - 0.005
BB445 0.011
0.025 0.006
RSF
3:507 20.292
0,345 0.00
0 B016
0.315 = 0.05
IL042 = 0,010
1.057 0.015
0,137 0.005
0.500 2
0.292 x 0.007
0,548 = 0004
0.530 = 0.005
IUI12 : 0.005
IUn19 2 0.806
0,011 2 0.104
ALD
0.642
0.211 0.084
0.762 0000
0.78 0.009
0.048 0.009
0.K7 0.006
0.042 0.033
CORNN
08788
0.007
0,752 nnlo
0.24K = 0.005
0.631 0.004
0.066 0.021
0.010
LegNorm
03359 0.111
0.775 = 0006
0 = 0007
D 0,014
1L115 =
Webbit
DrepStrre
0,877
0.01
6.774 RUIR
= 0.00%
IU11 9,009
IU129
DoMiLo)Nomo)
0.009
0.004
0.744 0.000
0,010
0.230 0.015
0.002
Decoltit
1937
0.055
0 0.055
0,764 0.005
D 0,015
0.195 0,000
0.015
GRM
0.100
0.252 0.007
0.767 n 0006
262 = 0.007
0.690 x 0,011
0.161 0.016
0.063 0.003
RSF
1 0,349
0,233 0.014
0.748 s 0011
0.242 = 0.12
0.647 : 0,015
0.77 0.021
0.074 0.008
CPe
0.288
0:70 0.008
0.78= 0.006
0,021 0,004
0.030
0,000 0.003
ALD
0.277 0.089
0,375 00012
0,588 = 00020
0,584 = 0.020
0.630 0,005
BJ33 = 0,012
0,128 0.006
CORNN
1043 00087
0.386 0.014
0.544 z n007
0,579 = 0.008
0.311 = 0.013
0.000 2 0.014
0.185 0.008
LegNNom
0.229 2 0.037
0,647 0.00
0.586 z n018
0.583 =: 0.088
IUI13 = 0,804
0.025 0.007
0.113 0.002
LogNor
Seeppurv
0.536 2
0.657 2003
0,587 = n.00
0.584 = 0.020
0.014 = 0.003
00030 0.013
0.117 a 0.006
DSMI Weiball)
0.38 00005
0.185 00012
0.40 + 0013
0.608 - 0013
0.08 0.011
BD44 0.011
0.009
DSMI agNam))
0.988 00088
6.992+0.04 0.014
0.9%% 0007
0,903 0.006
0.21 0.010
0.22 0.010
0.110 6.010
Dagiiti
0.761 0.117
0.566
0,539 = an3
0,57% = 0.000
0 0.048
0.048
0 0.008
0,005
0 00000
0 0.020
0,016 0.001
00223
1,077
0,400 0.008
0.36 0018
0.35 =
0.000
CAliwes
0:249
1013
0.588 0.014
0.014:00000
ALD
0.788 0.3364
0:011
6.916 JUI
6,86 0JMS
0,062 0.009
1.113 0.083
0.006
CORNN
0.400 0.045
0,486 0.012
0.916 0000
0888 = 0.17
D.VI 0,000
0.157 0,023
0.005
LegNerm
254 3.330
0,11 00077
0788 0.40
0.20 = 0.044
0.255 a 0,005
0.256 0.009
0 0.002
heavy
DeepSurv
2.14
0,559 0.009
0.746 z 0J:35
0.619 = 0.056
1 0.021
1.185 0,064
0.007 0.003
WeiiWiibult
1887 a 0.040
0,047 0.005
0,530 = 0011
3.766 = 0.019
0.656 2 0,009
012200 2 0.038
0.164 0.004
LogNoo))
1.90 2 0.039
0,047 0.005
0.786 = 01134
3.718 = 0.036
0.074 = 0,009
11.238 = 0.034
0,183 0.004
Dasglii
0.751 0.095
0.493 0.018
0.01 = 0007
01885 = 0010
0.067 : 0.011
01218 1 0.044
0.066 0.000
GBM
1619 2
0.042 0.003
0.670 z 0.007
0.812 = 0011
IUI31 = 0,009
10200 0.082
0.143 0.004
RSF
0.619 0.019
0,021 0.001
0.90% = 0.009
0U853 = 0016
0U072 = 0.021
1. = 0.063
0,070 0.003
0.309 0.101
0,025 0.000
0 = 0.013
0.826 = 0.024
0U033 = 0,005
IJ667 + 0,803
0.030 2: 0.102
ALD
a 0.071
0,861 6105
0,88% = BUISS
0.866 = 0.006
0.054 = 0.031
0.086 0.026
0.004
CORNN
0.257 0.014
0,854 = 0.005
0.862 = 0.004
0.000 1
0L093 0.013
0.019 0.001
LogNorm
5.524 2 1.179
00009
0.825 = 00028
0.792 = 0.032
0.195 2 0.002
02222 0.007
0.007
Nomm oeed
01009
Mishhllllll
0.006
0.755 0017
0.737 0.17
0.003
01235 0.087
0.005
DSMi LogNNo)))
1.893 0,000
0.109
0,655 = 0.020
0.657 = 0.028
00038 0,003
0.214 0,041
0.136 0.005
Despltit
0.928 0,009
0,504 0.11
0883 0.006
0.861 0.005
0.122 0,024
0.256 0,045
0.004
URM
1-434- > 0.007
0,097 0.005
0.57 a 0.001
0885 = 0.002
0 4 0.004
0.210 0.096
0 0.003
RSF
0.433 2 0,020
0,052 0.005
0.833 = 0.004
0.860 = 0.003
IUI13 2 0,007
BU149 2 0,814
0,035 0.101
0.634 + 0.077
0.067 0.005
0.87 + 0.005
0.852 + 0.005
0.6%0 + 0.004
0,050 0.004
0.017 6 0.081
35

---

Published as a conference paper at ICLR 2026
Datseet
Metlands
MAE
IBS
Harrel's C-imlex
Unn's C-indes
Average Calibration
Grup Calibration
Individul Calibration
ALD
2
0.103 2
0 0.013
0,872 0.003
QJT7 a 20054
0,111 0.037
0,027 00005
CQRNN
8.300 2 00,41
0.506 20.015
0,878 0.006
01870-10.00 0.005
0.036 = 0.015
01083 x 0.000
0.15 000002
LagNum
1.355 003303
0.551 00.01
0 0011
01842: 0.013
0.185 0.002
0.206 0.003
0.166 10.01
Nom
1904 DSM/Webblll
DecySury
0.26 2 0.000
2944:00.06
0.223 22002
0.726 0.18
0,882 IUI22
I1874 20.002
0.716 0.018
0.029 = 0.003
0.026 = 0.004
01234 0.038
BU056 1 0.007
0.017 20000
0.124 000003
DSVU on Nerm)
1.906 0.001
0.01-00.13
0.549 0.211
0.% 00.00
0.14
0.216 0.011
0.128 0.000%
Dorpiti
8983 0.056
0.681 0.024
0.876 0.00
0.66
0.117
0.251 0.004
0.91
CIRM
1.318 0.028
0.176 0.009
0.847
0,830
0,030
0.207 0.033
0.382 0.013
0,199 0.005
0.874
01866
0011200009
0.04 0.017
0.557
0.106
0.028 0.002
0.002
0.014 0.001
00,001
0.020
0,000 0.011
CORNN
0.334
0.374
0.037
0.004
Noro samee
DecpSurv
LogNum
0.255 0.004
0.26 0..49
0.797
0.752
0,894
0,887
0.006
0.025
0.027
0.056
0.013
0.04
0.002
0.007
Well/i
2.922 00000
0.006
0.737= 0.016
0.125 00,00
0.312 0.030
0.005
DSMILog
2147 00.04
0.177 0.006
0.665 0018
0.618 0.015
(1130 0.005
0.301 0.012
0. 1.176- 0.004
Dazpitia
1.274 0.057
0.566 0.632
0.882 0.003
0.526 0.021
0.02 0.007
0.187 0.046
0.083 0.005
GBM
1.546 0.077
0.141 = 0.004
0.877 0.009
0.779 0.005
0.858 0.005
0.277 0.033
0.101 0.004
RSF
0.71 a 0.024
Du76
a 0.003
0.674 0.004
1,821 0.010
0 = 0.012
11100 0.017
0.101 0.001
0
0.080
a 0,002
0 0U006
01827 0.007
0 0.004
01043 0.003
0.017 a 0.000
ALD
0:422 0.395
0.06 a 0,005
6,775 0.009
0.7.9) 6.019
0.037 = 0,229
0.58: + 0.044
0 0.008
CORNN
0.78 0.0021
0.203 0.032
0.766 8.007
0.722 0.020
0.174 00,05
0.294 0.014
0.113 0.01
LogNoom heary
LogNorm
0.597 00,042
0.399 : 0,010
0.619 0.04
0.578 0.036
0.024 00013
0.149 0.021
0.152 0.000
DeepSurv
0833 0.005
0.457 2
0.570 0031
0.445 0.028
0.179 0.019
0.291 0.003
0.047 2.OLD
0.77 00,01
0.124:00.000
0.687 0018
0.622 0.000
0.105 -
0.187 +0.015
0.156 200005
B.55 06.05
1:126 0.006
0.000
0.649
0.135 0.007
0.016
0.000
Decpitit
0.23 0:020
0.399 0.016
0,753 0.15
0.708 0.021
0.17% 0.01
0.323 0.013
0.184 0.015
CBM
0.666 0.016
0.119
0.660
0.01
0.200
0.100 0.016
0.706
0.016 0.729
0.195 0.009
0.272 0.015
0.317
0,098
0.002
0.020 0,003
0.008
0.041
ALD
0.20
0.175
CORNN
a 0.035
0.005
0,744
0.007
Loghoess rost.
LogNocm
0.49
0.459
0,096 0.03
0.671
0.025 0,007
0.000
DempSurv
864320.001
0.399 2
0.642 0.012
0.014
0.070
0.002
0.005
8638 0.003
0.221 00,00
0,068 0.00
0.665 0.005
0.016 0.004
0.015
0.002
DSMeLogNeem)
26442 0.013
0.213 0.007
0,719 0.004
0.694 0.008
0.944 = 0.005
0.167 0.015
0.179 0.003
Despiik
0.602 0,022
0.423 0,009
0,719 0.020
I.695 0.017
0.062 = 0.004
1.182 0.013
0.160 a 0,000
GBM
0.29 0.015
0.207 2 0,007
0,708 0.007
0.681 2 0.008
0.14 = 0.009
1.133 0.008
0.145 2 0,000
RSF
0.501 a 0.013
0.150 0.006
0.729 0004
0.700- 0.005
a.12 0.005
0.1 0.009
D081 0.003
0.257 0,002
0.173 a 0.007
0,749 = 00009
0,721 + 0.010
0.014 = 0.001
0/846 + 0.003
0 2 000002
ALD
0.143 0.826
0.306
: 0,013
0.726 B0088
0.715 1 0,010
aU12 = 0.005
BUC33 0.007
0 2 0.000
CURNN
0.15 00,00
0.334:0.000 0,010
0.720 0008
0.709 0.008
0L0SS = 0.007
0L074 0.012
0.030 0.0002
LogNonno Sptti
LogNorm
0 00017
0.806 0.005
0.712 0008
0.701 0.010
0.07 0.000
0.073 0.00
0.047 0.0002
DeepSSrry
0.00 000011
01817 0.002
0 0008
0.68 0.015
0.02 0.00
0.031 0.0003
DSM/Waibll)
B.23 0.013
0.385
0,644 0000
0.008
0,031 =0 00004
01158 0.01
0.0000
DSMILogNon))
0.358 0,013
0 0000
0.686 0.009
0.09 00000
01159 0.01
Decplikk
0.588 0.011
0.657 0.019
0,003 0.008
0.009
0.09 20000
0.130 0,014
0.002
0.016
0.005
0.017 = 0,005
0.01
D:443
0.317 0.006
0.715
0.002
0238 0,015
0.510
0 0,001
0U942
ALD
0.15 0.001
0.153
0,004
0,746
0.007
0 0.005
0,052 0.012
0.012 0.005
CURNN
0.386 0,044
0.170
0,747
0,705 0.015
0.029 0.005
0.000
0.0144 0.001
LogNocnn MATIA
LegNorm
0194 019110.000
0532 0.016
0,440 0.007
0.696
0.024
0,057 0.016
0.005
DerpSury
0.322 a 0.001
0512 0.007
6,745 0.111
017000 0.005
0.03 0.003
0.050 0.004
0.144 0.004
DSM//elbull
0U601 2 0,032
0.215 2 0,006
0.643 0.011
I.614 0.008
QU85T = 0,007
1.204 0.004
0,042 0.010
DSMILogNeem)
0612 0.033
0.215 0,005
0 0.020
0.656 0.015
Q.022 = 0,005
0,201 0.007
0.057 0.011
Desplikt
0611 0.112
0 = 0,013
0.617 0.075
0.609 0.051
0.035 = 0,015
0.141 + 0.025
0,056 0.011
GHIM
0.50:200000 0.028
0.196 = 0,005
0 0.11
0.661 0.009
0.010 = 0,004
0.150 0.005
01035 00000
RSF
0.399 2 0.015
0.166 0,006
0.727 8.12
0J6R2 0.009
0,050 = 0.006
DJOR3 a 0.011
0.172 0.009
0.230 0.004
8.149 a 0.003
0,738 0.007
0.701 0.005
0.014 a 0.002
BUN4 6.003
0 0.002
ALD
0.009
0.345 00010
0.866 0.16
06.57 1 0.001
0.136 0001
0.266 0.000
CURNN
1000420.044
0.42 0.013
0.644 0018
0.625 0.025
165 0.001
0.2/0 0.010
AECTABSCCC
DaupSurry
LogNorm
8998 0.014
1.187 0.000
0.907
0.516 0.025
0.575 0017
0.648 0008
0.640 0.005
0.549 0.013
0.005
0.275 0.01 5
0.281 0.009
DSM//Webull
00007
0258 0,005
0.611 0.016
0.027
0.155
0.275
NSMiLogNNem)
0.855 00000
0.277 0,005
0.616 0.16
0.054
00012
0.276
Deepiitt
0.003
03664
0.002
0,155 0.011
0.250
0.245 2.043
0,621
0.025
0.012
0.021
1.521
(1260
ALD
5410 2,095
0.199 : 0.008
0,816 0.12
0.81 0,011
105 - 0.090
0.290 0.000
CURNN
0.91 - 0.052
(1151 - 0.013
0.813 0013
n.336 0.014
(1144 - 0.021
0.343 0.001
LegNurm
1.23 0.146
0.634 0.014
0.614 0034
0.585 0.011
(L181 = 0.024
1.266 0.014
WHAS
DeMiWellull
DaspSurr
0,80 0.045
0,005
0.208 : 0,004
0688 = 0.013
0,702 0013
0,779 0011
0.787 0,012
0.634 0.012
0.101 0.029
0.247 00011
01281 0.022
1.205 0.018
DSMeLLogNeem)
1933 2 0,068
0.212 = 0.004
0 0.726 0006
0.763 0.015
0L245 = 0,010
0.2.7% 0.019
Docpliie
00006 a 0.060
999 0.621
0.805 - 0016
0.805 0.017
01132 = 0015
1.215 6.812
GBM
L111 : 00005
0.166 0,004
0.811 = 00009
0 0.012
0L157 = 0.004
0.245 0.0E3
BSF
86689 : 0.056
0.033
: 0,005
0,864 0014
01896 6.016
0.064 =
0.258 0.026
1.639 a 0.579
0.133 0.011
0.828 0017
n816 0.029
0.064 = 0.013
0.244 0.010
ALD
1.166
0.353 0.008
0.806 000
0.608
0.633 0.003
0.907 0.015 5
CORNN
B660 0.023
0.34) 0.009
0.809 0008
0611
0.008
0.174 0.005
0.216
LagNunm
0207:
0.766 0.013
0,588 0000
0.587 0.009
0.216 0010
0.261 0.012
SUPPORT
DeepSurr
0501
0.227
0.598 8000
0.011
0.135
0.100
DSM//eihilli
0573
0.313
0010
0,010
(1212
Neemm))
0.565
0.557
0.533
0008
(1170
0.210
GIM
0.425 0.006
0.258 0.001
0,595 (000
(1599
0L148 0.005
0.200 0.010
RSD
0.675 0.025
0,338 0.005
0.616 0000
01615 6,099
0L139 0.007
0.175 6 6,009
1.546 0.195
0.405 0.004
0,586 0.003
01596
0,240
0.283 0.007
ALD
1.266 0.177
0.273 0.011
0.673 0.13
0.666 0.012
0,201 0.017
11205 0.029
CORNN
a017 0,000
0.277 2 0.:08
0.676 1u114
IL667 : 0.012
0.204 0.003
1.315 a 0.019
LogNumm
1.324 0.074
0623 0.015
0 0.009
I1633 0.009
0.25% = 0.020
11315 a 0.028
CHBGG
DSMIll-gNeem)
DSM[Webull
DerpSur
8.708 4 0,835
1:000 0.090
1086 0.011
5.565 a 0.015
0.305 0.000
3036 0.010
0.638 0022
0.618 0.19
0.617 0.00
01630 0.019
0.610 0.017
01633-4 0.009
0,20 01007
01195 2 0.015
0233 - 0.007
0:795 0.007
0.271 0.027
0.296- 0.009
Decplia
8.755 > 0.085
0.49% = 0.021
0.649 0.15
0.644 0.014
0.165 = 0.012
1 0.010
GIM
0882 0.005
0.287 :
0.569 0.012
8162 0.011
0.201 = 0.013
0.259 0.011
RSF
0926 0.009
0.233 0,010
0.663 0.013
0.63 4 0.012
01192 =
0 0.007
1821 0.133
0.301 = 0,017
0.655 a 0.005
0U646 0,010
0.155 = 0.003
0.274 0.009
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Published as a conference paper at ICLR 2026
Dutasel
Methods
Harreli's C-indes
Uao's
Culibration
Group Calibratios
Individual Culibration
ALD
1.23 a 0,070
0.239 a 0.065
0.559 0.020
n,551 0,016
0.228 0.010
11275 a 0.015
CQRNN
2965500.000
0.246 a
2.01 0.113
0.546 1 0.021
0.229 00000
0.286 0.016
LogNorm
2,04
0.421 0,007
0,554 2.018 0.018
0.543 0.027
0.241 0.012
0.26: 0.026
DespSurv
2 00005
0,538 0:0220001
01527 2 0.013
2
0,250 0,0L0
Wibull
1006-10.017
0.246 x 0.006
0.547 0016
0.553 0.018
0.23 = 0.000
0.24) 0.020
DoMiLogNeem)
0953 2 0.017
0.345 0.006
0.511 0.20
0.521 0.004
0.23 00,00
0.251 0.022
Docphia
1148 0.334
0.398 : 0.003
0.558 0.22
0.551 0.004
0.239 : 0.013
0.264 0.022
GBM
0J78 0.019
0.241
=
0.573 00230
0.549 0.013
0.219 =
0.244 0.020
16566 0.041
0.268
0.100
2
0.225 0,000
0.574 00110
0.548 0,009
0.214 0,010
0,242 0.014
CQRNN
0.11
LogNocm
0.012
BreassMS
0.97
0.076
0.95
0.015
0.305 0.019
GBM
1.566 0.052
0J92 0.000
6,638
RSF
1679 a 0.159
0.87 0.003
0.630 0.53
0.561 0.020
(1273 0.024
0.305 0.037
2.81 0.196
0,084
8,000
0,272
ALD
1,55 0.334
0.106 0.011
0.786 0.131
0.728 0.034
0.175 : 0,050
0,348 0.034
CURNN
0.102
0.193 2
0,788 0.120
0,752 0.021
0
0,372 0.021
LogNorm
1052
0.399 0.012
0,785 0.17
0.727 0.045
0.150 0,000
0.2M1 + 0.027
DeepSury
081220..00
0.455 = 0.024
0. 0,722 0.48
01657 0.044
0115 2 0,000
0.362 1 0.007
DSM//ebbull
1073 0.0094
0.176 0.014
0.768 B.036
0.727 0.005
0.242 00016
0.335 1 0.009
DSMILogNemm)
0.989 00,091
0,585 0.041
0U618 0.057
0.245 00019
0.35) 0.016
Decpčlk
20.155
0.382 0022:0..00
0.772 0.027
0.726 0.006
0.253 = 0.024
0.317 0.003
GBM
0U.39 0.077
0.141 =
0.167 0.100
0.731 0.002
0,205 0.025
0.302 0.022
0.115
0.281 0.118
0.183 0,817
0 0.:38
0.763 6.047
6.118 0,823
0.257 0.007
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C.4
CASE STUDIES
Case Study I: Discretization, Crossing Quantiles, and Distribution Mismatch
Fig. iillsstrates a representative example comparing nonparametric and parametric ALD models. In the nonparametric approach (based the quantile form, i.e., AL(O,, o, q)), only a fixed set of quantile percentages {= are estimated independently. Due to this discretization, the cumulative distribution function (CDF) appears as a piecewise curve, with substantial gaps between neighboring quantiles. As shown in the figure, this leads to poor resolution in the distribution tail and visible quantile crossing, where estimates at higher quantile levels fall below those lower ones.
In contrast, the parametric ALD-based approach (based on the asymmetry form, i.e., AC x)) provides continuous, smooth estimate the conditional distribution. While this results in better global coherence and eliminates quantile crossing, the model can suffer from distribution mismatch, especially the distribution tails. this case, the ALD fit systematically underestimates the upper tail, failing capture the observed data spread. This illustrates the challenge of using single ALD to model highly skewed or heavy-taile distributions.
Norm linear: Individual Calibration Cases
Norm linear: Individual Calibration Cases (ID: 325)
1.0
Ecar
CALD
Ideal CDF
0.8
0.6
0.6
0
0.4
0.4
0.2
0.2
0.0
0.0
15
20
20
Time
Time
Figure 5: Illustration of limitations in nonparametric (left) and parametric (right) ALD approaches. Left: Discretized and crossing quantiles issues from nonparametric modeling. Right: Distribution mismatch issue in the parametric ALD-based model.
Case Study II: Overfitting and Asynchronous Convergence
While Theorem demonstrates that increasing the number of quantile samples improves the Monte Carlo approximation and enhances individual calibration, this benefit must be carefully balanced in practice. Sampling a larger number quantile percentages typically necessitates longer training, but prolonged training lead to overfitting, particularly datasets with limited size high noise (e.g., LogNorT: ba-bd datasets). As shown in Fig. 6 training for 2000 epochs results in degraded calibration performance due such overfitting effects. Specifically, while the training loss L Clssnn continues decrease, both the test loss Clesst ALD+CCa and test negative niglliill (NLL) L'esst begin to increase steadily after a certain point. This divergence indicates that the model starts to fit noise in the training data, thereby impairing its generalization capability. These observations underscore the importance of incorporating early stopping to maintain a proper trade-off between calibration quality and generalization performance. Fig. presents a case from the L LogNo rmm med dataset that illustrates the issue of asynchronous convergence encountered during training with the pre-calibration objectives in Equation 10 and Equation [2] This phenomenon typically arises in the early stages of training, when the model has not yet learned a meaningful approximation of the underlying distribution. At this point, applying the additional loss (i.e., Lea or Lcwr) too early may introduce noisy conflicting gradient signals that interfere with stable optimization. Specifically, the negative log- likelihood (NLL) loss CALD encourages the model to fit the global structure of the distribution, while the calibration loss enforces local alignment at specific quantile levels. When the distributional
ALD+Cal
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Training Loghlarm med
Than Lenk
Iran
Thei Col
NLLL
2.0
0.5
3.0
Epech
Figure 6: Training dynamics on the LogNorm med dataset with 2000 training epochs. Overfitting is evident in the increasing test negative log- log-likelihood (NLL) CALD and test loss curves, despite the stability of CAL loss (Lca), highlighting the risk of prolonged training under joint loss objectives.
LogNorm
Irain Loss
Test Loss
TSST N..
1.5
Teetl Cal
0.5
100
200
(a) Pre-calibration without warm-up calibration.
Training LogNorm,
Trin Loss
Train Col
10sT NLL
1.5
0.0
100
125
(b) Pre-calibration with warm-up calibration.
Figure 7: Comparison of pre-calibration training dynamics the LogNo rm__med dataset, with and without the proposed warm-up calibration strategy.
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parameters still unstable, such localized supervision may act more noise than constructive guidance, ultimately impeding convergence.
To address this, adopt the proposed warm-up calibration strategy, wherein training initially focuses solely on the NLL loss CALD. The additional loss (i.e., Ccat or Lcye is then gradually incorporated after fixed number of epochs, allowing the model first establish stable approximation of the distribution, As shown in Fig. bb) this strategy can stabilize training dynamics and improve calibration consistency. Notably, the final test NLL lower when using warm-up calibration strategy compared to direct pre-calibration, demonstrating improved generalization and more effective distribution fitting.
In contrast, the pasl-calibration ICALD models entirely bypass the issue of asynchronous convergence by applying calibration a post-processing step after the base model has been trained. This decoupled approach yields more stable and consistent calibration effect, free from the gradient conflicts introduced by joint training losses. As shown Table and Table 7 post-calibration ICALD models consistently achieve strong calibration performance across all metrics. These results underscore the practical advantage post-calibration in improving reliability, particularly on challenging datasets like LogNorm, which are more susceptible to instability under joint-loss training schemes.
Case Study I:: Interpretability Case Study via SHAP
We provide an interpretability case study to address the concern that introducing the quantile variable q may hinder factor-level insights. Although ICALD increases modeling flexibility through the continuous mixture mechanism, we can interpret predictions through scalar functionals of the mixed distribution (e.g., e, the mixed-ALD mean survival time), and then apply standard explanation tools such SHAP (Lundberg & Lee 2017) to these scalar targets.
SHAP Mes--Besedaae Attribution for Survival time Estimation IID: 308)
Rx)
1.42
20:00
24.00
0.805 x4
+4.45
-1.201 x2
+22 $2.89
-0.707 x5
1.991
4,009
-0.853 X3
-05112
-0.447 x1
-1031
10
12
144
16
18 -1531
SERX0)
Figure 8: SHAP-based interpretability for ICALD. Left: Individual-level SHAP waterfall explana- tion for representative test instance (ID: 308) on the predicted survival-time summary (mixed-ALD mean). Right: Cohort-style visualization (samples ordered by similarity) illustrating how feature contributions vary across individuals, enabling subgroup and dynamic analyses.
Individual-level explanation. Figure 8 (left) shows SHAP waterfall plot for representative test subject (ID: 308), where the explained target is the predicted survival time summary (here, the mixed-ALD mean). Starting from the population baseline E Rs 9.689, the model prediction is fx) Rs 9.96. Feature 2; exerts the largest negative contribution (-661), indicating that this covariate strongly shifts the prediction towards shorter survival. This effect is partially offiset by several positive contributors, notably I4 (+4.44), x2 (+2.89), and æt (+1.91), with smaller adjustments from T3 (-1.46) and 21 (+1.10). Overall, the explanation reveals a clear ppush-pull" structure: despite dominant risk-increasing factor (Ig), compensating protective factors (I T2, 2z)) recover the final prediction close to (and slightly above) the baseline. This illustrates that the model remains interpretable at the factor level even with the additional quantile input.
Cohort-level and dynamic insights. Beyond a single prediction, we can aggregate SHAP explana- tions across cohort to obtain population-leve summaries. For example, we compute the mean absolute SHAP value per feature to quantify global importance, and we can stratify the test set by predicted risk (e.g., low/mediu//high risk groups defined by the predicted median survival time)
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to examine how the ranking of covariate contributions changes across subpopulations. Moreover, because the mixed-ALD CDF admits a closed-form expression (Appendix (A.3), we can interpret time-dependen targets such as ((t at multiple horizons; repeating SHAP ((t(x) for grid of t yields dynamic visualizations that reveal how covariate effects evolve from short-term to long-term risk. Figure (right) provides example of such a cohort-style visualization for samples ordered by similarity, illustrating that the dominant contributors (e.g.. T2, T3, 25 in this example) can vary in magnitude and direction between nearby individuals, allowing nuanced subgroup analyzes.
Case Study IV: Calibration Performance with LPee AL.D+Cal vs.
LPee ALD+CCqr
Although both L DeCaa and Pre Cl. theoretically grounded in improving individual calibration (see Definition 1 Definition 2 and Theorem T their empirical performance differs markedly, likely as a result how they incorporate censored data and define their loss objectives (see Table 9
Table 9: Pairwise comparison of calibration performance between CAL.DCal and Che AL.aCgr across 21 datasets. The two sub-columns reflect settings with and without censored data. Each group reports the number of datasets where Lhe AL.--Cal performs better, worse, or the same. The final rows show total counts and proportions across 56 pairwise comparisons.
Metric
Train with censored data
Train without censored data
Average Calibration
13
0
8
1
19
Group Calibration
14
0
7
6
2
13
Individual Calibration
0
5
12
Total Proportion (%)
36
0
20
8
44
64.3
0.0
35.7
14.3
7.1
78.6
In the left half of Table g we compare the two methods under the standard setting where censored data is retained during training, and the quantile regression loss modified using the Portnoy estimator (Portnoy (003), as defined in Equation equation 10 Under this setting, CRI.D.Ga exhibits clear advantage: it outperforms Pre ALL+CCu in 64.3% of all comparisons and never underperforms.
To assess whether this performance gap stems from the presence of censoring or the estimator itself, repeat the comparison using the same datasets but exclude all censored samples during training (right half of Table 9 In this censored-free scenario, the two methods perform comparably in 78.6% of the cases, with only marginal advantages observed either side. This indicates that, in the absence of censoring, CPr ALDDCal and CPh
ALDxCCr behave similarly in terms of calibration, and suggests that the Portnoy-adjusted loss in LPr CALD+Cq may introduce bias instability when censoring is present.
In summary, these results highlight the sensitivity of CPre ALD+Cyy to the censoring mechanism. Its reliance on the Portnoy estimator may limit its effectiveness in capturing calibration signals under censored conditions. In contrast, ALDyCal appears to offer a more robust and stable calibration objective when applied to censored survival data. Notably, similar trend is observed as well the post-calibration setting.
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Case Study V: Sensitivity to the loss weight À and warm-up length L
Setup. We study the sensitivity ICALD (CALDaca) to the calibration loss weight À E {0.1, 0.3, 0.5, 0.7, 0.9} and the warm-up length L E {50, 100, 200, 400} (with the default L=200). In Tables IO and [T] each cell reports the percentage of datasets for which the setting on the left outperforms its comparator. Percentages not statistically significant by Student's f-test.
Table 10: Comparison across different À values (default A=0.1). Each column shows the percentage of datasets where A=0.1 outperforms the comparator À {0.3, 0.5, 0.7, 0.9} for each metric (higher is better for the comparison rate).
Metric MAE
0.1 vs 0.3
0.1 vs 0.5
0.1 vs 0.7
0.1 vs 0.9
52.4%
61.9%
66.7%
57. %
IBS Harrell's C-Index
61.9%
81.0%
76.2%
66.7%
76.2%
76.2%
76.2%
71.4%
Uno's C-Index
71.4%
81.0%
76.2%
66.7%
Average Calibration
57.1%
57.1%
57.1%
47.6%
Group Calibration
42.9%
33.3%
47.6%
47.6%
Individual Calibration
50.0%
57.1%
57.1%
50.0%
Total
59.3%
64.3%
65.7%
58.6%
Table 11: Comparison across different warm-up lengths (default L=200), Each column shows the percentage of datasets where L-200 outperforms the comparator L {50, 100, 400} for each metric (higher is better for the comparison rate).
Metric
200 vs 50
200 vs 100
200 vs 400
MAE
57.1% 1%
57.1%
57.1%
IBS Harrell's C-Index
52.4%
52.4%
52.4%
57.1% %
57.1%
61.9%
Uno's C-Index
52.4%
52.4%
57.1%
Average Calibration
66.7%
57.1%
66.7%
Group Calibration
66.7%
47.6%
66.7%
Individual Calibration
71.4%
57.1%
50.0%
Total
60.5%
54.4%
58.8%
From Tables 10 and we can draw the following key observations:
1. Overall robustness. ICALD is generally robust to variations in both and L. While differences are not statistically significant, the average trends (over five runs per dataset) show mild variations.
2. Effect of X. As À increases, salibration-orientel metrics tend to improve, suggesting stronger distributional calibration the cost of slight trade-offs in MAE, IBS, and C-indices.
3. Effect of warm-up length I. Changing I. has limited impact on overall performance. Metrics remain stable across I. (50, 100, 200, 400), likely because the calibration loss continues contributing during the post-calibration phase.
Case Study VI: Impact of the g-dimensionality in pre-calibration
Setup. To enhance the expressiveness of the calibration anchor, explore different choices of the g-dimensionality in the pre-calibration model Che ALLD-Ca Specifically, we consider {1, 11, 2, 4, 8 and evaluate whether increasing the dimension of g improves calibration performance. In Table 12 each cell reports the number datasets where d = 4 is significantly better worse the same as its comparator, with statistical significance assessed by Student's t-test (p 0.05, after FDR correction).
Table 12 shows that increasing the dimension of (e.g., d 2, 4) generally leads to improved calibration performance. More specifically.
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Table 12: Comparison under different g dimensions in pre-calibration. Each cell shows the number of datasets where d = 4 is significantly better / worse the same as the comparison.
Metric
d = 4 vs d =
d= 4 vs d =
d= 4 vs =
Average Calibration
1/0/20 20
0/0/21
2/0/10 19
Group Calibration
1/0/ 20
0/0/21
2/0/
Individual Calibration
0/0/ 14
0/0/14 14
2/0/ 12
Total Proportion (%)
5/0/54
0/0/56
6/0.50
3.6/ 96.4
0.0 0.0//.0// 100.0
10.7 / 0.0 89.3
For = vs. d The most significant improvements are observed on the SUPPORT dataset (where the feature dimension is 14), suggesting that higher g-dimensionality is particularly beneficial when the imput features are -igh-dimensional.
2. For d = 4 vs. d = 8. Significant improvements are found for the Exponential ial and LogNorm datasets, both with of dimension This indicates that increasing g-dimensionality beyond certain point does not necessarily yield additional benefits and may even degrade performance in lower-dimensional settings.
3. For = 4 vs. d = 2. Although there are no statistically significant differences, the average metric (mean over five runs) slightly better for across datasets.
In summary, our results suggest that increasing the g-dimensionality to moderate level (e.g., 2, 4) generally improves calibration, especially for datasets with higher feature dimensionality.
Case Study VII: Granular Analysis of Calibration and Accuracy
We performed an additional analysis on the synthetic Gaussian aarisiinnnniiii unw.niii. uniform dataset to assess how calibration and accuracy behave across different time horizons and quantile levels. We stratified test instances by the CDF of the true event time (() into four equal-probability bins, [0, 0.25), [0.25, 0.5), [0.5, 0.75), and 70.75, 1]. which correspond to early, mid-early, mid-late and late event
times. The results are shown Figures 9 and I0 where error bars indicate one standard deviation
Individual Calibration across bins true event-ime F:0
F-lti
0.25
(c.0.25)
[C (0.25,0.5)
(6.5.0.75)
Caterston1
10.73.11
f.o
%...
0.00
Loghoom
DeepSury
UsM]visbbuti
DSMLLggNomm)
Figure 9: Individual Calibration across bins of the true event-time CDF Fr(().
5 independent runs. From these results, we observe that: (0 ICALD consistently achieves the lowest individual calibration error across all Fr (t) bins while maintaining competitive or superior accuracy (IBS) in each bin; (6i) ICALD exhibits very small standard deviations across runs, indicating that its performance is highly robust; and (iii) ICALD performs particularly well in the late-event regime (t) e [0.75, 1), providing both better calibration and better accuracy than competing methods.
Overall, this stratified analysis confirms that ICALD maintains strong calibration and accuracy over extended time horizons and across different quantile levels.
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F-iti tins.
(3:25.0.5)
(3.5,0775)
10:75.11
0.4
0.3
KALO
Wgloorm
DeepSurv
LSM(Welbul)
OSMiLogNiomn
G3M
nS1
Figure 10: IBS across bins of the true event-time CDF FF(()
Case Study VIII: Scalability and Cross-Domale Evaluation
common concern for calibration-focused survival models is whether their reliability gains persist in challenging regimes that depart from the standard clinical benchmarks- in particular, (i) high- dimensional settings with limited sample sizes (large d, small n), and (ti) non-medical domains where censoring patterns and feature distributions can differ substantially from healthcare data. To partially address these external validity and scalability concerns, we conduct additional experiments on two widely used high- sigh-dimensional gene-expression datasets and four datasets from finance and engineering domains.
Higghddimensional cene-expression datasets. We consider: (i) DBCD (Dutch Breast Cancer Dataset, Rosenwald et al. 2002). which contains n = 295 patients with 4919 gene-expression features and censoring rate of 73.2%; and (ii) DLBCL (Diffuse Large B-cell Lymphoma, Van Houwelingen 240 patients with d 7399 gene-expression features and
et al. (006).. which contains censoring rate of 42.5%. These datasets are representative of a difficult regime for survival modeling: the feature dimension is orders of magnitude larger than the sample size, while censoring can be substantial, making both accurate distribution fitting and reliable uncertainty quantification nontrivial. As shown in Table 13 AAu achieves competitive or superior overall performance both datasets, with particularly strong results on calibration (average and group calibration) while maintaining competitive predictive accuracy (MAE/IBS) and concordance. This suggests that the post-calibrated ICALD framework remains effective even when the backbone must operate under severe d > n constraints.
Beyond the medical domain: finance and engineering. further evaluate generalizability, we additionally test on four datasets from the PySurvival package (finance and engincering applications). Table 14 reports the full results. Across these datasets, CPodt A continues to deliver competitive or superior performance, and the advantage is again most consistent calibration metrics. Notably, this behavior holds even when strong ensemble- based or neural baselines achieve high concordance, indi- cating that ICALD can improve the reliability of predicted survival distributions without sacrificing overall ranking performance.
Overall, these additional experiments support the practical robustness of ICALD under (i) high- dimensional, small-sample regimes and (fi) cross-domain shifts, and reinforce our central claim that the proposed calibration mechanism yields reliable survival distribution estimates in settings beyond the standard clinical benchmarks.
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Table 13: Full results table for the high-dimensional datasets, methods, and metrics. The values represent the mean + standard error for the test set over runs.
Dataset
Methods
MAE
IBS
Harrell's C-index
Uno's C-index
Average Calibration
Group Calibration
ALD
0.782 a 2..165 162
0.188 a 0.035
0.648 =: 0.114
0.663 1 0.054
0.259 2 0.021
0.689 : 0,236
OQRNN
1,539 0.281
0.480 + 0.044
0.688 + 0.062
0.628 0.051
0.324 + 0,074
0.698 0.128
LogNorm
1.129 0.137
1
0.668 0.086
0.608
DBCD
1SMM/Wehhll
DeepSury
3.263 + 0.349
1.128 + 0.154
0.143 + 0.020
0.503 # 0.057
0.676 + 0.061
0.572 = 0.051
0.541 0.070
0.662 0.074
0.351 0,021
0.426 0.045
01299- 0.002
0,556 2 0.089
DSMiLogNom)
3.341 0.195
0.150 0.021
0.688 + 0.053
0.647 0.084
0.370 0.014
0 0.002
DeepHii
1.129 0.137
0.343 + 0.063
0.681 = 0.038
0.668 0.086
0.193 0.046
0.608 0.135
GBM
0.146 1 0.018
0.583 a 0.016
0.591 1 0.060
0,343 a 0.031
0,513 4 0.035
RSF
2.225 0.544
0.133 1 0,019
0.660 : 0.08]
0.635 0.073
0.321 0.044
0.503 0.022
0.774 0.142
0.129 + 0.033
0.695 = 0.062
0.707 0.063
0.185 0.060
0.474 0.128
ALD
1,161 0.900
0.466 a 0.057
0.589 : 0.034
0.533 0.075
0,215 & 0.039
0.419 a 0.040
CORNN
0.050
0 1 0.077
0.607 : 0.010
0,515 0.127
0.253 0.083
0.444 0,036
LogNorm
2,894 0.983
0.812 0.079
0.554 =
0,239
0,435
DLBCI
DeepSurv
1,082 + D:489
0.743 + 0.089
0.546 + 0.002
0.579 + 0.084
0,254 0.071
0,442 + 0.139
DSM WeihWal))ll)
1,0.38 + 0.078
0.367 + +0033
0.601 = 0.089
0.532 + 0.138
0.259 + 0.024
0.373 + 0.018
DSMiLogNom))
0.069
0.033
: 0.085
0.526 146
0.248 0.029
Despliit
1.129 1 0.137
0.343 1 0.063
0.681 :: 0.028
0.668 0.086
0.193 0.016
0,508 0.135
GBM
0.587 0. 142
0.336 + 0.0.9
0.597 = 0.034
0.584 0.031
0.7 0.030
0.345 0.038
RSF
0,925 0.310
0.345 0.047
0.607 = 0.064
0.625 0.110
0.188 0.034
0.325 0.041
Clse
1.059 1 0.057
0,246 1 0.037
0.605 2 0.038
0,585 1 0.119
0.138 0.031
0.284 : 0.040
Table 14: Full results table for the finance and engineering datasets, methods, and metrics. The values represent the mean I standard error for the test set over runs.
Dataset
Melbods
MAE
IBS
Harrell's C-index
Uno's C-index
Average Calibration
Group Calibration
ALD
0.404 0.025
0.063 + 0.009
0.575 # (1.009
0 0.009
0.064 = 0.015
0.135 = 0.051
CQRNN
0,397 + 0.021
0,505 + 0.019
0.73 = 0.D13
0.902 0.017
D.77 + 0,014
0.157 D.046
LogNorm
0.400
0,793 + 0.047
0828 0.011
0.829 0.052
0.147 0.012
0.211 0.024
DeepSurv
0.397 : 0.025
0.713 0.033
0.703 = 0.038
0.652 0.06
0.033 0.009
0,099 0.035
Churn
DSi/Wwbbull)
1.257 0.048
0,237 1 0.022
0.722 2 0.015
0.702 a: 0.043
0,274 2 0,009
0,304 1 0.012
DSMiLogNom)
1,211 1 0.050
0.237 1 0.021
0,593 1 0.010
0.610 0.036
0,250 0.008
0.315 0.007
Deepliit
0.429 0:019
0.941
0.012
0.114
GHM
0.978 + 0.075
0.180 + 0.016
0818 + 0.020
0.816 0.027
0. 183 0.008
0.221 + 0.011
RSF
0,513 0.057
0.092 + 0.010
0.853 = 0.016
0.833 0.036
0.062 + 0.011
0.164 # 0.044
CPee
a 0.181
a: 0.018
0.023
0.048 0.016
0.119
ALDD
0.782 + 0.087
D.W2 + 0.009
0.722 + (1.20
0.714 + 0.035
D. 159 + 0.036
O. 0.333 + 0.017
CQRNN
1,043 1 0.065
0.255 1 0,830
0.691 : 0.013
0.693 0.045
0,208 0.031
0/417 0.022
LogNorm
0.065
0.022
0.610
0.018
0.080
CrediRRikk
DeepSurv
0.067
0.392 1: 0.020
0.660 :: 0.053
0.618 1 0.039
0,158 0.028
0.337 1 0.036
DSM(
1,273 0.005
0.107 + 0.012
0.564 = 0.011
0.512 +.031
0.219 # 0.019
0.374 # 0.056
DSMLLoNNom)
1.315 0.086
0.106 : 0.012
0.563 = 0.009
0.604 0.038
0.225 0.018
0.376 0.056
DezpHit
1,081 + 0.1
0.310 0.067
0.698 +
0.659 0.064
0.022
0.278 B.000
GBM
0,10] 1 0.013
0.680 : 0.031
0.649 1 0.034
D. 20201 0.021
0.336 0.015
RSF
1,363 0.13
0,000 + 0,013
0.706 = 0.012
0.709 0.014
172 0.034
0.356 DAUS1
1.339 0.547
0.064 0.011
0.638 0.021
0.137 0.020
0.234
ALD
0.181
0.018 : 0.002
0.933 1 0.00S
0.923 1 0.006
0.111 0.026
0.196 0.014
CQRNN
0,308 : 0.028
0.384 1 0.013
0,946 : 0.007
0.936 0.006
0.0F1 0.033
0.181 D.D48
Employee
LogNorm
2.466 0.400
1 0.009
0.155 1 0.010
0.012
0,289 0.013
0.339 0.014
DeepSurv
0.278 + 0.024
0.500 + 0.008
0.795 = 0..029
0.760 0.026
0.131 + 0.007
0.175 0.017
JSMeWeibull)
0.707 + 0.066
D.48 : 0.084
0.86 = 0.D13
0.815 0.009
0.210 0.016
0.291 DADUS
DSMU ogNom)
0.729 0.070
0 0.004
0.010
0.789 0.008
0.235 0.016
0.323 0.011
DespHtit
0.288 1 0.039
0.502 : 0.906
0.736 : 0.115
0.695 1 0.121
0.126 0.018
O. 0,304: : 0.046
GBM
1.041 a 0.015
0.058 1: 0.004
0.885 : 0.006
0.824 2: 0.007
0,210 0.002
0,316 2 0.001
RSF
0,250 0.032
0.019 1 0.001
0,941 1 0.007
0.943 0.005
0.368 0.003
0.419 0.010
0.175 = 0.015
0 : 0.003
0.954 = 0.007
0.927 0.004
0.056 0.055
0.167 0.071
ALD
0.079 a 0.016
0 + 0.002
0.957 = 0.012
0.946 0.021
0.204 0.021
0.246 # 0.019
CQRNN
0,033 0.004
0.065 1 0.021
0.978 a 0.003
0.976 0.004
0.053 0.012
0.133 0.026
Maintenince
LogNorm
1.129 0.137
+ 0.963
0.086
0
DeepSory
00019:20.00
0.667 + 0.016
0.975 = 0.019
0.976 0.019
D. 168 + 0.017
0.220 + B.011
DSM Webbull
0,315 1 0.015
0.015 1 0.063
0.675 a 0.026
0.656 0.031
0.084 0.010
0.301 0.056
DSMiLogNom))
0.026
0.555 0.029
0.087 0.011
0.313
Despliit
0.023 1 0.001
0.689 1 0.017
0.994 :: 0.003
0,993 0.003
0.226 0.015
0.463 2 0.017
GBM
0.270
0.005 + 0.003
0.334 = 0.024
0.838 + 0.023
0.064 0.010
0.313 + 0.026
RSF
0.013 0.001
0.012 0.000
0.992 0.003
0.992 0.003
0.365 0.011
0.435 0.007
0.041 4 0.020
0.013 1 0.004
0.969 à 0.010
0.960 1 0.018
0.075 0.009
0.173 4 0.039
45


Reviews:
[{"review_id": "EhMjHp6wEJ", "reviewer": "Reviewer_bEgR", "summary": "This paper proposes a method to learn survival distributions based on calibration of the asymmetric Laplace distribution. The ICALD model includes a ALD module as a backbone and an adapter module that refines the predictions further. The authors show that their ICALD model is probably approximately individually calibrated. They also propose pre and post calibration methods and via concentration show that as the number of quantile samples increases, the model's individual calibration performance improves. \n\nNote: This paper is far beyond my reviewing expertise and I have brought this up to the AC. I will defer to other reviewers on acceptance decisions.", "questions": "- Table 3 reports the proposed approach against a handful of baselines based on whether performance was better, worse or the same. This looks to me to be a little strange, why did the authors not simply report the average of whatever metric was chosen, as opposed to a qualitative approach like this? \n- The same goes for the following results, why were \"wins\" reported instead?", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper is well written and is somewhat accessible to an audience outside the area. The experiments seem to be thorough enough with sufficient abalations.", "weaknesses": "Disclaimer: this is outside my expertise and are issues that I am bringing up from a layperson's perspective. In terms of assessment, I will defer to other reviewers.\n\n- This paper appears to target a fairly niche audience in the ML community. The neural network architecture (to me) seems to be fairly simple, utilizing the standard reparameterization trick. This is not surprising since ALDs involve only a handful of parameters; part of the authors' contributions is to include a mixture of ALD (line 238).\n- Why was the distribution over $q$ the uniform distribution (line 242)? Was this prior chose for convenience or some other reason?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9bbF4fYcAq", "reviewer": "Reviewer_Ry6a", "summary": "his paper tackles the critical, and often overlooked, problem of fine-grained calibration in survival analysis models. The authors propose a novel framework, the Individually Calibrated Asymmetric Laplace Distribution (ICALD), which skillfully unifies parametric and nonparametric approaches built upon the Asymmetric Laplace Distribution (ALD).\n\nThe core contribution is a new training objective that combines the standard ALD negative log-likelihood (for global distribution shape) with a specific calibration loss (either a quantile regression loss, L_Cqr, or a direct CDF-based loss, L_Cal). This joint loss encourages the model to be \"Probably Approximately Individually Calibrated\" (PAIC), a property the authors provide theoretical guarantees for (Theorem 1).\n\nThe ICALD framework is notably flexible, supporting both:\n\nPre-calibration: Jointly training the model with the combined loss from scratch.\n\nPost-calibration: Using a lightweight adapter module to calibrate the outputs of a pre-trained base model, which is a more efficient approach.\n\nThe authors conduct an extensive empirical evaluation on 14 synthetic and 7 real-world datasets, comparing ICALD against 12 diverse baselines (including other parametric, non-parametric, and calibration methods). The results show that their method, particularly the post-calibration variant with the L_Cal loss, achieves state-of-the-art performance across metrics for predictive accuracy, concordance, and, most significantly, all levels of calibration (average, group, and individual).", "questions": "could it be used as a general-purpose calibrator for other survival models (e.g., DeepSurv, RSF, or DeepHit)?\nClarity on L_Cal for Censored Data: s applied to all data points, including censored ones. For a censored observation at time y, the true event time is unknown (T > y). How does the lossrovide a reliable calibration signal in this case, especially for quantiles q that may be far from the true (but unknown) event time? A more intuitive explanation of why this is robust to censoring would be helpful.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Addresses an Important Problem: The paper focuses on individual-level calibration, which is far more stringent than average calibration and is essential for high-stakes, personalized applications (e.g., clinical decision support), where the reliability of a prediction for a single individual is paramount.\n\nStrong Theoretical Foundation: This is not just an empirical \"trick.\" The authors provide formal proofs (Theorem 1) demonstrating that their joint loss objectives result in models that are Probably Approximately Individually Calibrated (PAIC). This theoretical grounding is a significant strength.\n\nComprehensive and Rigorous Evaluation: The experimental validation is exceptionally thorough. The use of 21 datasets, 12 baselines, and a full suite of metrics covering accuracy (MAE, IBS), concordance (C-Index), and all three levels of calibration (ECE, Wasserstein distance) makes the empirical claims highly credible.\n\nFlexible and Practical Framework: The authors provide two ways to use their method. The post-calibration approach (Section 3.4) is particularly valuable as it allows for the calibration of existing models with a simple, lightweight adapter, making it practical to implement and deploy.\n\nClear Empirical Gains: The results presented in Tables 1, 2, and 3 show a clear and consistent advantage for ICALD over all baselines, especially in its primary goal: improving calibration without sacrificing (and often while improving) accuracy and concordance.", "weaknesses": "The paper is very strong, and the authors are commendable for including a detailed \"Limitations\" section and several case studies in the appendix that proactively address potential issues. The primary weaknesses are:\n\nTraining Instability in Pre-Calibration: The authors note that the pre-calibration model can suffer from \"asynchronous convergence\" between the NLL and calibration losses, requiring a \"warm-up\" strategy (Appendix C.4). They admit this heuristic does not fully solve the mismatch, suggesting the joint optimization can be brittle.\n\nOverfitting Potential: The method's effectiveness relies on sampling many quantile percentages, which can necessitate prolonged training. The authors report (Section 5, Appendix C.4) that this can lead to overfitting, especially on heavily skewed datasets. This requires careful use of early stopping.\n\nSensitivity of the L_Cqr Loss: One of the two proposed calibration losses, L_ALD+Cqr (based on pinball loss), is shown to be highly sensitive to censored data (Appendix C.4, Table 9). While the authors correctly identify the L_ALD+Cal variant as the superior and more robust choice, this does highlight a fragility in the quantile-regression-based formulation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RcB5suKcGg", "reviewer": "Reviewer_z3uB", "summary": "The paper \"Learning Survival Distributions with Individually Calibrated Asymmetric Laplace Distribution\" introduces a novel framework called ICALD, which integrates parametric and nonparametric approaches using the Asymmetric Laplace Distribution (ALD) to enhance individual calibration in survival analysis. The authors provide a comprehensive methodology, theoretical foundation, and empirical validation of their approach.", "questions": "Some questions for the authors\n- Have you considered the challenges with computational complexity? The need to sample 2000 quantile percentages during training may increase computational complexity, especially for high-dimensional models\n- While detailed implementation is provided, the absence of open-source code limits reproducibility and integration into existing workflows. Are there plans to open-source the experiments?\n- The model's complexity may hinder interpretability compared to simpler methods, a concern for applications requiring factor-level insights. Have you explored intepretability criteria either through static or dynamic visual analysis\n-  Have you explored the model's ability to maintain calibration and accuracy over extended periods? InIsights into how errors propagate through the model and affect predictions at different quantile levels would enhance understanding, particularly for individual calibration.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The integration of parametric and nonparametric methods through ICALD is a significant advancement in survival analysis. Further, the emphasis on individual-level calibration addresses an important gap in existing survival models, particularly relevant for high-stakes applications like patient prognosis.\n- The paper establishes the model's Probably Approximately Individually Calibrated (PAIC) and Monotonically PAIC (MPAIC) properties, adding robustness to their claims.The claims were further backed by extensive experiments across 21 datasets demonstrate competitive performance against 12 baselines, with strong results in predictive accuracy, concordance, and calibration.\n- The authors provide strategies to address overfitting and asynchronous convergence during pre-calibration, enhancing the model's practicality.", "weaknesses": "Some of the challenged with the paper are as below:\n- The benefits of ICALD appear contingent on dataset characteristics, particularly for highly skewed distributions like LogNorm. This raises concerns about generalizability across diverse data types.\n- The experimental results are based on a specific selection of datasets, and validation against an independent, diverse benchmark could improve confidence in the model's broader applicability.\n- The paper does not extensively discuss scalability, which is crucial for large datasets or high-dimensional features in real-world applications.\n- While the authors propose solutions for capturing extreme quantiles and distribution mismatch in tails, it's unclear if these methods are sufficient for extremely heavy-tailed distributions.  The experiments primarily focus on medical datasets, limiting broader applicability to other domains unless further validation is conducted.\n- The methods described assume specific types of censoring, which may not be robust to all missing data scenarios. Further, while the paper discusses outlier robustness, it doesn't provide extensive testing, which could limit performance in real-world scenarios with outliers.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UVxBPR98Sq", "reviewer": "Reviewer_zNTr", "summary": "The paper proposes a survival analysis framework that augments the most recent asymmetric Laplace distribution-based model (Sheng & Henao, 2025) with calibration-oriented loss functions. The proposed model is demonstrated to have a profitable property of monotonic PAIC (probably approximately individually calibrated). Experiments on 14 synthetic and 7 real datasets show that it achieves substantial improvements in various accuracy metrics against conventional models.", "questions": "- I could not find a clear description of the better/worse/same evaluation procedure. Did you split each dataset into multiple folds for training and testing and then evaluate performance in a cross-validation manner?\n- How was the regularization parameter in X-CAL (\\gamma in the original paper) selected or tuned? Please specify the procedure.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "- The idea of the paper is clearly presented, and the paper is easy to read.\n- The validity of the proposed model was evaluated on a lot of (14 synthetic and 5 real-world) data sets.", "weaknesses": "- This paper largely follows the most related work of the asymmetric Laplace distribution (ALD) survival model (Sheng & Henao, 2025) in data sets, baselines, and protocol, and shows that tuning a calibration-oriented term/module in the objective further improves predictive performance. This is a reasonable but unsurprising outcome given ALD’s already strong empirical performance. The central idea of adding a calibration-encouraging term/module to the objective function has been established in prior works (e.g., X-CAL (Goldstein et al., 2020)). Thus, the contribution of this paper feels incremental from a technical viewpoint.\n- Although Section 3.1 highlights that the pinball loss can be interpreted as the ALD’s negative log-likelihood in quantile form, it is unclear to me what practical advantage this brings here (e.g., training stability, improved optimization landscapes). If the authors insist that the pinball loss can be evaluated analytically, then this is not a non-trivial idea: the same holds for simple parametric distributions beyond ALD. I may be missing/misunderstanding some important aspect here, so if this property plays an essential role in substantiating this paper’s contribution, please explain it clearly.\n- The ablation study in Table 1 is interesting, but the comparison of L_{ALD+Cal} vs. L_{X-CAL} does not seem to be appropriate. For example, we can consider ALD+X-CAL within the contribution of (Goldstein et al., 2020). \n\nMinor comments:\n- The Related Work section, explicitly discussing how this paper relates to ALD (Sheng & Henao, 2025) and X-CAL (Goldstein et al., 2020), should appear in the main text.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)