Learning Survival Distributions with the Asymmetric Laplace Distribution
Authors: Deming Sheng
Ricardo Henao
Venue: ICML 2025 poster (2025)
Keywords: Deep Learning; Survival Analysis; Asymmetric Laplace Distribution; Censored Data
Primary area: probabilistic_methods

Abstract: Probabilistic survival analysis models seek to estimate the distribution of the future occurrence (time) of an event given a set of covariates.
In recent years, these models have preferred nonparametric specifications that avoid directly estimating survival distributions via discretization.
Specifically, they estimate the probability of an individual event at fixed times or the time of an event at fixed probabilities (quantiles), using supervised learning.
Borrowing ideas from the quantile regression literature, we propose a parametric survival analysis method based on the Asymmetric Laplace Distribution (ALD).
This distribution allows for closed-form calculation of popular event summaries such as mean, median, mode, variation, and quantiles.
The model is optimized by maximum likelihood to learn, at the individual level, the parameters (location, scale, and asymmetry) of the ALD distribution.
Extensive results on synthetic and real-world data demonstrate that the proposed method outperforms parametric and nonparametric approaches in terms of accuracy, discrimination and calibration.

Learning Survival Distributions with the Asymmetric Laplace Distribution
Deming Sheng'
Ricardo Henao
Abstract
et al., 2012), and, notably, healthcare (Zhang et al., 2017; Voronov al., 2018; Lánczky & Gyōrffy, 2021; Emmerson & Brown, 2021).
Probabilistic survival analysis models seek to es- timate the distribution of the future occurrence (time) of an event given a set of covariates. re- cent years, these models have preferred nonpara- metric specifications that avoid directly estimating survival distributions via discretization. Specifi- cally, they estimate the probability of an individ- ual event at fixed times or the time of an event at fixed probabilities (quantiles), using supervised learning. Borrowing ideas from the quantile re- gression literature, we propose parametric sur- vival analysis method based on the Asymmetric Laplace Distribution (ALD). This distribution al- lows for closed-form calculation of popular event summaries such as mean, median, mode, varia- tion, and quantiles. The model is optimized by maximum likelihood to learn, at the individual level, the parameters (location, scale, and asym- metry) of the ALD distribution. Extensive results synthetic and real-world data demonstrate that the proposed method outperforms parametric and nonparametric approaches in terms of accuracy, discrimination and calibration.
Survival models can be broadly categorized into parametric, semiparametrie, and nonparametric methods, cach offering unique strengths depending on the characteristics of the data and the underlying assumptions. Parametric survival models assume that survival times follow specific statistical distri- bution, enabling explicit mathematical modeling of the sur- vival function. Common examples include the exponential distribution for constant hazards rates (Feigl & Zelen, 1965), the Weibull distribution for flexible hazards rate modeling (Scholz & Works, 1996), and the log-normal distribution for positively skewed survival times (Royston, 2001). Semi- parametric methods, such as the Cox proportional hazards model (Cox, 1972), assume proportional hazards structure without specifying baseline hazard distribution, which offers robustness and interpretability. Nonparametric meth- ods, including the Kaplan-Meies estimator (Kaplan & Meier, 1958) and the Welson-Aaler estimator (Aalen, 1978), rely solely on observed data, avoiding distributional assumptions while directly estimating survival and hazards (risk) func- tions. More flexible nonparametric models such Gradient Boosting Machines (GBM) (Dembek et al., 2014) and Ran- dom Survival Forests (RSF) (Ishwaran et al., 2008) have also been proposed, which learn survival quantities directly from data using ensemble learning techniques.
1. Introduction
More recently, neural networks have significantly advanced survival models across parametric, semiparametric, and nonparametric settings. In parametric methods, LogNorm MLE (Hoseini et al., 2017) improves estimation by fitting single log- log-normal distribution using maximum likelihood, while Deep Survival Machines (DSM) (Nagpal et al., 2021) extend this by modeling survival times as mixtures of mul- tiple parametric distributions (e.g., log- lor-normal or Weibull), allowing greater flexibility in capturing heterogeneous risk profiles. Semiparametric approaches, exemplified by Deep- Surv (Katzman et al., 2018), integrate neural networks to capture nonlinear relationships while preserving the struc- ture of models such as the Cox proportional hazards model. Nonparametric approaches, such as DeepHit (Lee et al., 2018) 8) and CQRNN (Pearce et al., 2022), leverage deep learning to directly estimate survival functions without re- lying traditional assumptions. These advances allow
Survival models (Nagpal et al., 2021), also known as time-to- event models, are statistical frameworks designed to predict the time until specific event of interest occurs, given set of covariates. These models are particularly valuable in situations where the timing of the event is crucial and often subject to censoring, which means that in some cases the event has not yet occurred or remains unobserved by the end of the data collection period. The ficxibility and adaptability of survival models have led to their widespread application in various fields, including engineering (Lai & Xie, 2006), finance (Gepp& & Kumar, 2008), marketing (Jung
1 Duke University. Correspondence to: Ricardo Henao <ri-
cardo.henao @ edk>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Learning Survival Distributions with the Asymmetric Laplace Distribution
survival models to handle complex, high-dimensional data with greater precision and flexibility.
Survival and hazards functions are two fundamental concepts in survival analy- sis. The survival function is denoted as S(t) = P(T > t), which represents the probability that an individual has sur- vived beyond time t. It can also be expressed in terms of the cumulative distribution function (CDF), F(t), which gives the probability that the event has occurred by the time I, as S(t) = F(t). The hazards function, denoted as X(t). describes the instantaneous risk that the event occurs at specific time 6, given that the individual has survived up to that point. Formally, it is defined as:
Survival and Hazard Functions.
Naturally, each approach has limitations that may affect its suitability for different applications. Parametric models rely on strong assumptions about the underlying distribu- tion, which may not accurately capture true survival patterns. Semiparametric models are dependent on the proportional hazards assumption, which can be invalid in certain datasets. Nonparametric models, such as DeepHit and CQRNN, tend to be computationally intensive and require large datasets for effective training, making them less practical in resource- constrained settings. Additionally, these models often pro- duce discrete estimates, which may compromise interpreta- tion and summarization flexibility compared to the continu- ous modeling offered predominantly by parametric models. To address these limitations, propose a parametric sur- vival analysis method based on the Asymmetric Laplace Distribution (ALD). Our contributions are listed below.
P(t ( T <t+ < A1T
A(t)
lim Ar-0
At
The hazards function is related to the survival function through:
d X(1) di log S(t), or S(() exp (-( M(u) Xw)) du
We introduce a flexible parametric survival model based on the Asymmetric Laplace Distribution, which offers superior flexibility in capturing diverse survival patterns compared to other distributions (parametric methods).
Furthermore, the probability density function (PDF), f(1,, which represents the likelihood that the event occurs at time t, can be derived as:
d -0---00 1(1) = - di =0(()=(00) =)0==)))
The continuous nature of the ALD-based approach of- fers great flexibility in summarizing distribution based predictions, thus addressing the limitations of existing discretized nonparametric methods.
These relationships establish unified framework linking S(t), F(t), A(t), and f(t), highlighting their interdepen- dence in survival analysis. Importantly, for the purpose of making predictions, we are interested in distributions conditioned on observed covariates, namely S(t]x), F(t)x), X(t|x) and (t)x).
Experiments on 14 synthetic datasets and 7 real-world datasets in terms of performance metrics demonstrate that our proposed framework consistently outperforms both parametric and nonparametric approaches in terms of both discrimination and calibration. These results un- derscore the robust performance and generalizability of our method in diverse datasets.
Survival Models. Survival models can be broadly classified into three main categories. Parametric models assume that the survival PDF follows specific probability distribution as described above. These models thus use a predefined closed-form distribution to describe f(t x) and F(t[x), for which a model estimating its parameters can be specified. Alternatively, semiparametric models, such as the Cox pro- portional hazards model (Cox, 1972), first decompose the conditional hazards function A(t x) = ((t)A(x), then estimate X()) from the data and specify model for X(x). In contrast, nonparametric models, such as DeepHit and CQRNN (Pearce et al., 2022) circumvent directly modeling conditional distributions by discretizing (t)xx (DeepHit, Lee et al., 2018), learning summaries of f(()x) such as (a fixed set of) quantiles Pearce et al. (CQRNN, 2022), or even learning to sample from f((lx) (Chapfuwa et al., 2018). More details can be found in Appendix A.2.
2. Background
Survival Data. A survival dataset D is represented as a set of triplets {(xm: yn: W where X,, Rd denotes the set of covariates in d dimensions, y,, min(On, C.) E IR+ represents the observed time, and e, is the event indicator. If the event of interest is observed, e.g. death, then o, < C, and the event indicator is set to e, = 1, otherwise, the event is censored and e, = 0. In this work, we make the common assumption that the distributions of observed and censored variables are conditionally independent given the covariates, i.e., o 1 x. Moreover, while we primarily consider right-censored data, less common types of censoring can be readily implemented (Klein & Moeschberger, 2006), e.g., left-censoring can be data handled by changing the likelihood accordingly (see Section 3.3 for an example of how the maximum likelihood loss proposed here can be adapted for such a case).
3. Methods
3.1. Asymmetric Laplace Distribution (ALD)
Definition 3.1 (Kotz et al. (2012)). A random variable Y is said to have an asymmetric Laplace distribution with
2

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Learning Survival Distributions with the Asymmetric Laplace Distribution
for censored events, for which = O, we optimize the parameters of the model to maximize the survival function SALD(( x) 1- FALD ( in (2). In this manner, the ALD objective below accounts for both the occurrence of events and their respective timing while explicitly incorporating the survival probability constraint for censored data:
FC; RaLU
EC, Esp
Staoed FC, ReLUU
FC, ReLU
PCC, Exp
FC. ReLU
PC,, Exp
K
Resdual Connection
Figure 1. The proposed neural network architecture for predict- ing the parameters of the Asymmetric Laplace Distribution AC(0, o, k)). Here, FC denotes the fully connected layer.
CCLLD
Σ nEDo
log fALD Xn.)
parameters (,, if its iDFis:
+
Σ nED
log SALD xn),
(3)
V26 w)). if 0, exp
(AD(U;,,,,K))
22
(1)
T##7
where Do and Dc are the subsets of D = Do U Dc for which e 1 and = 0, respectively. Detailed derivations of the objective in (3) be found in Appendix A.
exp
1 0)
if < -,
where 0, a > 0, and k. asymmetry parameters.
0, are the location, scale and
The simplicity of the objective in (3) is a consequence of the ability to write the relevant distributions, ftLD(t)x) and St x), in closed form. Moreover, make the follow- ing remarks.
Moreover, its CDF can be expressed as:
2 L2n (6- (0-
v)).
T=x exp
FAID(y;,,,,1)
(2)
The objective in (3) has the same form as the one used in other parametric approaches, for instance Royston (2001) for the log-normal distribution.
exp [y 0))
yx<0.
We denote the distribution Y as AL(y;,,o,,
We can readily adapt the loss for other forms of censoring, for instance, if events are left censored, we only have to replace the second term of (3) by SALD (f|x).
Corollary 3.2. The Asymmetric Laplace Distribution, denoted as AL k).. can be reparameterized as
AC(0, o, q) to facilitate quantile regression (Yu & Moyeed, 2001), where q E (0,1) is the percentile parameter that
We do not consider additional loss terms, as is usually done for other approaches, egg., DeepHit optimizes a form similar to (3), where the density function and cumulative distribution are replaced by discretized approximations, but also consider an additional loss term improve dis- crimination (Lee et al., 2018).
represents the desired quantile. The relationship between q
and K is given by 9 n²2((x2 1).
Additional details are provided in Appendix A.1.
3.2. Model for the ALD
Although the ALD in (1) has support for O, we have observed empirically that this unlikely to happen, as we will demonstrate in the experiments.
The structure of the proposed model is illustrated in Fig- ure 1, where shared encoder is followed by three inde- pendent heads to estimate the parameters 0, a, and of the ALD distribution. For the purpose of the experiments in Section 5 with structured data, we use fully connected layers with ReLU activation functions. The outputs of the model connected to 0, a and x are further constrained to be non-negative through an exponential (Exp) activation. In addition, residual connection is included to enhance gradient flow and improve model stability. See Appendix B.3 for more details about the architecture of the model.
3.4. Comparison between our Method and CQRNN
CQRNN (Pearce al., 2022) adopts the widely-used ob- jective for quantile regression, which is also based on the Asymmetric Laplace Distribution AC(O, o, q). and uses the transformation in Corollary 3.2. Specifically, they use the maximum likelihood estimation approach to optimize the following objective:
CoR 6g. 0. 9) = loo a - loglg(1 - 9)]
3.3. Learning for the ALD
g(y 0g),
y 2 0g
(4)
We propose learning the model for the ALD through maxi- mum likelihood estimation (MLE). Since the event of inter- est can be either observed or censored, we specify separate objectives for these two types of data. For observed events, for which e = 1, we directly seek to optimize the parameters of the model to maximize [ALD(())) x) (1). Alternatively,
(1 (0, v), if 0g.
Following the quantile regression framework, their approach optimizes a model to predict f,, for a predefined collection of quantile values, e.g., q = {0.11 0.2...., 0.9}. Effectively and similarly to ours, Pearce et al. (2022) specify shared
3

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Learning Survival Distributions with the Asymmetric Laplace Distribution
encoder with multiple heads predict g0 Note that the objective in (4) does not require one to specify a, which results in the following simplified loss:
intervals may lead coarser approximations that tend to affect model performance. This means that in some cases, the model is sensitive to the choice of the grid of g values.
In contrast, our approach enjoys a simple objective function resulting in parametric estimates of several distribution sum- maries such as mean, median, standard deviation, and quan- tiles without additional cost. Additional details of CQRNN are provided for completeness in Appendix A.2.
q(V 0g):
if y 2 0g.
Lou (v: og. 9)
(1 (0 - -y: if y fgy (y -uV-001 0,)(3 - T00 > v))
(5)
where T[-] is the indicator function. The formulation in (5)
is also known as the pinball or checkmark loss (Koenker & Bassett Jr, 1978), which is widely used in the quantile regression literature.
4. Related Work
Survival analysis is a fundamental area of study in statistics and machine learning, focusing on modeling time-to-event data while accounting for censoring. A wide range of mod- els has been developed that span parametric, semiparametric. and nonparametric methods.
Importantly, unlike in the objective for our approach in (3), CQRNN does not maximize the survival probability directly. Instead, they adopt the also widely used approach based on the Portnoy's 's estimator (Neocleous et al., 2006), which optimizes an objective function tailored for censored quantile regression. Specifically, this approach introduces a re- re-weighting scheme to handle the censored data:
Parametric models assume specific distribution for the Lime-to-even variable, providing a structured approach to modeling survival and hazard functions. Commonly used distributions include the exponential (Feigl & Zelen, 1965), Weibull (Scholz & Works, 1996), and log-normal (Roys- ton, 2001) distributions, as well as more flexible formula- tions based mixtures of these distributions (Nagpal et al., 2021). For example, the log- log-normal model assumes that the logarithm of survival times follows a normal distribution, en- abling straightforward parameterization of survival curves. In modern approaches (Hoseini et al., 2017), neural net- works are employed to learn the parameters of the assumed distribution, e.g., the mean and variance for the log normall This combination allows the model to leverage the power of neural networks to capture complex, nonlinear relation- ships between covariates and survival times, while keeping the interpretability and structure inherent to the parametric framework. However, these models face challenges despite their simplicity when the true event distribution significantly deviates from that assumed.
CCO og. q, w) = w (o og. 9) + (1 w) Lon("" 0).
(6)
where where y is a pseudo value set to be "sufficiently" larger than all the observed values of y in the data. Specif- ically, in CQRNN (Pearce et al., 2022) it is defined as y" 1.2 max; yi- However, Portmoy (Neocleous et al., 2006) indicates that y" could be set to any sufficiently large value approximating do. For example, Koenker (2022) sets y" = 1e6. This means that in practice, this parameter of- ten requires careful tuning based on the specific dataset, provided that different datasets exhibit varying levels of sensitivity it. In some cases, we have observed that small perturbations in y can lead to considerable variation on per- formance metrics. Consequently, optimizing this parameter can be non- non-trivial, making the use of CQRNN, and other censored quantile regression methods, challenging.
Semiparametric methods strike balance between flexibil- ity and interpretability. One notable example is the Cox proportional hazards model (Cox, 1972), which assumes a multiplicative effect of covariates on the hazard function. Building on this foundation, DeepSurv (Katzman et al., 2018), a deep learning- based extension, replaces the lin- ear assumption with neural network architectures to model complex feature interactions. DeepSurv has demonstrated improved performance in handling nigh-dimensional covari- ates while maintaining the interpretability of hazard ratios. However, semiparametric models face challenges in effec- tively handling censored data, particularly when censoring rates are very high. In such cases, the limited amount of usable information can lead to degraded performance and reduced reliability of the model's 's estimates.
The other parameter in (6) that requires attention is the weight w E (O, 1), which is defined as w- = (1-94)/(1-g.) qc). and where qe is the quantile which the data point was cen- sored (e- = 0, y c), with respect to the observed value distribution, i.e., p(o < c/zr.. The challenge is that qe is not known in practice. To address this issue, CQRNN proposes two strategies: a sequential grid algorithm and the quantile grid output algorithm. The core idea of both strategies is to approximate 9e using the proportion q corresponding to the quantile that is closest to the censoring value e using the distribution of observed events y. which are readily avail- able. Even with this approach, q. is an inherently inaccurate approximation. Its precision heavily depends on the initial grid of values, specifically, the intervals between consecu- tive q values. Consequently, smaller intervals provide finer granularity, but increased computational costs, while larger
The Kaplan Meier (KM) estimator (Kaplan & Meier, 1958)
4

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Learning Survival Distributions with the Asymmetric Laplace Distribution
is a widely used nonparametric method for survival analy- sis. KM estimates the survival function directly from the data without assuming any underlying distribution. The
Table 1. Dataset summaries: number of features (Feats), train-
ing/test data size, and proportion of censored events (PropCens).
KM estimator is particularly effective for visualizing sur- vival curves and computing survival probabilities. How- ever, its inability to incorporate covariates limits its ap- plicability in complex scenarios. To address this, more advanced nonparametric approaches have been developed. Tree-based ensemble methods such as Random Survival Forests (RSF) (Ishwaran et al., 2008) and Gradient Boosting Machines (GBM) (Dembek et al., 2014) 4) extend traditional nonparametric modeling by incorporating covariate informa- tion while avoiding strict distributional assumptions. These models are capable of capturing nonlinear relationships and interactions, and provide robust survival estimates even in the presence of high-dimensional features. In parallel, neu- ral network-based models such DeepHit (Lee al., 2018) and CQRNN (Pearce et al., 2022) have emerged, which directly estimate survival probabilities or quantiles with- out predefined hazard or survival function forms. These methods offer high flexibility and are particularly effec- tive in modeling complex, high-dimensional, and heteroge- neous datasets. Nevertheless, notable shortcoming of these models (e.g., DeepHit and CQRNN) is that they produce piecewise constant or point-mass distribution estimates, re- spectively, which lack continuity and smoothness. This can lead to survival estimates that are difficult to summarize, interpret, and use in downstream analysis.
Dataset
Feats
Train data
Test data
PropCens
Type 1 Synthetti target data with synthetic censoring
Norm linear
500
1000
0.20
Norm non-linear
500
1000
0.24
Exponential
500
1000
0.30
Weibull
500
1000
0.22
LogNorm
500
1000
0.21
Norm uniform
0.62
Norm heavy
2000
1000
0.80
Norm med
2000
1000
0.49
Norm light
4
2000
1000
0.25
Norm same
4
2000
1000
0.50
LogNorm heavy
8
4000
1000
0.75
LogNorm med
8
4000
1000
0.52
LogNorm light
8
4000
1000
0.23
LogNorm same
8
4000
1000
0.50
Type 2
Real target data with real censoring
METABRIC
9 1523
381
0.42
WHAS
6 1310
328
0.57
SUPPORT
14
7098
1775
0.32
GBSG
7
1785
447
0.42
1328
0.49
BreastMSK
1467
367
0.77
LGGGBM
510
128
0.60
sample sizes, and censoring proportions. Four of these datasets: METABRIC, WHAS, SUPPORT, and GBSG, were retrieved from the DeepSurv GitHub repository'. Other details are available in Katzman et al. (2018). The remaining three datasets: TMBImmuno, BreastMSK, and LGGGBM were sourced from cBioPortal for Cancer Ge- nomics. These datasets constitute a diverse benchmark across domains such as oncology and cardiology, allowing a comprehensive evaluation of survival analysis methods. Ad- ditional details of all datasets can be found in Appendix B.
5. Experiments
5.1. Datasets
We utilize two types of datasets, following Pearce et al. (2022): (Type 1) synthetic data with synthetic censoring and (Type 2) real-world data with real censoring. Table presents a summary of general statistics for all datasets. To account for training and model initialization variabil- ity, we run all experiments 10 times with random splits of the data with partitions consistent with Table 1. The reproduce the experiments pre-
5.2. Metrics
Predictive Accuracy Metrics: Mean Absolute Error (MAE) and Integrated Brier Score (IBS) (Graf et al., 1999), measure the accuracy of survival time predictions. MAE quantifies the average magnitude of errors between pre- dicted and observed survival times ÿ, and ys respectively, For synthetic data, ground truth values obtained directly from the observed distribution, while for real data, only observed events (e = 1) are considered. For the IBS calcu- lation, we select 100 time points evenly from the 0.1 to 0.9 quantiles of the distribution for in the training set.
source code required sented in this paper is available at: httpss//githmi com/demiiggen com//emingssheng washengggggggggggg ALD.
For synthetic observed data with synthetic censoring, the input features x are generated uniformly as x ~ (d(0, 2)d
where d represents the number of features. The observed variable o ~ p(o|x) and the censored variable e ~ p(c|x) follow distinct distributions, with each distribution param- eterized differently, depending on the specific dataset con- figuration. This variability in distributions and parameters allows for the evaluation of the model's robustness under diverse synthetic data scenarios.
Concordance Metrics: Harrell's C-Index (Harrell et al., 1982) and Uno's C-Index (Uno et al., 201 1), which evaluate the ability of the model to correctly order survival times in pairwise manner, while accounting for censoring. Harrell's C-Inexx is known to be susceptible to bias, when the censor- ing rate is high. This happens because censoring dominates
For real target data with real censoring, we utilize datasets that span various domains, characterized by distinct features,
htnp://gghhhcooo//eeeeeee/ee//er fmnps://wwwcchhppotaa..rrg
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Learning Survival Distributions with the Asymmetric Laplace Distribution
the pairwise ranking when estimating the proportion of cor- rectly ordered event pairs. Alternatively, Uno's C-Index adjusts for censoring by using inverse probability weighting, which provides more robust estimate when the proportion of censored events is high.
waran et al., 2008): A nonparametric ensemble model that builds multiple decision trees estimate survival functions without strong distributional assumptions. GBM (Dembek et al., 2014): A tree-based gradient boosting model adapted for survival analysis, capable of modeling complex nonlin- carities in covariates. DeepHlt (Lee et al., 2018): A deep learning-based survival model that predicts piece-wise prob- ability distributions over event times using fully neural network architecture. CQRNN (Pearce al., 2022): A censored quantile regression model that employs neural network architecture, and whose objective is based on the Asymmetric Laplace Distribution. Together, these base- lines span parametric, semi- cammiiaaametric, and nonparametric survival modeling paradigms and include both traditional statistical models and contemporary neural architectures, thereby providing a comprehensive benchmark for perfor- mance evaluation. The implementation details, including model selection, of our method and the other baselines can be found in Appendix B.3.
There are several metrics to assess calibration. We consider summaries (slope and intercept) of the calibration curves using the predicted PDF f(()x) or the survival distribution S(t)x). Moreover, use the cen- sored D- D-Calibration (CensDcal) (Haider et al., 2020). For the former, [t x)]. 9 prediction interval widths are considered, e.g., 0.1 for [0.45, 0.55]. 0.2 for [0.4, 0.6). etc. These are used to define the time ranges for each prediction, after which we calculate the proportion of test events that fall within each interval. The calculation of the proportion of censored and observed cases follows the methodology in Goldstein et al. (2020), with further details provided in Ap- pendix B.2. This calibration curve of expected vs. observed events is summarized with an ordinary least squares linear fit parameterized by its slope and intercept. A well-calibrated model is expected to have unit slope and a zero intercept. For the survival distribution, Ca]]S(t|x)], we follow a sim- ilar procedure, however, we consider 10 non-overlapping intervals in the range (0, 1). i.e., (0, 0.1]. 00.1, etc and then calculating the proportion of test events that fall within each interval. The calculation of Dens starts with that of (al[S(((xx)] which is followed by computing the sum of squared residuals between the observed and expected proportions, i.e., 0.1 for the 10 intervals defined above.
Calibration Metrics:
5.4. Results
Table 2 provides a comprehensive summary of the com- parisons between our model and the eight baselines in 21 datasets and 9 evaluation metrics, which is 189 comparisons in total. When assessing the statistical significance of the different metrics we use a Student's t test with p < 0.05 considered significant after correction for false discovery rate using Henjamini-Hochbeeg (Benjamini & Hochberg, 1995). These results underscore several key insights:
Overall Superiority: Our model is significantly better than the baselines consistently more often. For instance, our model significantly outperforms CQRNN in 23% of the comparisons while the opposite only occurs 12%, and these proportions are higher for the comparisons against the other baselines, namely, 41 %, 52%, 55% 60%, 69%, 71% and 73% for DeepSurv, GBM, RSF, LogNorm, DSM (Weibull), DSM (LogNorm) and DeepHit, respectively.
These three groups of metrics provide robust framework for evaluating predictive accuracy, calibration, and concor- dance in survival analysis. For the results we calculate averages and standard deviations for all metrics over 10 ran- dom test sets. The metrics that require a point estimate, i.e., MAE and C-Index are obtained using the expected value of ft(xx) which can be calculated in closed form. More details about all metrics can be found in Appendix B.2.
Our model demonstrates strong improvements in predictive accuracy. In particular, it achieves superior MAE performance on 10 out of 21 datasets relative to LogNorm, 11 datasets compared to GBM, DSM (LogNorm), and DSM (Weibull), and 12 datasets compared against DeepHit. AI- though performance in MAE is mixed when compared to CQRNN and DeepSurv, our method still performs competi- tively. Moreover, it consistently outperforms the basclines on nearly every dataset when evaluated with the IBS metric. This consistent superiority in IBS underscores our model's 's ability to provide accurate and reliable predictions over the entire time range, not just at specific time points. Table 4 and Figure 4 in the Appendix further support this, showing that our method achieves significantly lower IBS values, which reflects its effectiveness in learning from censored data without exacerbating bias in survival estimates.
Accuracy:
5.3. Baselines
We compare the proposed method against eight baselines representative of related work to evaluate performance and effectiveness. LogNorm (Royston, 2001): A classical para- metric survival model that assumes event times follow a log-normal distribution. DSM (LogNorm/ Weibull) (Nag- pal et al., 2021): A neural parametric model that represents survival times as mixtures of either log-normal or Weibull distributions, allowing flexible modeling of complex event- time distributions and adaptable hazard dynamics through deep learning. DeepSury (Katzman et al., 2018): A semi- parametric survival model based on the Cox proportional hazards framework, leveraging neural networks to model nonlinear covariate effects on the hazard function. RSF (Ish-
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Learning Survival Distributions with the Asymmetric Laplace Distribution
Table Summary of benchmarking results across 21 datasets, Fach column group shows three figures: the numher of datasets where method significantly outperforms, underperforms or is comparable with the baseline indicated. The last two rows summarize the column totals proportions simplity comparisons. reference, the total number comparisons 189.
CQRNN
LogNorm Worse
DeepSury
DeepHit
Metric
Better
Worse
Same
Better
Same
Better
Worse
Same
Better
Worse
Same
MAE
8
3
12
IBS
19
1
21
0
0
21
Harrell's C-Index
4
2
15
10
3
8
6
13
15
0
Uno's C-Index
2
3
16
9
2
10
6
14
15
0
CensDeal
8
4
9
10
1
10
8
15
Cal (S(Cx) (Slope)
0
0
13
O
21
0
7
(Slope)
Cal [ftt (Interepp))
4
0
11
Total
189 0.228
124 189
113/ 189
189
189
27 189
85
138 189
7/ 189
44/
Proportion
0.116
0.656
0.598
0.048
0.354
0.407
0.143
0.450
0.730
0.037
0.233
GBM
vs. RSF
DSM (LogNorm)
DSM (Weibull)
Metric
Better
Worse
Same
Better
Worse
Same
Better
Worse
Same
Better
Worse
Same
MAE IBS
11
7
3
9
6
6
11
5
17
1
3
14
2
5
19
1
19
Harrell's C-Index
14
2
5
16
2
3
17
0
15
0
Uno's C-Index
13
1
14
5
0
CensDeal
2
19
6
11
0
(S (Slope)
0
12
Cal (Iterret))
0
10
11
Cal (Slope)
15
Cal [/(t (Intercpp))
11
0
10
8
13
Total
98 189
13/ 189
78 / 189
104 189
16 189
69 189
134 189
7/ 189
48 189
130 189
6 189
53/189
Proportion
0.519
0.069
0,413
0.550
0.085
0.365
0.709
0.037
0,254
0.688
0.032
0,280
Our model shows performance comparable to CQRNN for Harrell's and Uno's ' C-indices, yielding sta- tistically indistinguishable results on most datasets (15 and 16 ties, respectively). However, when compared a broader of baselines, including LogNorm, GBM, RSF, DeepHit, and DSM variants, method consistently achieves greater concordance. In particular, for Harrel''s C-index, model attains better results on 10 datasets compared to LogNorm, 14 compared to GBM, 15 compared to DeepHit and DSM (LogNorm), 16 compared to RSF, and 17 datasets compared to DSM (Weibull), respectively. A similar trend evident for Uno's C-index, with our method outperforming Log- Norm datasets, GBM on 13, RSF and DSM (Weibull) on 14, DeepHit 15, and DSM (LogNorm) 16 datasets. These findings underscore the robustness of our model in capturing relative risk and generating accurate survival time rankings across a variety of baselines, ranging from classical parametric approaches to contemporary neural methods.
Concordance:
subtle. These results indicate that our model achieves better calibration of the predicted survival probabilities, en- suring a closer alignment between predictions and observed outcomes in both the survival CDF S(t)x) x) and PDF ft(x).. This underscores the reliability and robustness of our model in accurately capturing true survival behavior across diverse datasets.
For illustration purposes, Figure 2 shows the performance of all models and datasets using Harrell's C-index and C- cal. Specifically, Figure 2(a) provides comparison of Har- rell's C-index, highlighting of our proposed model. ALD, alongside other baseline mod- els. The x axis lists all the datasets, both synthetic datasets (e.g., Gaussian linear, exponential, etc.) and real-worle datasets (e.g.. METABRIC, WHAS, etc.), while the axis indicates the range of corresponding mean C-index values, with error bars representing standard deviation across 10 model runs. Our method demonstrates consistently strong performance across both synthetic and real-world datasets, frequently achieving higher or comparable C-index values in relation the baseline models. Among these, CQRNN stands out as the most competitive alternative, achieving similar levels of performance on certain datasets. However, in most cases, our model outperforms CQRNN, reflecting its robustness and superior discriminative capability. In par- ticular, ALD excels under scenarios with high censorsing
discriminative performance
Our model demonstrates strong performance in calibration metrics, particularly in CensDeal, Cal ((( x)] (slope) and Ctl]f(t|x) (slope). The results for CensDcal highlight its ability to effectively handle censored obser- vations. Furthermore, our model shows significant supe- riority in slope- and intercept-related metrics, particularly compared to LogNorm, DecpHit, RSF, and DSM variants. However, the improvement over CQRNN remains relatively
Calibration:

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
Ligham Dvernt
5%
Landers
play
Star
Sesvengg
LAG0200
Ment
(a) Concordance metric (Harrell's 's C-index).
CLRMM
ORN
denglee!
chitery
Relbol
Aigiiboo dpor
Abdee
AEMEON
6861
(b) Calibration metric (CensDeal).
Figure Performance on discrimination and calibration metrics. (a) concordance and (b) calibration. Reported are test averages with standard deviations over 10 runs
LOGGEM -alifcc
Using a similar comparison framework, Figure 2(b) presents the calibration results using CensDeal. Concisely, the pro- posed model achieves consistently better (lower) CensDeal figures across most datasets, reflecting superior calibration performance compared to the baseline models.
cate
Complementary to the CensDcal calibration metric, the slope and intercept summaries of the calibration curve pro- vide a more intuitive (and graphical) perspective of the calibration results. Figure 3 presents the best (first row) and worst (second row) results from our model on real-world data. For comparative analysis, we include results from CQRNN, LogNorm, DeepSurv, and DeepHit. The left and right columns represent the curves for Cal (S((x)) and for Cfl[f(t)x)), respectively. The gray dashed line represents the idealized result for which the slope is one and the inter- cept is zero. A full set of calibration visualizations across all datasets and methods is provided Appendix C.2.
SUPOORT CHURTT [K]
S-PPOIT
Brotr parr
Figure 3. Examples of best and worst calibration curves. Slope and intercept the linear are shown the legend.
The proposed model demonstrates exceptional performance the TMBImmuno dataset for (a][f(S|x)) summaries, as well as on the LGGGBM dataset for Cal(f sum- maries indicating robust calibration across both versions of the calibration metrics. In contrast, the performance on the SUPPORT dataset is relatively weaker. This discrepancy can largely be attributed to our method's 's reliance on the as- sumption of the ALD, which may not be appropriate across all datasets. This limitation is particularly evident in datasets
rates. For example, on Norm heavy (PropCens: 0.80), Norm med (PropCens: 0.49), LogNorm heavy (PropCens: 0.75), and LogNorm med (PropCens: 0.52), ALD consistently out- performs other models. This performance highlights ALD's ability to effectively handle challenging scenarios. Such ro- bustness under high censorship further underscores ALD's reliability and adaptability in various survival analysis tasks.
8

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
like SUPPORT. Notably, the SUPPORT data exhibit high skewness with relatively small range of y- Specifically, Figure 12 in the Appendix shows the event distribution for the SUPPORT data, from which we can see that it is heavily skewed. Such an skewness manifested as the concentration of events close to O makes it challenging to achieve good calibration in that range, i.e., 1 - O. Similarly, our method attempted to predict smaller values for the initial quantiles but still allocated disproportionately large weight to the first two intervals because of the small and highly concen- trated predicted quantiles, which significantly reduced the capacity for the remaining intervals and ultimately degraded calibration performance. However, the calibration results for this dataset remain within reasonable ranges and more importantly, comparable to those from the baselines. De- tailed results for all datasets are provided in Appendix C.I. Overall and consistent with the summary results in Table 2, our model demonstrates a clear advantage on the slope and intercept metrics, consistently achieving better performance compared to the baselines.
FALD max; y vx)x) < 1 for some x. We further validate this behavior using synthetic datasets with known ground- truth CDFs, Figures 10 and 11 compare the worst-estimated instances within each cluster for our method and Deep- Hit. Across clusters, our model consistently achieves lower Wasserstein distances, demonstrating superior accuracy and robustness in modeling heterogeneous survival distributions.
Case Study 3: Alternative Distribution Summaries. We also explore other distribution summaries, i.e., the mode and me- dian, to evaluate their impact on the performance of MAE and C-index. As shown in Table 5, different summaries can offer improved performance on specific datasets, demon- strating the flexibility of our probabilistic formulation to adapt the summary statistic depending of the downstream evaluation needs.
Case Study 4: Empirical Behavior of (O)x() Finally, recognizing that the ALD has support for t < 0, we sum- marized the empirical quantiles of the predicted (0)x)) i.e., the probability that events occur up to = 0. Interest- ingly, Table 6 indicates that this is not an issue as in most cases (O x) - 0 for the majority of the predictions made by the model on the test set.
Case Studies To further support our empirical findings, we include additional case studies in Appendix C.3, providing robustness analysis and deeper understanding of the behavior of our method in various scenarios.
6. Conclusion
Case Study 1: Robustness under High Censoring and Quan- tile Extremes. We assess the robustness of our model un- der varying levels of censoring and at key survival quan- tiles (25%, 50%, 75%). following the experimental proto- col of Nagpal et al. (2021). Figures and 8 report time- dependent concordance scores for the METABRIC and SUP- PORT datasets, which exhibit distinct censoring rates and distributional properties. Our method maintains strong per- formance across all settings and demonstrates particular robustness at higher quantiles on METABRIC, where cen- soring and skewness are more severe. Although perfor- mance is comparatively lower on SUPPORT, which likely reflects its deviation from the ALD assumption, our model remains comparable to DSM-based baselines, indicating stable behavior even under distributional mismatch.
In this paper, we proposed a parametric survival model based on the ALD and provided a comprehensive comparison and analysis with existing methods, particularly CQRNN, which uses the same distribution. Our model produces closed-form distributions, which enables flexible summarization and in- terpretation of predictions. Experimental results on a diverse range of synthetic and real-world datasets demonstrate that our approach offers very competitive performance in rela- tion to multiple baselines across accuracy, concordance, and calibration metrics.
Limitations. First, our method relies on the assumption of the ALD, which may not be universally applicable. This lim- itation was particularly evident in certain cases, such as with the SUPPORT dataset, as highlighted in Section 5.4, where the performance of our method faced challenges, especially in terms of calibration. Second, while our approach facili- tates the calculation of different distribution metrics, such as mean, median, mode, and even distribution quantiles, selecting the most suitable summary statistic for specific datasets applications remains non-trivial task. In this study, we selected the mean as the main summary statistic, which results in relatively balanced performance metrics. However, it does not offer an advantage, for example, in terms of C-index and MAE when compared to CQRNN. Nevertheless, considering other summary statistics as part of model selection, which we did not attempt, may improve performance on these metrics for certain datasets, detailed in Appendix C.3.
Case Study 2: Capturing Diverse Survival Patterns. To evaluate the ability of our model to capture various survival behaviors, we perform a clustering analysis on the predicted cumulative distribution functions. Figure 9 presents six rep- resentative patterns obtained through K K-means clustering on the estimated parameters from our method and DeepHit on seven real-world datasets. Although both models cap- ture multiple behaviors, the CDFs produced by DeepHit tend to converge to one at 120% of the maximum observed event time (1.2 max, y:), thus suggesting that all individ- uals eventually experience the event by that time horizon, which in practice is unlikely. In contrast, our model bet- ter reflects long-term survival characteristics by keeping
9

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Acknowledgments
sis. Advances in neural information processing systems, 33: 33:18296-18307 2020.
This work was supported by grant IR61-NS120246-02 from the National Institute of Neurological Disorders and Dis- cases (NINDS).
Graf, E.,, Schmoor, C., Sauerbrei, W., and Schumacher, M. Assessment and comparison of prognostic classification schemes for survival data. Statistics in medicine, 18(17- 18):2529 1999.
Impact Statement
Haider, H.,, Hoehn, B., Davis, S., and Greiner, R. Effective ways to build and evaluate individual survival distribu- tions. Journal of Machine Learning Research, 21(85):
The proposed survival analysis method utilizes the Asym- metric Laplace Distribution (ALD) to deliver closed-form solutions for key event summaries, such as means and quan- tiles, facilitating more interpretable predictions. The method outperforms both traditional parametric and nonparametric approaches in terms of discrimination and calibration by optimizing individual-level parameters through maximum likelihood. This advancement has significant implications for applications like personalized medicine, where accurate and interpretable predictions event timing are crucial.
1-63, 2020.
Harrell, F. E., Califf, M., Pryor, D. B., Lee, L., and Rosati, R. A. Evaluating the yield of medical tests. Jama, 247(18):25432544 1982.
Hoseini, M., Bahrampour, A., and Mirzaee, M. Comparison of weibull and lognormal cure models with cox in the survival analysis of breast cancer patients in rafsanjan. Journal of research in health sciences, 17(1):369, 2017.
References
Aalen, O. Nonparametric inference for a family of counting processes. The Annals of Statistics, pp. 701-726, 1978.
Ishwaran, H., Kogalur, U. B., Blackstone, E. H., and Lauer, M. S. Random survival forests. The Annals of Applied Statistics, pp. 841-860, 2008
Benjamini, Y. and Hochberg, Y. Controlling the false dis- covery rate: a practical and powerful approach to multiple testing. Journal of the Royal statistical society: series B (Methodological), 57(1) (7(1):289-300 300, 1995.
Jung, E.-Y,, Baek, C., and Lee, J.-D. Product survival analysis for the app store. Marketing Letters, 23:929- 941, 2012.
Kaplan, E. and Meier, P. Nonparametric estimation from incomplete observations. Journal of the American statis- tical association, 53(282):457-481, 1958.
Chapfuwa, P., Tao, C., Li, C., Page, C., Goldstein, B., Duke, L. C., and Henao, R. Adversarial time-to-event modeling- In International Conference Machine Learning, pp. 735-744. PMLR, 2018
Katzman, J. L.,, Shaham, U., Cloninger, A., Bates, J., Jiang,
T., and Kluger, Deepsurv: personalized treatment rec- ommender system using cox proportional hazards deep neural network. BMC medical research methodology, 18: 1-12, 2018
Cox, D. R. Regression models and life-tables. Journal of the Royal Statisticul Society: Series (Methodologicul), 34(2):187-200 1972.
Dembek, K. A., Hurcombe, S. D., Frazer, M. L., Morresey, P. R., and Toribio, R. E. Development of a likelihood of survival scoring system for hospitalized equine neonates using generalized boosted regression modeling. PLoS One, 110)cc192222, 2014.
Klein, J. and Moeschberger, M. Survival analysis: techniques for censored and truncated data. Springer Science & Business Media, 2006.
Koenker, R. Quantile Regression, 'quantreg'. : Pack- age. Cambridge University Press, 2022. doi: 10.1017/ CBO978051 1775499..
Emmerson, J. and Brown, J. Understanding survival analysis in clinical trials. Clinical Oncology, 32-1): 2021.
Koenker, R. and Bassett Jr, G. Regression quantiles. Econo- metrica: journal of the Econometric Society, pp. 33-50, 1978.
Feigl, P. and Zelen, M. Estimation of exponential survival probabilities with concomitant information. Biometrics, pp. 326-838, 1965.
Kotz, S., Kozubowski, T., and Podgorski, K. The Laplace distribution and generalizations: a revisit with applica- tions to communications, economics, engineering, and finance. Springer Science & Business Media, 2012.
Gepp, A. and Kumar, K. The role of survival analysis in financial distress prediction. International research journal of finance and economics, 16(16):1334, 2008.
Goldstein, M., Han, X., Puli, A., Perotte, A., and Ran- ganath, R. X-cal: Explicit calibration for survival analy-
Lai, C. D. and Xie, M. Stochastic ageing and dependence for reliability. Springer Science & Business Media, 2006.
10

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
Lánezky, A. and Györffy, B. Web-based survival analysis tool tailored for medical research (kmplot): development and implementation. Journal of medical Internet research, 23(7):e27633, 2021. 2021.
Lee, C., Zame, W. Yoon, J., and Van Der Schaar, M. Deep- hit: A deep learning approach survival analysis with competing risks. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.
Nagpal, C., Li, X., and Dubrawski, A. Deep survival ma- chines: Fully parametric survival regression and repre- sentation learning for censored data with competing risks. IEEE Journal of Biomedical and Health Informatics, 25 (8):1633-315,, 2021.
Neocleous, T., Branden, K. V., and Portnoy, S. Correc- tion to censored regression quantiles by portnoy, 98 (2003), 1001-1012. Journal of the American Statistical Association, 101(474);800-86,, 2006.
Pearce, T., Jeong, J.-H., Zhu, et al. Censored quantile regression neural networks for distribution-fuu survival analysis. Advances in Neural Information Processing Systems, 35:7450-7461, 2022.
Royston, The lognormal distribution as model for survival time in cancer, with an emphasis prognostic factors. Statistica Neerlandica, 55(1):89-104, 2001.
Scholz, F. and Works, B. Maximum likelihood estima- tion for type censored weibull data including covariates. Technical report, ISSTECH-96-022, Boeing Information and Support Services, 1996.
Uno, H., Cai, T., Pencina, M. J., Agostino, R. B., and Wei, L.-J. On the c-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Statistics in medicine, 30(10):1105-1117 2011.
Voronov, S., Frisk, E., and Krysander, M. Data-driven battery lifetime prediction and confidence estimation for heavy-duty trucks. IEEE Transactions Reliability, 67 (2):623-633) 6399, 2018.
Yu, and Moyeed, R. A. Bayesian quantile regression. Statistics Probability Letters, 54(4):437-447, 2001.
Zhang, W., Lc, T. D., Liu, L., Zhou, Z.-H., and Li, J. Min- ing heterogeneous causal effects for personalized cancer treatment. Bioinformatics, 33(15):2372-2378 2017.

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Learning Survival Distributions with the Asymmetric Laplace Distribution
A. Analytical Results
This section provides the analytical results. Detailed proofs for the Asymmetric Laplace Distribution Loss can be found in Appendix A.I, while the analysis of all the baselines, including CQRNN, LogNormal MLE, DeepSurv, and DeepHit, is presented in Appendix A.2.
A.1. Proofs for the Asymmetric Laplace Distribution Loss
Theorem 1. If AC((,, G, it), where AC denotes the Asymmetric Laplace Distribution with location parameter 6, scale parameter a > and asymmetry parameter R O, then the ALD loss given by:
CLLD +
Σ DED
log SALD(Un xn))
Σ BED
log (1 FALD(UN xn)),
where Do and De are the subsets of D for which = 1 and e 0, respectively. The first term maximizes the likelihood fALD(E x) for the observed data, while the second term maximizes the survival probability SALD(T x) for the censored data. To achieve this, the parameters 0, a, x predicted by multi-layer perceptron (MLP) conditioned on the input features, x, enabling the model to adapt flexibly to varying input distributions. The observed component L. (y: 0, o, x) is defined as:
V2 kLy 0), if 2 0,
Lo k) = log a log
K
+
(8)
x² +1
(0 y), if < A.
The censored loss component Le 0, or k) is computed using the survival probability function:
log(n² + 1) 22 xc(y 0),
y 2
Le(y;0,a,k))
(9)
log(n> + - 1 + x² (1-
V2 (- v) -)))]
if y <0
Proposition 2. (Mean, Mode, Variance of Y) The mean, mode, variance of Y are given by:
Fly 0 + V2
k)
(10)
Mode Y = 0,
(11)
0² 1
Var
+ x2
(12)
2
R2
Proposition 3. (Quantiles of Y) Let OALD denotes the q-th quantile of Y. Then, the quantiles can be expressed as:
+ OK log
1+x² x²
if E
(0,
x²
/2
It*
@LL
(13)
0
log [1 + (11
9) if I+n7
2²
/2a
A.2. Analysis of All the Baselines
CQRNN. CQRNN (Pearce et al., 2022) combines the likelihood of the Asymmetric Laplace Distribution, fALD(t x). with the re-weighting scheme introduced by Portnoy (Neocleous et al., 2006). For the observed data, CQRNN employs the Maximum Likelihood Estimation (MLE) approach to directly maximize the likelihood of the Asymmetric Laplace Distribution AC a, 9). The likelihood is defined over all quantiles interest. For censored data, CQRNN splits each censored data point into two pseudo data points: one at the censoring location y = c and another at a large pseudo value y". This approach enables the formulation of a weighted likelihood for censored data, resulting in the following loss function:
12

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
LoOr Loly:0, + (.
(14)
where Lo represents the negative log- leelikkel for observed data, and Le accounts for the weighted negative log-likelihood of censored data using the re-weighting scheme. Expanding this, the loss can be expressed as:
LoQR =
Σ nED,
log (UL x..) - Σ mED
[log fALD(Un x.,) + (1 - w) fALD(V" xx)]
(15)
where Do and De are the subsets of 'D for which = and O, respectively. Here, CQRNN utilizes the Asymmetric Laplace Distribution AL a, Q) to model the data. The Asymmetric Laplace Distribution, denoted as AL a, k). can be reparameterized as AA, a, q) to facilitate quantile regression within a Bayesian inference framework (Yu Moyeed, 2001), where y E (0, 1) the percentile parameter that represents the desired quantile. The relationship between & and given by:
x²
(16)
+
Thus, the probability density function for ~ AA((0, 0,4) is:
exp (2 (00-y) ((-y)),
if 2 0,
fALD g(1-g)
(17)
exp للدي (y -
y <0.
And the cumulative distribution function is:
1 - (1 - 9) exp (2 - y))
y 2 8
FALD 0,
(18)
qexp ( 0)):
if < 0.
Thus, the negative log-likelihood LOR(U: 0, a, q) then can be explicitly derived as:
1 q(v 0), if (rr 0,, 0, = log log[a(1 - 4)] + о (1 (0 v). if <
(19)
In their implementation, the scale parameter is omitted, and the percentile parameter q predefined, typically set values such (0.1, 0. 0.9). A multi-layer perceptron (MLP) in CQRNN, conditioned on the input features x, predicts 0g for the predefined quantile values, corresponding to the location parameter 0. The negative log-likelihood COR (y; 0, a, q) is then further simplified as:
q(v 0g);
if 2 Og.
(oR 0g:9)=
= = - -
- 0--00--11100 I[00, > vl).
(20)
(1 - ()g - v), if < 0g-
This formulation is also referred to the pinball loss or "checkmark" loss (Koenker & Bassett Jr, 1978), which is widely used in quantile regression to directly optimize the q-th quantile estimate. For censored data, CQRNN adopts Portnoy's ' estimator (Neocleous et al., 2006), which minimizes a specific objective function tailored for censored quantile regression. This approach introduces re-weighting scheme to handle all censored data, with the formula defined as:
Lely. u²; 0g q, w) wook 0g. 9) + (1 LoR 00: 9),
(21)
13

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
where y" is pseudo value be significantly larger than all observed values of y in the dataset. Specifically, it is defined as y" 1.2 max; yi in CQRNN (Pearce et al., 2022). The weight parameter is apportioned between each pair of pseudo-data points as:
qe
(22)
qe
where de the quantile which the data point was censored (e = 0,y = c) with respect to the observed value distribution, Le., p(o x). However, the exact value of ge not accessible practice. To address this issue, CQRNN approximates de using the proportion q corresponding to the quantile that closest the censoring value c, based on the distribution of observed events y, which readily available.
LogNormal MLE. LogNormal MLE (Hoseini et al., 2017) enhances parameter estimation using neural networks for NogNormal distributions. Specifically, a random variable Y follows a LogNormal distribution if the natural logarithm of Y, denoted as iY))). follows a Normal distribution, Le., In(Y) ~ (NH,, Here, μ represents the mean, and is the standard deviation (SD) of the normal distribution. The probability density function of the LogNormal distribution is given by:
(In y - 11)
fLogNorma (3:
exp
(23)
2m 2T
211²
where y > 0 and 1y > 0. The cumulative distribution function is expressed as:
[In(y) 10
F FLogNormat (U,
(24)
where ((2) is the standard normal cumulative distribution function:
12
(x)
exp
(- dt.
(25)
/2T
The maximum likelihood estimation (MLE) loss with censored data is then defined as:
LogNNrnal
log fLogNormal x..) log (1 - LoggNoraa (Un x,)). Σ nED.
(26)
nED
A multi-layer perceptron (MLP) in LogNormal MLE, conditioned on the input features x, is used to predict the mean μ and the standard deviation of the corresponding normal distribution. The quantiles 8 ogogNomlal for the LogNormal distribution can be expressed as:
@LogNormal =ex(u + 7/up-~110 (g)),
q
(27)
where t (q) is the inverse CDF (quantile function) of the standard normal distribution.
DeepSury, DeepSurv (Katzman et al., 2018) is semi-parametric survival model based on the Cox proportional hazards framework, leveraging deep neural networks for feature representation. A multi-layer perceptron (MLP) in DeepSurv, conditioned on the input features x, is used to predict the log hazard function h(x):
X(t x) ((tt)
(28)
where At) is the baseline hazard function. The hazard function is defined as:
P(t 5 < T
A(t x) = lim
(29)
Ar-40
At
14

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
This can be rewritten as:
X(t x) ds(t x)/dt. S(1 x)
(30)
where S(t x) P(T > & / x) is the survival function. By integrating both sides, we have:
I A(t x) xt= /
d5(t x) S(t xx)
(31)
which simplifies to:
A( | x) log S(l x) + C,
(32)
where C' is the constant of integration and A(t x) is the cumulative hazard function:
A(t x) (t)
(33)
where (t) is the baseline cumulative hazard function. For survival analysis, C is typically set to 0 when starting from L == 0. Thus, the survival function can be expressed as:
S(1 x) e (((x) - (1))
(34)
where (t) is the baseline survival function, typically estimated by the Kaplan-Meier method (Kaplan & Meier, 1958) using the training data. The cumulative distribution function (CDF) can then be derived as:
DeeepSure (1 x) = S(L x) = 1 [So(1) A00
(35)
The quantiles 6, &DeepSurv for DeepSurv can be obtained from the inverse CDF DecpSurv (t x) (quantile function).
DeepHit. multi-layer perceptron (MLP) in DeepHit (Lee et al., 2018), conditioned on the input features x, is used to predict the probability distribution [L x) over event times using fully nonparametric approach. The quantiles Deeptit
can be obtained from the inverse cumulative distribution function Fe (t x). where FDeepHir
x) Σ fDxepHiet x).
Gradient Boosting Machine (GBM). GBM for survival analysis (Dembek et al., 2014) is nonparametric, additive ensemble of decision trees trained to minimize a loss function tailored to censored data, such as the Cox partial likelihood or the Brier score. Similar to DeepHit, the model directly learns a discrete probability distribution f(t x) over event times. The quantiles 8GBM can be obtained via the inversion of cumulative distribution function FG (t x). where FOBM(t (t x) foBM x).
Random Survival Forests (RSF). RSF (Ishwaran et al., 2008) is nonparametric ensemble method that extends Breiman's 's random forests to survival analysis. Each decision tree is trained on bootstrap sample of the data and uses the log-rank test statistic to determine optimal splits. The survival function SRSe(t x) for given input x is estimated by aggregating the survival estimates from all trees in the forest:
B
1
StSE(t xx)
Sb x),
(36)
B
where So(t x) is the survival function from the b-th tree and B is the total number of trees. The corresponding cumulative distribution function is FRSE(! x) = SRSF(t | x). The quantiles ORSF then can be obtained via the inversion of cumulative distribution function Fast(( (t. x), where FRSF(t x) = Σ fRSF(t x).
15

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
Deep Survival Machines (DSM). DSM (Nagpal al., 2021) is a neural parametric model that represents the survival distribution mixture of K components from chosen parametric family (e.g., LogNormal, Weibull). Given input x, the model outputs both the parameters {uA, Iik km for each component and mixture weight Ty (x) using a neural network. The probability density function is expressed as:
K (x(()) p0s, mk).
fDSM(t
x) = Σ
(37)
1=1
and the cumulative distribution function is:
K (tx
FDsM(t x)
(38)
where f& and Fk are the PDF and CDF of the k-th parametric component (e.g., LogNormal or Weibull). The quantiles 0DSM are obtained from the inverse CDF @DSM = FDSM(S (q x).
g
q
Summary Comparative Advantages. In summary, our proposed method offers a unified and efficient solution that bridges the strengths of both classical and modern survival modeling approaches. Compared traditional parametric models, it retains closed-form, differentiable PDF and CDF formulations that facilitate stable neural optimization and achieve robust empirical performance across diverse data settings. Compared to nonparametric models, it avoids the drawbacks of temporal discretization and excessive memory usage, providing compact, continuous, and interpretable estimates of survival quantities. Finally, relative to mixture-based models such as DSM, our method is simpler, faster, and more robust-requiring fewer parameters, introducing less architectural complexity, and demonstrating greater stability across datasets and censoring regimes. These advantages collectively establish the ALD framework as a compelling and scalable alternative for survival modeling both synthetic and real-world applications.
B. Experimental Details
This section provides additional details about the experiments conducted. The experiments were implemented using the PyTorch framework. Detailed information about the datasets, metrics, baselines and implementation details can be found in Appendix B.1, Appendix B.2, and Appendix B.3, respectively.
Hardware. All experiments were conducted on MacBook Pro with an Apple M3 Pro chip, featuring 12 cores (6 performance and 6 efficiency cores) and 18 GB of memory. CPU-based computations were utilized for all experiments, as the models primarily relied on fully-connected neural networks.
B.1. Datasets
Our datasets are designed following the settings outlined in Pearce et al. (2022). The first type of dataset consists of synthetic target data with synthetic censoring. In these datasets, the input features, x, generated uniformly as x 21(0, 2) where D denotes the number of features. The observed variable, distinct distributions, with their parameters varying based on the specific dataset configuration. Table provides detailed descriptions of the distributions for the observed and censored variables. Additionally, the coefficient vector used in some datasets defined as 3 [0.8, 0.6, 0.4, 0.5, -0.3, 0.2, 0.0, -0.7].
p(o x), and the censored variable, c ~ p(c x), follow
The other type of dataset comprises real-world target data with real censoring, sourced from various domains and character- ized by distinct features, sample sizes, and censoring proportions:
METABRIC (Molecular Taxonomy of Breast Cancer International Consortium): This dataset contains genomic and clinical data for breast cancer patients. It includes 9 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): This dataset focuses on predicting survival following acute myocardial infarction. It includes 6 features, 1310 training samples, and 328 testing samples, with a censoring proportion 0.57. Retrieved from the DeepSurv Repository.
16

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Table Characteristics of synthetic datasets encompassing the number of features, parameterized distributions of observed variables, and
censored variables, utilized in the experimental framework.
Synthetic Dataset
Feats (D)
Observed Variables o ~ p(o x)
Censored Variables c ~ p(c) x)
Norm linear
(2x + 10, (x+ + 1)
W44x + 10, (0.8x + 0.4)3)
Norm non-linear
N (xxsi((2x) + 10, (0.5x + 0.5)
M( + 29)
Exponential
Exp(2x +4)
Exp(-33 + 15)
Weibull
Wcibull(xsin(2x - 2) 10,5)
Weibull(-33) 3x + 20, 5)
LogNorm
LogNorm((x 100--]))- 11),,x))
(400, 10)
Norm uniform
N (2xcos(2xx) + 13, (x+ 0.5)²)
((00, 18)
Norm heavy
4
(3xx + +x] x3 2sin (xxxs) 6, (x + 0.5)3)
U(0, 12)
Norm med
4
U(0, 20)
Norm light
U400, 40)
Norm same
4
Equal to observed dist.
LogNorm heavy
LogNorm Di=1 (ix, 1)/10
2((0, 0.4)
LogNorm med
2((0, 1.0)
LogNorm light
8
2((0, 3.5)
LogNorm same
Equal to observed dist.
SUPPORT (Study to Understand Prognoses Preferences Outcomes and Risks of Treatment); This dataset provides survival data for critically ill hospitalized patients. It includes 14 features, 7098 training samples, 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 (Rotterdam & German Breast Cancer Study Group): Originating from the German Breast Cancer Study Group, this dataset tracks survival outcomes of breast cancer patients. It includes 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): This dataset predicts survival time for patients with various cancer types using clinical data. It 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 the cBioPortal.
Derived from the Memorial Sloan Kettering Cancer Center, this dataset focuses on predicting survival time for breast cancer patients using tumor-related information. It includes 5 features, 1467 training samples, and 367 testing samples, with censoring proportion of 0.77. Retrieved from the cBioPortal.
BreastMSK:
This dataset integrates survival data from low-grade glioma (LGG) and glioblastoma multiforme (GBM), frequently used for model validation in cancer genomics. It includes 5 features, 510 training samples, and 128 testing samples, with a censoring proportion of 0.60. Retrieved from the cBioPortal.
LGGGBM:
B.2. Metrics
We employ nine distinct evaluation metrics to assess model performance comprehensively: Mean Absolute Error (MAE), Integrated Brier Score (IBS) (Graf et al., 1999), Harrell's C-Index (Harrell et al., 1982), Uno's C-Index (Uno et al., 201 I), censored D-calibration (CensDeal) (Haider et al., 2020), along with the slope and intercept derived from two versions of censored D-calibration (Cal [S(())) (Slope), Cal (Interep) Cal (()))())))))) and (a)][(() These metrics provide holistic evaluation framework, effectively capturing the survival models' predictive accuracy, discriminative ability, and calibration quality.
<MAE:
N
1
MAE-
y;
7el.
(39)
N
1=1
where y/ represents the observed survival times, T. denotes the predicted survival times, and N is the total number of data points in the test set.
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Learning Survival Distributions with the Asymmetric Laplace Distribution
IBS:
F(L F()))))) xx)) (w S S4, 1) + x;) (1 t)
(S(()= Σ
1 N
F(t
(40)
G(vi)
G(t)
IBS 1
yy dy.
(41)
t1
where BS(t) represents the Brier score time t, and 100 time points evenly selected from the 0.1 to 0.9 quantiles of the y-distribution in the training set. F(L xx) denotes the estimated cumulative distribution function of the survival time for test subjects, I(() is the indicator function, and e; is the event indicator (e; = if the event is observed). x; represents the covariates, and G-) refers to the Kaplan-Meier estimate (Kaplan & Meier, 1958) of the censoring survival function.
Harrell's C-Index:
Ch = P(o: > $3 Di < y4 = Eie [[0: > 05) + 1(:; = 6z) < Wi)es
(42)
ΣiS} T(y < yj)es
where Φ; = Syy x,) = F(t x,) represents the risk score predicted by the survival model. For implementation, function from the sksurv metrics module, as documented
we utilize the concordance.indxx.coettt in the scikit-survival API.
Uno's C-Index:
Cu P(o: > Dj I1 vb Di (-)
E Ej= G(y)) - > bz) 1(0, = o;)l < Mj < <y)ee
(43)
Σ プド G(v))) vj. < yr)e,
where y- is the cutoff value for the survival time. For implementation, we utilize the concordance index ipew function from the sksurv.metric. module, as documented in the scikit-survival API.
CensDcal:
10
(at+) 41) - =²
CensDeal = 100 x
Σ jml
(44)
where Ç is defined by (Goldstein et al., 2020)
(gj+1 9:) T)) < y: S 6 Bg
+ (9,+1
95 [5)[[ < 4g
100.. < ys S &
+ iE
(45)
q
1 q
Sobervad
Here, the percentile parameter gú is predefined |0.1, .., 0.9] at the outset, and q, is the quantile which the data point was censored (e 0, y = e) with respect to the observed value distribution, iee, p(o e x). Oz.g, represents the estimated qth quantile of yı.
Slope & Intercept: The Slope and Intercept metrics evaluate the calibration quality of predicted survival quantiles relative to observed data under censoring. We utilize the np. polyfit function from the NumPy module, docu- mented in the NumPy API, to the 10 points { (0.15. E, +c,)
10 and subsequently obtain the Slope and Intercept Cal[S(+|x)] (Intercept). (tl]s(t) and
metrics. Two versions of the Slope and Intercept (Cal[S(t)x)) (Interpt) are calculated, differing in how the quantile intervals are defined:
Version 1 (Measuring S(t x)): The predicted survival probabilities are divided into intervals based on the target proportions, i.e., q [0.1, 0.2, ... 0.9, 1.0]. For each quantile interval, the proportion of ground truth values (observed survival times) that fall within the corresponding predicted quantile N is calculated. For example, the ratio for 0.1 (j = 1) is calculated within the interval [0, 0.1], and for 0.2 (j = 2), within [0, 0.2). Thus, the horizontal axis represents the target proportions 0. 1j, while the vertical axis represents the observed proportions Σ3 & derived from predictions. In the end, this metric is suitable for evaluating the Survival Function S(t x) (or CDF F(t] x)).
18

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Version 2 (Measuring S(L x)): Narrower intervals centered around target proportions are used, i.e., q (..., 0.4, 0.45, 0.55, 0.6,--.. For each quantile, the observed proportions calculated within these narrower intervals. For example, the ratio for 0.1 is calculated within the interval [0.45, 0.55), and for 0.2, within [0.4, 0.6]. In the end, this metric is ideal for assessing the probability density function (PDF) f(t x).
B.3. Implementation Details
Baselines. We compare method against four baselines to evaluate performance and effectiveness: LogNorm (Royston, 2001), DeepSurv (Katzman et al., 2018), DeepHit (Lee et al., 2018), and CQRNN (Pearce al., 2022). All methods were trained using the same optimization procedure and neural network architecture to ensure fair comparison. The implementations for CQRNN and LogNorm were sourced from the official CQRNN repository (GitHub Link), The implementations for DeepSurv and DeepHit were based on the pyсox. meth module (GitHub Link).
Hyperparameter settings. All experiments were repeated across 10 random seeds to ensure robust and reliable results. The hyperparameter settings were 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
Batch Norm: FALSE
Our Method. We incorporate residual connection between the shared feature extraction layer and the first hidden layer to enhance gradient flow. To satisfy the parameter constraints of the Asymmetric Laplace Distribution (ALD), the final output layer applies an exponential (Exp) activation function, ensuring that the outputs of the 8, and branches remain positive. Fach of the two hidden layers contains 32 hidden nodes. A validation set is created by splitting 20% of the training set. Early stopping is utilized to terminate training when the validation performance ceases to improve.
CQRNN. We followed the hyperparameter settings tuned in the original paper (Pearce et al., 2022), where three random splits were used for validation (ensuring no overlap with the random seeds used in the final test runs). The following settings were applied:
Weight Decay: 0.0001
Grid Size: 10
Pseudo Value: y" 1.2 X max, V/
Dropout Rate: 0.333
The number of epochs and dropout usage were adjusted based on the dataset type:
Synthetic Datasets:
-orrm 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:
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Learning Survival Distributions with the Asymmetric Laplace Distribution
METABRIC: 20 epochs with dropout disabled.
:HHAS: 100 epochs with dropout disabled.
SUPPPRR:: 10 epochs with dropout disabled.
GBSG: 20 epochs with dropout enabled.
TMBImmmun: 50 epochs with dropout disabled.
BreastMSK: 100 epochs with dropout disabled.
LGGGGM:: 50 epochs with dropout enabled.
LogNorm. The output dimensions the default neural network architecture are 2, where the two outputs represent the mean and standard deviation of 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 dropout usage were adjusted based on the dataset type follows:
Synthetic Datasets: The same settings described above for CQRNN.
Datasets:
METABRIC: 10 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 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 employed to terminate training when the validation performance ceased to improve. Batch normalization was applied.
DeepHit. We adhered to the official hyperparameter settings from the pywox.methaa.. mods 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 employed terminate training when the validation performance ceased improve. Batch normalization was applied, with additional settings: num_durat.in ions 100, alpha = 0.2, and s gma 0.1.
class from the sksurv. ensemble module.² The model was configured with n est iisttoorss 100, learning rate = 0.01, and max.depth= following standard practices for tree-based boosting in survival analysis.
GBM. We implemented GBM model using the
inttoosttiggSuvve.
RSF. The RSF model was implemented using the RandomSurvivalE srrrriiiiiFooeestttto orest class from sksurv ensemble. We used the default configuration with n.ss imators 100, which has been shown to provide good trade-off between performance and efficiency.
DSM. We adopted the Deep Survival Machines (DSM) model from the auton auton--urviin survival library using the DeepSurvival 1achiia class. The neural network consisted of two hidden layers, each with 32 hidden units. For the LogNorm variant, we used & = 10 mixture components, as higher values of & were observed to degrade performance due to overfitting and instability. For the Weibull variant, we followed the library's 's default configuration with = 100, which was necessary to maintain sufficient model capacity. Training was performed using observed event times and censoring indicators, and the final predictions were computed by evaluating the learned mixture distribution over fixed 1000-point discretized time grid to estimate the cumulative distribution function (CDF).
20

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Learning Survival Distributions with the Asymmetric Laplace Distribution
C. Additional Results
This section presents additional empirical results to support the evaluation of our method. The results include comprehensive comparisons across datasets and metrics, as well as calibration plots and case studies to illustrate key behaviors. Full benchmarking results are provided in Appendix C.I, calibration curve visualizations in Appendix C.2, and detailed case analyses in Appendix C.3.
C.I. Overall Results.
Table 5 summarizes the full results across 21 datasets, comparing our method with baselines across metries. Figure 4 visualizes these results for more intuitive comparison. In Table 5, the best performance is highlighted in bold. Figure 4 provides a graphical representation of nine distinct evaluation metrics to comprehensively assess predictive performance, including Mean Absolute Error (MAE), Integrated Brier Score (IBS), Harrell's C-Index, Uno's C-Index, Censored D- calibration (CensDeal)), and the slope and intercept derived from two versions of censored D-calibration (Cal [S((t (Slope)) (al[S(t)x]][ntee)))
Cal (f (f)x) (Slope),)) and Call (Ittttt))))))))))))))))) Specifically, the following transformations were applied to enhance the clarity of the results:
MAE and CensDeal were log- teggsraanformed to better illustrate their value distributions and differences.
For Cal(S(t)))) (Sllope and Call (t]]((t)(]]) (Slopp)) |1 - (alpp(t and |1 (txl were computed to measure their deviation from the ideal value of
and [t]x(( ([iterrppp)))))))) [a][((t)x)][nnerccr and [[al]((t))]][(eeerepp)) were com- puted to measure their deviation from the ideal value of O.
For (al[S(t)x)
These transformations allow for a more intuitive comparison of the performance differences across metrics and models. In the end, each subfigure in Figure 4 provides a comparison of its corresponding metric. The x-axis lists all the datasets, both synthetic datasets (e.g., Gaussian linear, exponential, efc.) and real-world datasets (e-g., METABRIC, WHAS, etc.), the y-axis indicates the range of corresponding its metric, and error bars represent standard deviation across 10 model runs.
Table 4. Full results for all datasets, methods, and metrics. The values represent the mean + standard error for the test set over 10 runs.
Dataset
Method
MAE
IBS
Harrell's C-index
Uno's C-indes
CeneDiaa
CallS(e
Cal Sirx (Intercopt)
Call (allSetii(SSsppe)
Call (onteeee))
ALD
0,66 a 1.337
0,278 0.000
0.655 0014
0,585 0001
0.407 0.545
1.025 a 0,016
0.005 0,00
1.027 00042
00016 2 0.001
CORNN
0.326 0.144
8.65720.00
0.15
1 0.062
-0,013 0.126
1807 0,039
2.020 0,047
NomuJacu
LogNom
0.222
0.709 0.025
0.02200016
0.645
0 0.399
0.965 2 0.024
0.005
20.41
0.065
DecpSurv
0.676 0.005
0.667
0661 00007
0,139 0.071
0.83 0.018
0.006
1.07
-0.05 0.014
Deeplit
0.527
0.56 0:005
0.65 0024
0,25 0.025
6.300 1.458
0.97 : 0.008
$009$20.000
1,216 00055
-0,30 0.009
GBBM
8631 0.084
0.305 00007
0.611 0012
0633 x 0.00
0.146 + 0.078
0.99 a 0.003
8UK0 0.018
1000 + 0.014
-0.014 - 0.014
RSF
1 0.120
0.328 0.009
0,588 0014
0,581 0.013
0,594 2 0.256
0386 2 0,015
0,025 2 0.017
0.800 2 0:021
0,069 = 0.020
DSMeLegNeen)
1880 a 0.946
0.324 a 0.007
0662 0000
0.646 x 0.088
2.514 x 0.523
1028 a 0.017
.0,016 0.016
0.968 0.019
0,071 = 0.022
DSM(Wehum
0.050
0.323 0.000
0,644 0.08
0.647 0.007
0.593
0,015
0.004 0.017
1068 0,011
ALD
2243200000
0.212 1.31220.006
0015
0,644 0.06
0,406 0.179
1.072 0,001
0,011 0.015
1 0.025
00016 2,000
CQRNN
0:241 0.099
0.012
Nocaaand
LogNomm
0.442
0,005
0.08
0.617
2.136 à 3,886
1000 à 0.052
1.99 0.060
-0099 0.009
DecpSury
8 0.047
00013
0670 0015
0.650 0.014
0.196 0.128
1.05 0.019
0:007 0.016
1819 0.002
-0.007 0.006
Desphii
0.130
0.515 00,49
0660 0.01
0.596 x
3,886 : 3,682
1L999 2 0.864
.0.007 a: 0.061
1064 0.067
-0.16 0.004
CIIM
0.323 0.007
0.226 a 0/00
0.63 0018
BA65 e 0.06
0.179 a 0.104
1004 a 0.017
0.04 0.018
1,812 6.627
-0.133 0.031
RSF
0.459 2 0.045
0.242 02
0:022
0,504 2 0.012
0.910 0.300
0385 a 0.023
0,017 0.012
0.890 2 0.019
0.065 0.025
DSMLLogNoemi
0.310 0.08
0.244 0:005
1.99820016
0.565 0.06
2683 0.242
0.988 a 0,017
1 0.004
0.035
DSM/Webull
0.029
0
0,13
0.610 0.016
3.561 0.316
0,015
0.155 0.016
1.157 0,002
-0.067 0,006
ALD
0.344
0.44 0.0002
0
0.019
0.115 0.030
0,020
0.016
1016 0.015
JU106
CQRNN
0535
0,706
0.141
1,022
-0.00
Noro nattraa
LogNomm
17.079 23,333
0.003
0.115
0.575 0.003
3,799 a 0.354
0.951 a 0.059
0.159 0.043
1.156 2.016
..129
DeepSurv
0516 0.1800
0.781 0014
0.701 + 0.020
0 : 0.149
1.034 + 0,017
0.022 0.00L
1,089 00.012
0.051 0.016
Despiii
1466 a 0.458
0.364 00048
0,758 0.033
0.685 x 0.028
3.150 1.142
1.015 a 0.045
0.024 a 0.847
1.128 a 0.051
-0.209 = 0.027
CIIM
1.154 0.065
0158 a 0:003
0.799 0019
0.674 e 0.05
0.101 à 0.054
1L091 à 0.013
0013 0.009
1,013 0.012
-0.00 - 0.017
RSF
+.111...11
0.55 - 0:003
0.65 0022
0.21 x 0.019
0-12 +.1.+00..10 0.101
0.915 x 0.011
-0,012 0J007
01930 0.013
0.943 0.012
DSMLLogNocmi
0.63 0:003
0/12
0,691 0018
6,693 1.280
0.930 0,015
0.019
1,089 0.005
0.035 = 0.022
DSM/Wehbul
1.352 0.020
0.063 0.003
0,768 0017
0.683
3.66 0.655
0,020
0.058 0.011
1.041 0.015
00,019
ALD
2942 2.389
0.309 20018
0,560 0.08
0.560 0
0A322 0.405
0.978
40,015 0.014
0.049
0.016
CORNN
0.013 0,297
0.033
0.557 0.011
0,305 0.129
(1976 a 0.066
0.043
1001 0.027
-0000 0.019
Expaceiial
LagNomm
0.455 0.000
0008
0.52% 0.038
0.19 0.141
0983 + 0.006
0/42
105 0.002
-0.01 00001
DeepSurr
0.866 0..69
0.558 00007
0.558 +
0.119 : 0,066
[1986 : 0,003
0.009 0.0022
1,00 0.018
0L808 = 20,15
Despiii
26266 a 2.759
0.41 0.012
0.526 0.022
0.525 x 0.031
1.205 : 1.060
0.960 0.007
.0.012 2 3.021
0.907 1 0.005
0.127 2.127=0066
GBM
2.099 a 0.249
0.295 a 0,007
0.541 0011
0,540 0.010
0.137 2 0.055
1.977 2 0,034
0U007 0.025
0.995 2 0.019
2.104 2 0,017
RSF
8.22 1222.4000 0.308
(1343- 0.015
0
0.513 + 0.014
0.77% + 0.31
0.885 + 0.006
-0.009 + 0.011
(189) 0.004
0.066 - 0.000
DSMLLogNoem)
23344 0.229
0.294
0013
0.595
19935 0.548
01976 2 0,025
+ 0.016
1
0.028 0,024
DSMuWebolly
0.230
0.292
0007
0.557
1,571 0,315
A 9.622
0.000
000022
6.092
21

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Dutrnd
Method
MAR
IBS
Harrellss Cánlex
Unn's Cáindes
Candeaal
ALD
5 9.533
0,219 = 0.028
0.677 0.00
0.763
0.68: 0.511
1044
.0003 0.033
0:093 00049
0.021 0.060
COKKN
6.850 0098
0.60 =: 01000
0.78 = 00,05
0.69 0.005
0:366 0.131
0999 e -0057
-0,01 + 8.012
0,59 + +0.022
-0,015 + 9.023
LogNorm
0.840 0.021
a 0.006
0.767 0.006
0.598 0.172
0.933 =0,036
0.029 + 0.315
1,000 2 0.028
.0.053 2 0.035
Webbol
DeepSarr
0.381 0008
0.969 + 0.019
0.72 00044
0.766 0.006
1.118 0.049
0199 0.021
0.005 00012
1,005 x 0.01K
-0.009 c.121
Desptitt
1.97
0.612 0.032
0.709 e 0.005
0,763 : 00007
5.030 1.157
0.998 e 0,854
0.122 a 0,071
1
0,187 + 0,009
GBM
1467=
0.254 0.006
0.76 : 0.005
0.759 : 0.006
0.359 0.343
1005 : 0.022
0,012 a 0,020
1.063 0.025
.0,057 0.0022
MSB
0.15
0.235+000 0.01
0.747 0000
0.240 0.100
0.966 0,299
0165 - 0.04)
-0009 0.014
0:386
0.066 0.0021
2.08800073
0.328 0.007
0.745
0.743 0.005
2.713 0.460
1.015 =
0,04 0.022
0,981 0,003
0,046 0.025
0.529
0.745
0.742 0.004
3,500 0.485
1.00 0.055
0.020
0,018 0.023
ALD
0.36 0098
6,5% 00113
5.588 0014
0.565 0014
0.256 0.150
1006 0.521
0,006 0.011
1.011 0.00%
20004 (003))
CQRNN
0.00
0 0.01
0016
0316
1.00 0.00
0,019 e 0.004
0.044
0,000 0.031
LogNoons
DeepSors
LogNorm
6.367 0082
0.44 0.021
0.658 0009
0.588 0015
0.889 0.816
0.544 0.05
0.886 01816
IL103 0,020
01332 0,009
0996 0.021
0.006 a 00000
0.001 6.020
0:597 0.025
1.016
0/00 0.001
0.010
DeegHiit
0.50
0.588 0.025
0.551 0.032
0,582 0.031
2,088 1,666
0.988 0.031
0,006 2 0.050
0.892 + 0.002
0.162 4
GBM
1.034:0000
0.555 0.010
0.579 0015
0.575 = 0.015
0.163 0.062
0.999 = 00,04
.0.002 a 0.018
0.999 0.018
-0,003 x 0.016
RSF
0.77
0.428 0.022
0,542
0.541 = 0.018
0.907 0.2-7
0.851 = 0.619
.0.01 0.015
0.587 0.019
0,062 a 0,020
DSMM agghim))
0.919 -0050
03586 0011
0.506 + 0.000
0.566 - 0000
1 0.385
1066 0.020
-0.006 + 0.016
0,088 0.021
0,020 0.0000 0.000
WeibaWl
0,762 0.05
0.382 0.01
0.00 00013
0.501
2153 0,450
= 0.019
0,010 0.011
0.017
0,11 0,015
AI
0.667-019
6.019 1J001
0.919 1.007
0870 1.:29
1.136 0.817
1009 0.005
-0:004 0.004
1,001 0.000
-4,022 0.000
CORNN
0.574
0.5% 0.06
0914=
0.63
0.099
1,000
.0.002 0.007
1,000 0.012
.0.044 0.011
Nomb
LogNorm
12:001
0411
0.781
0.09
0,,00
1222
0,001 0.029
1.111 0.131
0,774 4-0002
Serp
1.662 0.17
0 0.07
0.76 0.03
0.582
0.577 0.067
1.000 0.006
.0002 e 0.009
1.065 a 0.011
0,065 &00.L0
Doophii
0814-0 0:00
0445 0.07
0913
0856 0.34
0.374
051 0.044
0.065 00.05
1.139 0.027
.0.12 0.005
GBM
1.82=
00002
0.756 0.084
0B33 0.01
100 0 0.09
0.011 00000
1.015 x L.11
0,015 + 00007
RSF
0.607 0016
0.000
0.001
0.510 = 0.008
0.802 : 0.111
0.83 0.016
1.055 0.006
0.002 a 0.005
.0,005 0.0L0
18MI (gNNom)
1,915 0037
0.47 0.00
0.752 = 0003
0.206 0.087
7B4 0.555
0997 = 01:00
0,023 + 0.007
1,02 + 0.000
-0,015 + CUENS
DSM//Wibult)
1.906 0.037
00002
0.827 0.014
0,771 = 0.032
6:120 0.605
1.008 0.000
0,006 a
L013 a 0,005
0,005 0.0005
ALD
0.238 0.036
6.47 0.003
0,894 0.005
0.872 0,004
0.157 0.044
1058 = 0.012
00,035 a 0.0L1
0.947 a 0,012
2
CIQRNN
0.312 0.33
0.988 0.010
0888 0005
01877 0.00
089? 0.045
(1954 0.036
0.301 a 0,013
0,589 + 0.019
0,007 + 0,000
Norm med.
DeepSare
LogNorm
0.16 0.25
7,300 2,59
0.430 0.019
0.712 0.022
0810
0.83
0.777 0,44
0.004
8.192 00000
0.111
0,75 0.003
1.05=
0.008 0.015
0,350 0.002
1.06 +
1 0,021
-0,051
0.192 0.136
Dopplit
0.56+
0586
09633
0.00
0.012
+0,19
GBM
0 0.004
0000
0.000
0.045
0.11
0,015 0.009
LOST 0.011
0,033 0.001
RSP
0.02
0.081 = 0U0G
(188 000M
0161 = 0.00
IU122 0,835
0.066
0.003 000000
0.002
0
DSMLLoNNum)
1.95
0.118 0.006
0692 = 0.023
0.662 = 0.027
1404 0.176
1.019 0.014
0.030 2 0.009
1,055 0.009
.0,036 a. 0.012
Werhall)
1,920 0018
0.117 + 0.06
0757 -
0.788 - 0.17
1.174 0.191
1031 + 0.018
0.009 + 0.013
1.07 0.010
-0,010 CL01
ALD
6.236 0051
0.090 2 IU00
0.852 : 0.004
0.874 0.004
0.339 0.076
1.057 0.017
.0,000 0.011
1.99120.014
0.005 a 0.021
CQRNN
0.32
0.671 0.013
0.579 0.012
0871 = 0.002
0,149 0.097
0,999 = 0.827
0,014 0.017
0 2 0,022
0.004 0,827
Nona Light
LagNorm
2:54
0.548 0.023
0832 : 0.02
0831 0.02
12.84 =: 1.700
0804 0.162
0.358 0.126
1,351 x 0.018
-0.55 0.038
SrepSure
0016=00166
0.911 00024
0.882 0.002
0.74 0.002
0
1.08 =0014
0.0019
1,05 0.0233
0,00 0.023
DespHii
0,059
0,001 0.030
0.875 0.004
0.867 0.004
1854 0.46]
0.002
1.159 0.003
0,157 0.003
1.309
0.89
0.250
0.02 0.013
RSF
00000
0.00
0.66 2
0.133 0.053
0014=0014
0,001 0.13
0.11
0013
DSMiLagNNm))
0055
0.227 0.014
0.20=00000
0.643 0.030
0888 0.119
0.01
0.024 0:008
0.002
40045 0.014
1.926 0.0
0.222 e 0.014
0.22=0006
0717=000 0016
053 + 0.190
1044 0016
00.01
+.001
40014 0.001
ALL3
0,405 0.09
0/066 = 0.00
0.900 =
0847 0.000
0.14 0.006
1007 = 0004
0.000: 20.012
20,010
CQRNN
0.30120002
0.568 = 0.018
0.36 = 0.006
0841 = 0.166
0.147 0.109
0.58 0.028
0,003 2 0.014
0
4,807 2 8.031
Nort sime
Leghiorm
0.379 0200
0.770 + 0.09
(RN = 0.05
0850 a 0,00
0:00 0.501
8.994 = NUA32
0.017 + 0.000
1.00 0.187
-0.04% 1.10
SeepSure
6,254 00136
0.787 0.015
a559 2 0004
0337 = 0,025
0.227 0.053
1.033 = 0.014
0,014 0.011
0,01
0.00
DuggHii
1.303 0.12
0.572 0.032
0832 0.006
0.82 0.077
1.7% = 0.770
1.41 0.049
0,060 2 0.047
1.142 0.041
40.14 0.035
GRM
1.534 0052
0114 0006
0838 0.0
7588- 0.00
1 0.043
(191 0.019
0,029 0.011
1,0% 0.010
-0,011 C1009
0,444 0.00
0
0.08
0.28 0.000
1.004
0,001 0.009
1,00 0.016
0.008
Logh
0.05
0.179 0000
0011
0015
0.911 00015
0,009 + 0.005
1:052 00013
0.001
DSM(Welball
2,182
0.178 0.006
0.73 0017
0.699 0.03
3.834 0.328
0001
0,011 0.007
1,021 0.001
4,8007 8.011
ALD
0.193
1000
0.777
0.77 # 0.021
0043 0.019
1 0.014
0,00 : 0,005
0.998
4,003
LagNombeory boory
LogNorn
CQRNN
8.17= 0.07
0.194
0.436 0.035
0.401
0.767 0.000
0.643 0.053
0718 0.08
0.609 0,066
0.235
0.066 0.056
1.018=00112
0.036
.0,002 2 0.005
.0007 a 0.013
LO11 0.019
0.948 2 0.032
.0,003 2 9.020
0,019 0.035
DicpSarr
0.42 0019
0.459 0.03
0.407 = 0004
0.465 0.29
0100 0.066
1.031 = 0.008
.0,010 0.007
1.015 0.017
-0.01& C.015
Deeptiit
0.724 0022
0,402 + 0.12
0.756 z 0012
0 = 0.119
0282 0.121
1 = 0.030
0,012 + 0.006
0.00
-0045 + 1.0LS
GBM
2.68
0118:00.00
0.666 = 0.652
0618 z 0,029
IM30 0,016
1.00 0.009
0,001 2 0.003
1.001
-0000 0.0012
RSP
0.85 0018
0.09 0.06
0715 - 0016
01663 0018
0631 0.000
106 0.00
0.002 0,094
0.011
-0.005 0.012
0.125 0016
0.655 0.054
0.64 0,051
18733 1.302
0.94 0,010
0004 0.005
0,012
0
DSM/Wiillll
0.08
0.00
0.69
18844
0.97
0.004
0.013
ALD
6.17 0J146
6.174 JUN6
0.747 0000
0718 nul0
00,022
1.08 0007
.00044 0.010
L,02 0.09
20001 0.012
CIRRNN
0540 0.09
0003
0.36
0.16
0.106
0.955 0.000
.0,001 6/07
0,994 0015
0,008 0.0)
LogNom ated.
LogNorm
0.549=0.100
0.422 0.022
0.694 0.04
0.66
h.085 0.067
0.06
0.007 = 0.011
1,00 + 0.022
0.000 0.026
Decp/Srr
0654 0.099
0.545 0.15
0638
0.50 0011
81 0.004
1000 0.012
.0.15 0.009
0:094 0.016
0,005 0.018
Deeptiit
0.00=0001
0.000
0.29 0019
0.00 0005
0.54
1006 00018
0003 2 0.017
0.000
0,033 0,000
GBM
0016
0.205
2 0.08
0711 0000
0664 0.008
0.067 0.012
1.006 0.017
0.008 2 0.011
1.007 100011
0.014 0.0014
RSP
0.08 0016
0.180 000
0733 0000
200 = 0.0s
IU00 0.024
1015 - 0.17
0.002 00012
000400000000
-00015 4..14
DSMI NggNmm))
0.643 0014
0.225 0.06
0.700 - 0.06
(1604 0808
7:000 0.465
(1956 0.023
.0:003 + 0.014
0.003
0,025 x 0.015
DSM Wetballl
0014
0.221 0000
0010=0011
0.48 0,00
7,371
1.099 0.021
0.014
0.002
0.000
ALD
6.184 0u33
6,300 1il
0.725 nuMr
0213 0.008
0.165 0.095
0.85 e 0.015
0.000 0.000
0.004
4,801 à 8.0017
CORRN
0073
0.418 0.045
0.725 0.007
0.14 0.008
0.76 0.,00
0.988 0.077
0012 0.045
0.962
0.04 0.072
LagNorat Light
LagNonm
0311=00022
0.7% 0.026
0.709
0.68 0.100
0.231 0.150
0.92 0.027
00007 0.014
2.064
0.029 0.041
Serp
0.403
0.833 0.025
0.055
0.70 0.11
0211 0.123
1000
00,00 = 0.022
1.004 0.011
0,005 2 0.015
DaspHii
0.511 0.18
0.654 0.077
0,702
0.62 0.08
0255 0.174
0.000
.0,013 a 0.016
0:074 0.001
0.043 0.025
GBM
a616=001
0.363 01000
06933 0000
0163 0.006
01104 0,042
1007 = 0.02
0.009 00,017
00000
0019 2.015
RSF
044220001
0.517 0.009
0714= 0.006
0,703 = 0,077
0.114 0.082
1.025 0.020
.0,000 8.012
0.018
.0,005 2: 0.020
DSMI ag/mm
0.43 4004
536 0000
0.08 - 0007
06688 0.000
1800 0.275
0.997 a 01129
0.002 + 0.017
0:88K + 0.01K
0006 + 0.019
0.014
0.384 0.001
0.10=00000
0.639 0000
2.292 0.343
1.037 02 0.023
0.013 0.016
0.995
0,023 0,019
ALD
6.191 0.044
0.154 0.086
0.739 =z 0.009
0.697 = 0.008
DU07 0.057
1012
.0,001 2 8,008
LO09 0.011
0,005 2 2.00L
CORNN
0.319 0.079
0.500 s: (.049
0.740 : 0.088
0168 : 0.000
0.767 0.336
(1966 0.096
-0:003 1 0.040
6:971 a 0.058
0.041 a 0,056
Lagionns state
LogNorm
0.275 0198
0.518 (1017
01336
0.655 0.100
01213
0192 0005
0006 + 0.010
0.563 0.128
0,033 4 0,000
Serp
0.362 0.026
0.511 0.012
0.743 0,010
0.700 11007
0.138 0.040
0.07=00013
0000 0.012
1.04 0.014
0.001 0.013
Dosphii
0.560 0.08
0.585 0.12
a.52
1265
0915 0077
.0010 0.011
0.025
0088
GBM
0.511 0.023
0.096:00006
0.16
0.66 00011
[J64 0.831
000000000
0,003 + 0:009
1,00 0.010
0,017 0.007
0.397 0.019
0.167 0.007
0011
0612 0000
0.123 0.045
1.06=
0.024 0.012
L.82 a 0.010
.0,009 0.000
DSMI aggNmm)
0.005 0005
0.255 + 0.006
0678 = 0025
0646 0.300
4.6 0,674
0.9% 0.014
40000 a 0.007
0.978 0.01.5
0028 0.012
DeseWebbolly
0.907 0.024
0,216 + 0.06
0.39 = 0013
0612 0010
4.44 0.520
1006 - 0.11
.0,005 + 0.007
0.083 + 0.009
0,000 + 0.005
22

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
Datawe
Method
MAF
IRS
Harrell's C-ndes
Unn's C-ndees
CenDDeal
(Ieterr()
ALD
1836 194
0:345 0001
B657 0.001
0.63
0.29%
1 0.633
-0.011 0.016
0.991 +0.0%%
-0.00 - 0.004
CQRNN
0.998 0.074
0.027
0.622 0.0017
0630:00:00
0.641 = 0.391
(.572 2 0,045
1.00 9.042
0.022 0,051
METABRIC
LogNorm
1.329 0.041
000055
0.609 0.00
0.613 00095
0,19 e: 20.247
0.964 a 0.006
.0.024 e: 0.011
0.93) 00001
0,040 0.015
DespSurv
0,533 0.009
0.645 9.016
0.65 0.835
0,159 2: IU175
1809 0,011
.0.00 20.013
1,033 0.002
2,010 0,023
Dapphi
0:46:+000 0.055
0.006
0.563 0.044
0.57 -0000
0.699 + 0.212
1.700 00,002
-0.0%6 0.004
10118 0.015
-0004 + 0.000
GBM
a 0,043
0:207:
0.01
0.632
0.199
0,025
-0018 0.021
1,09
0.241
0.0022
0,004
+.0000
4000
NSMiLogNiem))
0.02
0.265
0.613 000024
0.00
11300
1.002 a 0.026
00006
18034
0001
DSM(Welllll
0.002
0:366
0,60 0.005
0.588 0.024
4,998 0.510
1042 0.023
40.005 0.014
0.622 0.011
0.016 00019
ALD
0:134+0000 0.12
00013
0.23 0.006
0.83 00014
0.198 IUI
0
0:003
0.61 0.001
0.009 0002
CORNN
0.798 0.090
0.65 0.018
0885 0.016
0.546 = 0.016
0.564 :: 0.248
0.974 a 0.060
IUM2 0.022
0.998 a 0.057
2.024:00000
LogN-or
0.64 0.0225
0:042
0,575 = 0.009
0.581 :: 01233
1 0.002
0:361 + 0.021
0.994 0.0032
-0,02 = 0,003
WHAA
DespSurv
0.567 a
0,699 x
0.711 0.014
0.637 a 0.025
0.228 = 11101
1.997 a 0,020
0,003 a 0.018
1.012 0.020
2.019
Decpliii
1 2
0,004
0.811 00.15
0.269 0L.172
01963 0,036
0
1,008 0.005
0.015 0,031
GBB
0.165 0.009
0.01
0,363 0.174
0.034
0.000
1076 0.002
-0.044 0.023
XSF
11,283 3,013
3177 0.275
1,00 +0.001
2.65 0.175
0:311 +000
0.77 0.006
0.788 0015
11:200 1.181
0.98 0.006
0,020 0.071
LO15 0.005
-0.011 a 0.027
DeMtWeibull
0.112
0.205 000000
0.765 0.002
07.01000017
10.569 2 1,355
(.907 2 0,025
0.022
1001 0.024
0.00=000020
ALD
1.121 0.107
0.362 0.003
0,565 0.015
0,572 00.05
2.197 = 0.667
188420.003
0,113 a 0.023
0,000 0.000
00,0000.000
CORNNN
0.047
0.344 0.007
B612 x 0.005
06613 = 0.005
0.724 + 0.428
11334 a 0.066
-0.019 0.033
0.992 + 0.051
0.009 = 000000
SUPPORT
DerpSury
LogNorm
0.110
0.021
0,299 0.014
0.020
0,599 0.06
0.597 10011
0.97 0.009
0.97 = 0.011
0 0.136
2.792 + 0.912
0.989 à 0.016
0.942 + 0,040
-0,003 0.011
0,114 +: 0.008
0.709 0.004
0.966 0.012
0.169 = 00000
6,006 0.014
Decplili
0.024
0.009
0.00
0.009
01213
11291 0,013
0.056 00.02
GBM
0.277 0.006
0.350x00.00
L0K5 0.018
40,010
LoK5 0.010
0.019 00015
RSF
0.027
0.339 = 0.005
(U619 0.006
6668
0.122 0.058
a 0.013
.0.02 3.009
1,006 0.000
0.037=00015
DSMILeg Normm))
0:512
0.379 0.00
0.007
0.70 0:007
16.644 1.442
1,142 à 0.011
-0.145 0.010
0928 0.000
0.057 00012
DeMtWelbul
0.581 0.003
0,375 0.007
0.563 0,0000
0,509 2 0,000
14.725 : 1.066
1.299 2 0,010
0,147 + 0.010
0141 0.000
0.056 0.012
ALD
0.200
0.279 + 0.014
0,67 0.0113
0.665 = 0.013
0,283 = 0.106
1J00 2 0.035
0,018 =
0.77 0.005
0.014 0.034
CORRNN
0.965 + 0.070
0021
0L690 1 0.015
0.672 : 0.814
0.575 + 0.577
0.953 + 0.045
-0.008 + 0.016
0.967 00,000
6.002 = 0.949
LugNunn
0.015
0.600 0.012
0.653 = 0.012
0,817 0.303
1.968 a 0,025
+0,057 a 0.011
0.386 0.025
0,886 = 0.035
GBBSG
DerpSury
0.709 à 0.066
0.569 0.016
0.017
01602 0.015
0,180 0.126
0.001
-0,003 0.013
(L996 0.013
0,008 0.018
Decpliii
0.373 0.027
0,495
0.649
0.644
2,020 1.450
0,049
2
GBM
0337 0.017
0.289
+0.013
0,622 0.012
0.165
= 0.002
1011 00020
0.00000:000
RSF
0.052
0.283 0.011
0.65 000010
0.648
0.192
1,955 a 0.029
.0.007
0.944
0.02
0:308 +000
0.6 +0.03
0.699 - 00000
7,079 0.364
1068 0.017
.0.05 0.010
0990 0.001
0.044
WishWll
1.075 + 0.076
0.09
0168 0.01
0.632 0.01
8.608 + 0.718
1.00 + 0.015
-0,077 0.000
0.968 0.014
0.031 00016
AI
3:02. 497
0.345 0005
0.561 0.017
0:547 + 0.000
0,835 + 0.604
1 + 0.045
-0.0%% + 0.075
IL994 6.621
0.00% - 0.025
CURNN
1005 : 0.053
0.272 0.013
0.567 0.002
0.557 0.017
0.251 = 1L123
11967 2 0,007
0.011 0.026
0.968 0.027
0.009 00020
LugNuun
0.155
6:20 0.001
0.561 0.028
0,557 : 0,025
0.617 = 0.196
0.940 2 0.025
.0.027 a 0.019
0.133 0.025
0,066 0.027
DerpSury
0.097
0.995 0.02
0.563 0000
0.226 0.039
0,346 0.168
+ 0,000
40,001 0.023
1,00 000000
-0.83 0,018
Desphin
0.144
0555
GBM
0 0.032
0.343
0.576
0:55 0.023
0.04 0.279
1.330 0.067
0,019
1.009 0.00
0.029
RSF
1,007 0.069
0:363
0.550 0.01
0.5% 0.01
0.448 0.12
: 0.046
40.018 0.009
1017 0.002
0.01
DSMILogNeen)
0.556.20.018
0,245
0.009
0.518 0.003
0,522 00003
7.414 0.791
1UMS = 0.022
0,028 = 0.019
01933 0.016
0,085 00015
1.000 0.030
0.345 0.09
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23

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Comparison LogiMAE) the Models
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24

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Learning Survival Distributions with the Asymmetric Laplace Distribution
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25

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Comparison xiXIntercepti Across All the Models
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C.2. Calibration Plots.
Figures 5 andd 6 present the complete calibration curve fitting results, offering a comprehensive and visual assessment of model calibration performance. Specifically, Figure 5 compares our method (ALD) with CQRNN, LogNorm, DeepSurv, and DeepHit, while Figure 6 extends the comparison to GBM, RSF, and both DSM variants (LogNorm and Weibull). Each plot shows the observed event probabilities against the expected target proportions [0.1, 0.2, ... 0.9, 1.0]. The ideal calibration corresponds to the identity line, where predictions align perfectly with the empirical outcomes. Deviations from this line highlight calibration errors. These plots provide intuitive insight into the degree and nature of miscalibration across models.
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26

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Learning Survival Distributions with the Asymmetric Laplace Distribution
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27

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Learning Survival Distributions with the Asymmetric Laplace Distribution
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28

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Learning Survival Distributions with the Asymmetric Laplace Distribution
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29

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Learning Survival Distributions with the Asymmetric Laplace Distribution
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Learning Survival Distributions with the Asymmetric Laplace Distribution
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Learning Survival Distributions with the Asymmetric Laplace Distribution
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Learning Survival Distributions with the Asymmetric Laplace Distribution
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33

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Learning Survival Distributions with the Asymmetric Laplace Distribution
C.3. Case Studies
Case Study 1: Robustness Under High Censoring and Quantile Extremes
To further assess the robustness of model under varying levels of data censoring and across different event-time quantiles, we conduct experiments following the setup proposed by Nagpal et al. (2021). Specifically, we simulate additional censoring by uniformly sampling censoring times from the interval [0, I and applying them to a randomly selected subset of uncensored training samples, reducing the proportion of uncensored data to approximately 50% and 25%. Test splits remain unchanged to ensure unbiased estimation the time-dependent concordance index (Ctd.) This procedure is applied the METABRIC and SUPPORT datasets, which differ markedly in their censoring rates and event time distributions.
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Figures 7 and illustrate Cldd values across three quantiles (25%, 50%, 75%) under default, 25%, and 50% censoring regimes. On the METABRIC dataset, our method demonstrates strong performance across settings, particularly at higher quantiles where skewness and right-censoring are more pronounced. This suggests that our model is well-suited for capturing long-term survival signals under substantial information loss due to censoring.
On the more challenging SUPPORT dataset, our model's performance is comparatively lower, especially under extreme censoring. This outcome can be attributed to the SUPPORT data's distributional properties, which deviate significantly from the ALD assumption and exhibit heavy skewness and short survival times. Nevertheless, our method still matches the performance of DSseb baselines, indicating that it retains stable and competitive predictive behavior even under distributional mismatch and extreme censoring.
34

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Overall, this analysis confirms the resilience of method both reduced observed events and variability in event-time horizons, with particularly strong performance on datasets aligned with ALD assumptions.
Case Study 2: Capturing Diverse Survival Patterns
To assess the expressiveness of our model in characterizing heterogeneous survival behaviors, we perform a clustering- analysis on both real-world and synthetic datasets.
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Figure visualizes six representative survival patterns identified via K K-means clustering the estimated parameters from our model and DeepHit across seven real-world datasets. Each curve represents the average cumulative distribution function (CDF) within cluster. Although both models demonstrate the ability to represent distinct survival behaviors, the CDFs produced by Hie consistently converge to one at 1.2 max; which implies that all events are predicted to occur by that time with high probability (close to 1). This behavior is unlikely in the presence censoring and indicates a limited capacity model long-term survival. contrast, our model more accurately reflects plausible survival behavior under right-censoring, with CDFs that appropriately plateau before reaching one.
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13
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155
50
23
20 125
23
104
155 15
130
80 15
30
Figure 10. Worst-estimated instance in each cluster for Gaussian_nonlinean dataset (best synthetic dataset).
We extend this analysis to synthetic datasets where ground-truth CDFs are available, allowing a more rigorous evaluation. In Figures 10 and 11 11, we compare the worst-estimated instance (based on Wasserstein distance) within each of six clusters for the best-case dataset (Gaussian_nonlinea and most challenging dataset (LogNorm_ respectively. In both cases, our method consistently yields lower Wasserstein distances compared to DeepHit, highlighting its robustness and precision in modeling diverse survival distributions across various difficulty regimes.
These results collectively demonstrate our model's 's enhanced ability to recover realistic and heterogeneous survival patterns, both in real-worke settings and under controlled synthetic evaluations.
35

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Learning Survival Distributions with the Asymmetric Laplace Distribution
Loghorm med
Sar. 2 Ditaein E01
Antoraii Disarce 0.01 CTP
Bule Nocker Vissenio Shiiee 0317
Saxe haluen Ditaaio 1
Tare Assoersai car
Sann fellin faiio Bitrrs
4TTP 20
83-
83-
6.4
022
81
E0
03
83
05
42
04
34
36
12
02
04
60
13
12
02
34
65
03
18
03
04-
05
14
32
36
62
C4
04
M
Renurre 2M4
1M1
C4 LI Dicoai
Ceivese OK (irogrt)
Ca0 37 Juccer
Coleree CV Stepls
08
38
04
17
09
12
Figure 11. Wosst-estimated instance in each cluster for LogNorm_med datasct (most challenging synthetic datasct).
Case Study 3: Alternative Distribution Summaries
To examine the flexibility of our probabilistic model, we evaluate its predictive performance under three different distribution summaries: mean, median, and mode. While the mean serves as the default choice in our main experiments, Table 5 shows that using the median or mode can lead to improved performance on certain datasets, particularly in MAE and concordance metrics. This highlights the adaptability of our approach, where different summary statistics can be selected based on the requirements of downstream evaluation. For example, in datasets with high skewness (e.g.. Exponential, L, the median often outperforms the mean in terms of MAE. Conversely, the mode may yield better alignment with the underlying survival dynamics on the real datasets. This case study emphasizes the benefit of having access to a closed-form survival distribution, allowing flexible downstream use of different summaries without retraining.
Table 5. Full results table for all datasets, the ALD method (Mean, Median, Mode), and metries. The values represent the mean * 1 standard error on the test set over 10 runs.
Dataset
Methed
MAE
Harrell's C-Index
Uno's C-Index
CensDial
Sit (Siope)
Call Calt
(t (Sllppp)
ald (Mean)
0.565
0,653 0.014
0,645
Nurm lincer
(Malian)
0.887
0.276 0.008
0.62 0.037
0.407 0.343
1.027 0.042
-0.16 0.057
1.025 0006
0,005 0.030
(Modd)
0 0.088
0 0.00
ald
0,000
0.670 0.055
0,644 0.016
Norm non
(Modian)
0L006
0.077
0,552
+ 0.025
0011 0,000
1.072
0,011 0001S
(Madr)
0.438
0.633 0.060
0.573 0.054
(Meur)
0.785
0,703 : 0.020
Norm miffrr
(Modian)
0.392 0.196
0.045 0.002
0,885 1
0,695 à 0.013
0,115
1,016 + 0.014
-0.006 0.21
1.019 0000
0/002 + 0.016
ald (Modz)
0.613 0.:118
0.785 = 0.012
0,695 2 0,014
ad (Mean)
2.942 2.288
0,500 2 0,088
0.560 2 0,007
Expoeniill
al (Mallian)
1 0.348
0,309 0.018
0,559 : 0.000
0.553 x 0.000
0,432 0.405
0.064 = 0.049
0.016 = 0053
0.97% = 0.47
2.015:000044
ald (Mode)
5,009 0.233
0,001
0,555 + 0,020
ad (Meon)
0,765 : 0,099
0,763 + 0,010
Wabhll
al (Maian)
0.484 0.859
0.129
0.767 x 0.006
0,601 a 0,023
0.648 0.511
0,993
0.021 = 0.060
1.044 00033
-0,03 a 0.033
ald (Modo)
1.163 0.310
0.700
0,009 à
add (Man)
0.363 0.066
0,585 : 0.014
0.3X5 x 0,014
LegNeen
(Median)
0.531 0:007
0.055
0,589 à 0.015
0
0.256
1,011 0.02
-0009 0022
1,005 0002
1.733
0,549
0
ald
0..39
0.007 0.11
0,570 0.029
Norm kony
0.061
CL001
0.256 150
1.011
-0.04 0.22
1.005 0002
0,005 0.011
(Mode)
0.502 +
(Meun)
0.844 0.005
0.572 +
Numn mod
(Maslian)
0.047 0L003
0,580
0.568 a 0.011
0.157
0.012
0.004 0014
1.058 0012
-0,086 0.011
ald (Modi)
0.985 0/0N7
0,884 00000
0,889
(Mzas)
0.236 0.851
0,822 :: 0.094
0.874
Norm light
akd (Modian)
0.255 0.016
0.000 0L0007
0,800 =: QU00S
0383 0.017
0,539 =: (L076
0,99% + CL014
(UIOS = 0021
0017
-0.000 & (LUFT
ald (Modz)
0.325 a 0.029
0,876 : 0.003
0.550 4 0.017
add (Mean)
0.404 0.075
0,890 = 0.085
0.547 4 0.008
Norm same
ald (Mialian]
0.261 a 0.822
0.066 OLDOS
0J88 + 0.006
0,86 à 0,094
0.116
1.004 + 0L01N
0,010 + 0072
0000
0,005 + 0.012
ald (Mode)
0.515 0:005
0,581 = 0.008
0.000
add (Meun)
0.385 4 0.193
0.777 +: 0.012
0.727 4 0.022
LogNorn heavyy
aid (Molian]
0,244 0,042
0.005 a 0L006
0,779 e 0.011
0,549 x 0.011
0.043 0.019
0.99% e 0L014
-0.002 0014
1.003 0004
-0,055 &. CL0GS
0.305 0/045
0.129
0,724 0.012
36

---

Learning Survival Distributions with the Asymmetric Laplace Distribution
Datane
Method
IBS
Harrellys Giidee
Uni's Condees
CenaDeal
CalSS( (Slppe)
Call =(((eetereppt)
Call (1/]][[ntereppp))))
(Mran)
a 9,046
0.218 00007
LogNorm med.
old (Median)
0.247 2: 0.024
0.174 0.006
0.768 0.0004
0,789 0.001
0,087 = 0.052
1,002 0L009
0,001 2 0.012
LOO0 + 0.017
0,004 & 0.OLD
sid (Milk)
0.596 0.082
0.723 0.013
0.709 0.012
old (Meun)
0.184 2 0.005
0.725 0.000
0.713 0.000
LogNorn light
old (Median)
0.221 & 0.0004
0.11
0.725 a 0,0007
0.690 + 000029
0 0000
1,001 = 001
0,000 + 0,015
0,985 + 0,015
0,00S a 0,009
zid (Maxls)
0,921 0.005
0.702 0.014
0.697 0.015
old (Meon)
0.391 + 0,044
0.739 0,000
01697 + 0,000
LegNorn
ald (Median)
0.289 01062
0.154 0.006
0.240 0..10
1,751 0.014
0,075 0:007
1,009 0011
-0.05 0,010
4,012
-0,001 A CLOOS
zdd (Modo)
0,943:00000
0.007
0.001
old (Mron)
1,66 1.14
06577 0.001
RA33 0.631
METABRIC
(Mediun)
0.005
0015
06,01
0.583 0.000
0.2125
0.953 1A02I
0,088 0.024
1,01 0.003
0,112 A 0L0L6
zld (Mode)
9.00
+00021
+0,015
(Mizn)
2,196 01612
1.,016
(Median)
0:115
0,13
0764
00.017
0.094
0.981 00021
0.009 A 0.023
0972
0,00 A
(Mixle)
2.10010.101
0.015
(Mazs)
1,121 0.000
0.568 0.015
0.015
SUPPORT
ald (Median)
0.856 0100
0,582 = 0.013
0.572 0.015
0.361 +0.015
0.667
0U900 = 0.056
0.884 A 0,005
1X64 0.045
zid (Mide)
0.221 2 0.051
0.552 0.016
0.522 1 0.0044
add (Meut)
1.713 0.220
0 0.014
IU665 0.013
GRSG
akd (Millian)
1.161 + 0L004
0.278 - 0014
RU672 0.010
0.590 0.005
0,283 01 106
1097-0002
0.14 0.094
1.000 0.03S
-0O0S% CLOUS
zid (Mode)
0,664 + 0.072
0.657 0.023
01.54 0.002
odd (Meun)
3,002 +: 1.497
0.561 0.037
01547
2 0,040
TMBlnmun
ald (Maliaa))
1.065 0.119
0.245 0.015
0.562 a 0.882
I1.485 0,030
0.835 0.604
0,904 =: 0.021
0.004 = 0.025
1,033 a 0.045
-0,08% a 0.025
zid (Mode)
0,409 0.069
0.566 0.004
0,0055
(Mean)
2,593 0.229
0.,03
0.563 0.036
BeeaMMKK
(Meliun)
0.3394
0.086 0.008
0.003
0,066
0,993 8.020
0.003 0.001
-0.007 0.0LO
07728 0.001
LGGCBM
(Median)
0.545 0.239
= 0.011
0.785 0.030
0.750
0.267
0.996
0.008 a 0,040
0.995 a 0.047
0,003
(Mile)
0,497 x 0.100
0.777
0.730 0.058
Case Study 4: Empirical Behavior of (0)x)
Given that the Asymmetric Laplace Distribution (ALD) has non-zero support over the negative time domain, we investigate whether the model assigns substantial probability mass to implausible event times & < O. Specifically, we analyze the empirical distribution of FO x) the predicted probability that an event occurs before time zero.
Table The 50th, 75th and 95th percentiles of the CDF estimation for 0. F0 x)) under the Asymmetric Laplace Distribution.
Dataset
50th Percentile
75th Percentile
95th Percentile
Norm linear
0.0001
0.0007
0.0018
Norm non-linear
1.9878-006
0.0001
0.0007
Norm uniform
2,9879c-05
0.0028
0,0124
Exponential
0.0194
0.0665
0.1204
Weibull
0.0015
0.0032
0.0046
LogNorm
0.0031
0,0109
0.0134
Norm heavy
1.1804c-06
2.6128c-05
0.0007
Norm med
4.22 4.2222c-06
3.5778c-05
0.0004
Norm light
1.1978-005
0.0001
0.0009
Norm same
7.8051c-07
4 18624c-06
0.0001
LogNorm heavy
0.0001
0.0014
0.0142
LogNorm med
0.0001
0.0007
0.0082
LogNorn light
0.0004
0.0024
0.0150
LogNorn same
0.0004
0.0021
0.0123
METABRIC
0.0068
0.0123
0.0292
WHAS
0.0046
0.0151
0.0507
SUPPORT
0.0957
0.1393
0.2035
GBSG
0.0248
0.0394
0.0668
TMBImmuno
0.0523
0.0681
0.0878
BreastMSK
0.0006
0.0008
0.0130
LGGGBM
0.0570
0.0842
0.1356
Table 6 reports the 50th, 75th, and 95th percentiles of (O x) across 21 datasets. The results indicate that this quantity remains close to zero in the vast majority of cases. Even at the 95th percentile, most values remain below 1-2%, suggesting that the model rarely assigns significant mass to invalid time regions. This empirically confirms that, despite the ALD's unbounded support, the learned parameters stay consistent with the temporal constraints of survival analysis.
Case Study 5: Limitations of ALD on SUPPORT
The model's s performance on the SUPPORT dataset comparatively lower than on other benchmarks. This shortcoming
37

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Learning Survival Distributions with the Asymmetric Laplace Distribution
largely be attributed to the assumption of the Asymmetric Laplace Distribution (ALD), which may not be universally suitable across datasets. In particular, the SUPPORT dataset exhibits pronounced skewness and narrow range of event times, as visualized in Figure 12. A substantial concentration of events occurs near zero, resulting in extreme quantile values and rendering the distribution challenging to calibrate, especially in the early time horizon (i.e., + 0).
SUPPORT
SUPPORT
2000
400
Count
Count
300
1000
200
500
100
0
0.0
0.5
1.0
1.5
2.0 y.train
2.5
3.0
3.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
y_test
Figure 12. Distribution of the SUPPORT dataset for the training and test set.
In response to this skewness, the model assigns high probability mass to early intervals while allocating limited capacity to later time points. This imbalance compromises calibration accuracy across the full temporal range. Nonetheless, the resulting calibration metrics remain within acceptable bounds and are generally comparable to those of the baselines. Furthermore, shown in Appendix C.1 and Table 2, our model still achieves competitive performance slope and intercept-based calibration metries, reaffirming its robustness despite the sppeset- limitations.
38


Reviews:
[{"review_id": "opCpFG5Yw7", "reviewer": "Reviewer_jgfN", "summary": "- Authors introduce a parametric survival analysis model, which utilises the Asymmetric Laplace Distribution in the Quantile Regression inspired loss function. \n\n- Inspired by the mean absolute deviation loss function, which models the hyperplane with median distance from all points, the quantile regression is a modelling approach where we use weighted absolute value as a loss function to model the hyperplane which separates p and 1-p observations in the optimal way. To estimate the parameters of the Asymmetric Laplace normal, authors present their architecture based on the shared encoder, residual skip connection  and 3 heads that predict the parameters of the Asymmetric Laplace Distribution. \n\n- The findings are supported by the experiments on the real data.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive and insightful feedback. \nWe appreciate the time and effort spent evaluating our work and will address the concerns and questions below.\n\n**R1[Other Comments Or Suggestions]:**\nIn response, we would like to emphasize that our method introduces a novel but simple loss formulation based on the ALD for survival modeling. \nCompared to non-parametric methods, our approach offers great flexibility and interpretability, as it enables closed-form expressions for key survival summaries (*e.g.*, mean, median and quantiles) and continuous cumulative distribution functions.\nIn contrast to existing (semi-)parametric methods, our approach is easy to train and offers balanced performance metrics, making it well-suited for modeling diverse survival patterns across datasets. \n\nFor the experiments, we conducted extensive evaluations in 14 synthetic and 7 real-world datasets to comprehensively assess our method under diverse survival scenarios.\nWe compare our approach against 8 competitive baselines (4 more added), covering a broad spectrum of survival models — (semi-)parametric and non-parametric —as well as both neural and non-neural architectures.\nOur evaluation spans 9 performance metrics, capturing the model's accuracy, discrimination, and calibration.\n\nAs a result, Table 2 summarizes the superiority of our method, and the Appendix further analyzes the details of the other baselines for a better understanding of the difference between our methods from both theoretical and empirical aspects. \n\nThanks to the insightful feedback from all reviewers, we have included additional results and case studies to highlight the following:\n\n1. The superiority of our method compared to mixture-based models (*e.g.*, DSM with Log-Normal and Weibull components) and non-neural survival models (*e.g.*, Generalized Boosted Models and Random Survival Forests) [https://anonymous.4open.science/r/ICML25/Fig1.png ];\n\n2. The flexibility of our method in capturing diverse survival patterns in real-world datasets [https://anonymous.4open.science/r/ICML25/Fig4.png ] and synthetic datasets [https://anonymous.4open.science/r/ICML25/Fig5.png; https://anonymous.4open.science/r/ICML25/Fig6.png ], complementing our synthetic benchmarks that already span a wide range of structural variations distinct from ALD;\n\n3. The robustness of our approach under challenging real-world conditions [https://anonymous.4open.science/r/ICML25/Fig2.png; https://anonymous.4open.science/r/ICML25/Fig3.png ], such as pronounced skewness and heavy censoring, provides further evidence of its reliability beyond the varied censoring and skewness levels considered in our initial experiments.\n\nWhile time constraints limited the extent of results we could include in this rebuttal, we would be more than happy to conduct and report additional experiments or case studies if the reviewers find them helpful.\n\n**R2[Questions For Authors 1-2]:**\nWe did not explore alternative architectures in our work for the following reasons:\n\n1. Dataset simplicity. Most of our datasets are low-dimensional (the highest being SUPPORT with 14 features). \nGiven this, we focused on fully connected architectures, which offered sufficient capacity while avoiding overfitting. Empirically, we found them to generalize well across datasets with varying feature sizes and censoring levels.\n\n2. Comparability with prior work. Most recent neural methods for survival analysis, such as CQRNN, DeepSurv, DeepHit and DeepSurv, also use fully connected architectures (see Appendix B.3). \nTo ensure a fair comparison, we adopted the same architectural choice.\n\nWhile an exhaustive architecture search was beyond the scope of this work, we fully agree that this is a valuable direction. \nWe would be happy to explore alternative configurations in future work and greatly appreciate any concrete suggestions regarding alternative architectures.\n\n**R3[Questions For Authors 3]:**\nThe point about domain shift is excellent and domain shift is indeed an important and valuable direction for future research.\nWhile such an analysis falls outside the scope of our current work, we fully acknowledge its significance. \nIn future extensions, we plan to incorporate experiments under domain shift scenarios. \nWe will also add a discussion paragraph in the final version to explicitly reflect this limitation and outline potential directions for addressing it.\n\n**R4[Questions For Authors 4]:**\nThank you for this valuable question. \nYour concern aligns closely with the feedback from **Reviewer 6hhK**, and we have further conducted additional case studies following the evaluation settings used in DSM. As illustrated in Fig2 and 3 [https://anonymous.4open.science/r/ICML25/Fig2.png; https://anonymous.4open.science/r/ICML25/Fig3.png ], our method consistently demonstrates superior or competitive performance across different censoring regimes and event-time quantiles. The full analysis can be found in the reply for **Reviewer 6hhK**.", "claims_and_evidence": "Claims are well supported. Evidence is provided in the form of:\n-  the derivations, \n-  experiments,\n- relevant citations.", "methods_and_evaluation_criteria": "Yes. \n\n - Authors carry experiments on the synthetic datasets and 7 real-life dataset related to survival statistics task to support their claim. \n\n - Authors vary the number of the features, sizes of the datasets and various  censored fractions of the datates. \n\n - Each experiment is repetead 10 times.", "theoretical_claims": "Verified Corollary 3.2 in Appendix A.1. Seems correct.", "experimental_designs_or_analyses": "Authors compare the performance against other 4 benchmark models: CQRNN, DeepSurv, DeepHit and LogNorn", "relation_to_broader_scientific_literature": "The model design seems to be novel.", "essential_references_not_discussed": "- All relevant text books coverring Quantile regression are cited.\n- The benchmark models are well described and discussedI .", "other_strengths_and_weaknesses": "Well written paper, easy to follow, methodology is great.", "other_comments_or_suggestions": "Not convinced that the results itself is strong enough as a topic to publish the paper (however methodologically is the paper well composed and written). Happy to be convinced though.", "questions_for_authors": "1. Looking at the size of the architecture and datasets: when it comes to selection of the architecture, how can you achieve that given the sample and architecture size, the networks are generalising reasonably well rather than memorising.\n2. Did you experiment with different architectures? \n3. Did you consider experiments containing shifts of the test sets to verify how well the selected method generalises to domain shift?\n4. How does the model behave for the edge cases, i.e. is your methodology suitable for the extreme quantiles?", "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": "oSVW1lSbTF", "reviewer": "Reviewer_rnhn", "summary": "This paper proposes a parametric survival analysis model which uses asymmetric Laplace distributions (ALDs) to represent survival distributions, where the non-linear dependence of ALD’s parameters on static covariates is modeled by neural networks. The experiments on synthetic and real-world data confirmed the model's validity.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful and constructive feedback, and for generously suggesting many references that we had not previously considered.\nWe added 4 more baselines and the results are available in Table 2 [https://anonymous.4open.science/r/ICML25/Fig1.png ].\nBelow, we provide a detailed discussion comparing our proposed method with the baselines you mentioned and address your specific concerns.\n\n**R1[Claims And Evidence]:**\n\n**Compared to parametric methods, our approach offers two key advantages:**\n\n1. For our method, both the PDF and CDF of the ALD are available in closed form and are smooth to their parameters.\nWhile it is possible to use any distribution to model survival functions, some commonly used alternatives, such as the Gamma distribution, often suffer from gradient instability during training. \nSpecifically, the Gamma distribution lacks a closed-form expression for its CDF, which instead relies on numerically approximated special functions (*i.e.*, the incomplete Gamma function). \nThese operations introduce significant numerical instability when computing gradients via backpropagation, leading to unstable training dynamics or poor convergence. \nIn contrast, the ALD offers closed-form expressions with stable gradients, making it well suited for optimization in neural network-based survival models.\n\n2. Experimental results on 14 synthetic and 7 real-world datasets across 9 metrics show that our method consistently achieves strong performance, even under high skewness and heavy censoring. \nAs shown in Table 2, it outperforms LogNormal in 60\\% of 189 settings, with worse performance in less than 5\\%, highlighting its robustness and generalizability.\n\n**Compared to nonparametric methods, our approach offers two key advantages:**\n\n1. Our method provides continuous and closed-form estimates of CDFs and summary statistics, avoiding the discretization inherent in many nonparametric models.\nNotably, these discretization CDFs can introduce significant approximation error, especially when the interval between time steps is large, which limits their precision in capturing fine-grained temporal dynamics.\n\n2. The parametric nature of the ALD enables faster training and inference, with fewer parameters and lower memory overhead compared to nonparametric methods. \nFor example, our model only needs to estimate 3 parameters per instance. \nIn contrast, nonparametric approaches, such as DeepHit, require computing and storing hundreds of discrete CDF values to achieve comparable resolution, resulting in significantly higher computational costs.\n\n**Compared to the mixture models, our approach offers two key advantages:**\n\n1. DSM models the survival function as a mixture of $K$ Log-Normal or Weibull components, leading to increased computational complexity as $K$ grows. \nIn contrast, our ALD-based model is simple and estimates only 3 parameters per instance, enabling faster training and inference.\n\n2. In practice, DSM is highly sensitive to the choice of parameter $K$ and architectural design, partly due to its loss function, which aggregates multiple distributional components. \nThis can make it vulnerable to instability when some mixture components fail to capture the underlying survival patterns effectively.\nIn contrast, our model relies on simple neural network hyperparameters and delivers more stable performance across diverse settings, without the need for extensive tuning.\n\n**R2[Experimental Designs Or Analyses, Essential References Not Discussed]:**\nWe will include and discuss these important works in the related work section. \nWe have implemented both GBM and RSF, as reported in Table 2. \n\n**Compared to these two standard nonparametric approaches:**\n\n1. Our method outperforms GBM in 52\\% of the 189 benchmark settings while underperforming in fewer than 7\\% of them. Similarly, it surpasses RSF in 55\\% of the settings and performs worse in fewer than 9\\%, demonstrating strong robustness and generalizability across diverse survival scenarios.\n\n2. Ensemble-based nonparametric methods such as GBM and RSF typically require training 100 or more decision trees, which can be computationally intensive and memory demanding, especially for large datasets or when tuning hyperparameters.\nMoreover, once trained, our method enables fast evaluation of continuous survival functions in closed form, whereas tree-based models require traversing ensembles for each prediction.\n\n**R3[Questions For Authors]:**\nSemiparametric models like the Cox proportional hazards model incorporate censored samples only indirectly through the risk set in the partial likelihood. \nAs a result, when censoring is high, the model relies on very limited observed events, reducing its statistical efficiency. \nIn contrast, our parametric approach models both censored and uncensored data explicitly through a full likelihood, allowing it to make better use of all available information and remain robust under high censoring.", "claims_and_evidence": "The primal contribution of this paper is to introduce asymmetric Laplace distributions (ALDs) to represent survival distributions. But it seems not to be clear why ALDs should be adopted or why ALDs are superior to other parametric/nonparametric approaches. For example, do ALDs have any advantages over mixture models (e.g., mixture of log-normal or gamma distributions) in terms of accuracy or computational cost?", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense.", "theoretical_claims": "Theoretical claims seem to be correct.", "experimental_designs_or_analyses": "The compared reference models seem to focus heavily on neural network-based models. More standard nonparametric approaches in survival analysis (e.g., random survival forest) should be included in the comparison. See below about essential references.", "relation_to_broader_scientific_literature": "This paper provides an accurate survival analysis, which could be beneficial especially in clinical fields.", "essential_references_not_discussed": "To overcome the limitation of Cox proportional hazards model, a vast number of non-neural network survival models have been proposed that replace the log-linear parametric function with a non-linear one. Especially, generalized boosted models [a], random survival forests [b], and Gaussian process models [c,d,e], which are not cited in the paper, should be discussed in Related Work. Some of them should be included in the comparative experiments (e.g., generalized boosted models can be easily implemented via gbm(distribution = ’coxph’) in R).   \n\n[a] Ridgeway G, Developers G (2025). gbm: Generalized Boosted Regression Models. R package version 2.2.2, https://github.com/gbm-developers/gbm.\n\n[b] Hemant Ishwaran, Udaya B. Kogalur, Eugene H. Blackstone, and Michael S. Lauer. Random survival forests. The Annals of Applied Statistics, 2(3):841–860, 2008.\n\n[c] Tamara Fernández, Nicolás Rivera, and Yee Whye Teh. Gaussian processes for survival analysis. In Advances in Neural Information Processing Systems 29, 2016.\n\n[d] Minyoung Kim and Vladimir Pavlovic. Variational inference for Gaussian process models for survival analysis. In Uncertainty in Artificial Intelligence, pages 435–445, 2018.\n\n[e] Hideaki Kim. Survival Permanental Processes for Survival Analysis with Time-Varying Covariates. In Advances in Neural Information Processing Systems 36, 2023.", "other_strengths_and_weaknesses": "Strength\n- The validity of the proposed model was evaluated on many real-world data.\n\nWeakness\n- See above.", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "I cannot follow the sentence on pp. 4, \"However, semiparametric models face challenges in effectively handling censored data, particularly when censoring rates are very high.\" Could you explain why semiparametric approaches work poorly in high-rate censoring?", "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": "RUksbjmeY3", "reviewer": "Reviewer_QAQg", "summary": "The paper proposes a parametric survival analysis method based on asymmetric Laplace distribution. It enables predicting continuous distribution-based predictions, unlike existing discretized nonparametric methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank you for your careful reading and constructive feedback. \nWe added four more baselines shown in Table 2 [https://anonymous.4open.science/r/ICML25/Fig1.png ] and more comparisons (See the reply for **Reviewer rnhn**) will be added.\nBelow, we will address your concerns.\n\n**R1[Claims And Evidence 1]:**\nUnlike discretized nonparametric approaches, our method estimates only 3 parameters of the ALD per instance, providing computational efficiency and smooth, closed-form CDFs.\nIn contrast, models such as DeepHit must balance computational cost and accuracy by discretizing time, which introduces approximation errors, especially, under coarse time grids, and limits their ability to capture fine-grained survival dynamics.\n\nBeyond theory, across 14 synthetic and 7 real-world datasets, our method outperforms DeepHit in 73% of 189 benchmark settings and underperforms in less than 4% (Table 2), underscoring the strength of continuous modeling. \nThis advantage is most evident in the IBS metric, where our method achieves superior performance across all 21 datasets. \nNotably, IBS evaluates both discrimination and calibration over time, making it especially sensitive to distributional smoothness and continuity.\n\n**R2[Claims And Evidence 2]:**\nIndeed, any parametric model makes certain assumptions about the underlying distribution.\nWe adopt the ALD assumption primarily due to its strong theoretical and practical connections to quantile regression.\nThe widely used pinball (checkmark) loss, foundational in methods like CQRNN, is essentially derived from ALD. However, such methods lack full distributional modeling and continuity.\n\nOur work bridges this gap, retaining the empirical strengths of quantile-based approaches while introducing a fully parametric, continuous, and interpretable survival model. As noted by **Reviewer 6hhK**, this represents a principled evolution from loss-based to distribution-based modeling.\nOther parametric options (*e.g.*, Gamma, Log-Normal) are possible but often suffer from the absence of closed-form CDFs, unstable gradients, and reduced robustness (see the reply for **Reviewer rnhn** and Appendix A.2). In contrast, ALD offers stable training and strong empirical results, making it a compelling choice.\n\n**R3[Methods And Evaluation Criteria 1]**:\nTo the best of our knowledge, there is limited prior work exploring the use of Fourier or Taylor series expansions for continuous survival distribution modeling, making this a novel and interesting direction. \nWe would greatly appreciate any specific references or formulations the reviewer has in mind for further consideration, and we are open to exploring this line of research in the revision.\n\nWhile many alternatives exist, our goal is to balance simplicity, interpretability, and performance. The ALD-based framework achieves this effectively, as demonstrated in our results.\n\n**R4[Methods And Evaluation Criteria 2, Experimental Designs Or Analyses]:**\nWe now provide new analyses to support this claim. \nFig 4 [https://anonymous.4open.science/r/ICML25/Fig4.png ] shows the flexibility of our method and DeepHit in capturing diverse survival patterns in the 7 real-world datasets.\nSpecifically, we applied K-means clustering to the estimated parameters from our model, with the number of clusters set to 6.\nThis allows us to visualize 6 representative survival patterns, where each CDF curve corresponds to the average parameters within a cluster.\nAs shown in Fig 4, both our method and DeepHit are capable of capturing diverse survival patterns in real-world datasets. \nHowever, the patterns generated by DeepHit often appear less realistic. \nIts CDFs consistently converge to 1 across all identified patterns, suggesting that every individual eventually experiences the event. \nThis outcome contradicts the existence of censoring in the data, where a non-zero censoring rate implies that a portion of individuals should remain event-free over time. \nAlthough the true underlying distribution is not observable in real-world settings, this behavior illustrates DeepHit’s limited ability to model long-term survival.\n\nTo further compare what survival patterns can the proposed method succeed in and when it fails, we leverage the fact that ground-truth CDFs are available for each instance in the synthetic datasets. \nThis allows us to perform the same clustering analysis as in the real-world setting. \nSpecifically, Fig 5 [https://anonymous.4open.science/r/ICML25/Fig5.png ] presents the worst-estimated instance (based on Wasserstein distance) within each of the 6 clusters for the best-performing synthetic dataset, comparing our method and DeepHit. \nFig 6 [https://anonymous.4open.science/r/ICML25/Fig6.png ] repeats this for the most challenging dataset.\nThese results demonstrate that our method consistently yields lower Wasserstein distances across different survival patterns, indicating a more reliable estimation of survival distributions, even in difficult scenarios.", "claims_and_evidence": "The paper mentions several limitations of the existing methods 40-54, such as strong assumptions about the underlying distribution, only predicting discretized distribution.\n\nThe experiments show the proposed method is significantly better than the selected baselines. However, there seems to be not many discussion about how important each module is. Specifically, \n1. the paper claims predicting continuous distribution is an advantage. I acknowledge this and it is achieved obviously through the theoretical foundation. However, how many benefits are brought by this?\n2. The paper claims the strong assumptions about the underlying distribution. Why does the proposed method release the assumption and what is some evidence to show that?", "methods_and_evaluation_criteria": "The proposed method predicts the parameters of the asymmetric Laplace distribution, and thereby predicts the survival distribution. The corollary 3.2 provides the justification of using asymmetric Laplace distribution, that tells the relationship between it and quantile regression. This method makes sense and works as expected, but the contribution is limited, mainly because of these:\n1. There are many ways to model a continuous parametrized distribution other than Asymmetric Laplace Distribution. For example, predicting the coefficients of a Fourier transform, or Taylor series. They are also continuous in nature. What are the unique advantages of ALD?\n2. The paper claims the proposed method offers superior flexibility in capturing diverse survival patterns. But there seems no experiments supporting this claim.", "theoretical_claims": "The theoretical claims make sense to me. But the detailed math is not carefully checked.", "experimental_designs_or_analyses": "The experiment statistics looks good, but as mentioned in previous sections, it does not sufficiently support the claims made by the paper. If possible, I would recommend adding more experiments:\n1. baselines of predicting the coefficients of a Fourier transform or Taylor series.\n2. compare what survival patterns can the proposed method succeeds and when it fails.", "relation_to_broader_scientific_literature": "I am unsure about this question.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "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": "kRlvYa1U0k", "reviewer": "Reviewer_6hhK", "summary": "The paper introduces a novel parametric survival analysis framework based on the Asymmetric Laplace Distribution (ALD). By formulating the survival problem through ALD, the authors derive closed-form expressions for key distributional summaries (e.g., mean, median, mode, and quantiles) and propose an architecture where a shared encoder feeds into three independent heads to predict the ALD parameters. The model is trained via maximum likelihood estimation (MLE) with tailored loss functions for both observed and censored data. Comprehensive experiments on 14 synthetic and 7 real-world datasets demonstrate that the proposed approach often outperforms established methods such as LogNorm MLE, DeepSurv, DeepHit, and CQRNN in terms of predictive accuracy, discrimination (using C-index metrics), and calibration.\n\n**Updates after Rebuttal**\n\nI thank the authors for their rebuttal. Although I am not sure if including additional results in anonymous githubs is legitimate, the revised experiments look much more promising and well-address my concern on empirical evaluation. I will increase my score by one point.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank you for your comprehensive evaluation and insightful comments. \nWe also appreciate the detailed review of our theory, experiments, and connections to the broader literature. \nBelow we respond to the specific points raised:\n\n**R1[Claims And Evidence, Other Strengths And Weaknesses (Weaknesses), Questions For Authors 1]:**\nWe would like to emphasize that our benchmark covers a diverse range of survival scenarios with varying censoring and skewness levels. All 21 datasets used in our experiments differ in their censoring rates, and for synthetic datasets (*e.g.*, multivariate Normal, Log-Normal), we explicitly vary the censoring levels to create more challenging conditions. Additionally, distributions like LogNormal and Exponential inherently introduce strong skewness. Importantly, these synthetic datasets are distributionally distinct from ALD, yet our method performs well across them, demonstrating robustness beyond the assumed distribution.\nThe latest Table 2 [https://anonymous.4open.science/r/ICML25/Fig1.png ] (4 more baselines added) highlights the superiority of our method compared to a broad spectrum of baselines, including (semi-)parametric and nonparametric models, covering both neural and non-neural architectures. \nAlthough our method shows relatively weaker results on the SUPPORT dataset, the same trend is seen in several baselines. Nevertheless, our performance remains competitive and stable across metrics.\n\nFollowing your suggestions, we added case studies inspired by DSM to evaluate robustness under high censoring and at key quantiles (25%, 50%, 75%). Fig 2 and 3 [https://anonymous.4open.science/r/ICML25/Fig2.png; https://anonymous.4open.science/r/ICML25/Fig3.png ] present results on METABRIC and SUPPORT datasets, which differ significantly in their censoring and distributional properties. \nOur method consistently achieves strong or competitive time-dependent concordance scores across censoring levels. \nOn METABRIC, it performs especially well at higher quantiles, where skewness and censoring are more pronounced. \nAlthough our method shows relatively weaker results on SUPPORT, this is likely due to the substantial deviation of its underlying distribution from the ALD assumption. However, our method still performs comparably to DSM-based baselines without significant degradation.\n\nWe also appreciate the point and agree that scaling to higher-dimensional datasets is important. \nWhile our current experiments focus on structured survival datasets with up to 14 covariates (*i.e.*, SUPPORT), our method is lightweight, requires only 3 estimated parameters of ALD per instance, and leverages compact neural architectures. \nIt does not rely on large time grids or mixture components, making it computationally efficient. \nIntuitively, we expect our method to retain its advantages even in higher-dimensional settings due to its compact parameterization and stable training behavior. \nWe are currently exploring extensions to such settings and plan to evaluate their scalability and performance on high-dimensional survival data in future work.\n\n**R2[Experimental Designs Or Analyses, Essential References Not Discussed]:**\nIn our current version, we have already reported results for DeepCox (referred to as DeepSurv in our initial submission), and have additionally incorporated new baselines such as DSM with Log-Normal and Weibull components, as well as classical non-neural models like GBM and RSF.\nFor other models such as Neural Frailty Machine and Deep Cox Mixture, we will include a detailed discussion in the Related Work section (similar to the additional comparisons presented in our response to **Reviewer rnhn**) and are happy to incorporate further empirical evaluations in the revision if needed.\n\nHere, we would like to highlight that, compared to these mixture-based models, our approach offers two key advantages:\n\n1. Computational efficiency. DSM models the survival function as a mixture of $K$ Log-Normal or Weibull components, which increases computational complexity as $K$ grows. In contrast, our model requires estimating only 3 parameters per instance, enabling faster training and inference with a much simpler architecture.\n\n2. Performance stability. DSM is sensitive to $K$ and architecture choices. Misaligned components can lead to instability. In contrast, our model relies on straightforward neural network hyperparameters and exhibits stable performance across diverse datasets without extensive tuning.\n\n**R3[Questions For Authors 2]:**\nWhile we acknowledge that more sophisticated loss functions or architectural enhancements could potentially further improve performance, our current focus is on maintaining a simple and interpretable design.\nDespite its simplicity, our approach already achieves strong empirical results. \nWe highly agree that integrating more advanced components is an interesting and valuable direction and plan to explore this in future work.", "claims_and_evidence": "* While the paper asserts that the ALD-based method is broadly effective, its reliance on the ALD assumption is problematic in cases with highly skewed event distributions (e.g., the SUPPORT dataset). The evidence does not fully demonstrate robustness when the underlying survival distribution deviates significantly from an ALD. This point requires more convincing ablation studies or experiments.\n* The paper makes claims regarding the efficiency and stability of the proposed model, bolstered by the use of a shared encoder and residual connections. Still, the experiments do not sufficiently address how the approach scales to very high-dimensional data or extremely censored datasets --- especially some heavily censored datasets.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes I chwecked all theoretical proofs and claims.", "experimental_designs_or_analyses": "I have checked all experiment designs and analyses. Essentially I think there are several baselines missing, such as DeepCox, Deep Survival Machine, Neural Frality Machine, and Deep Cox Mixture.", "relation_to_broader_scientific_literature": "The paper’s contributions are deeply rooted in—and extend—the established literature on survival analysis and quantile regression. In particular, it builds on classical parametric models (such as exponential, Weibull, and log‐normal models) by introducing a flexible, ALD‐based approach that provides closed-form expressions for key survival summaries (mean, median, quantiles). This idea is an evolution of earlier work in quantile regression where the asymmetric Laplace distribution was used as a working likelihood (see Koenker and Bassett, 1978; Yu and Moyeed, 2001) to capture the conditional quantiles of the response.\n\nFurthermore, by integrating a neural network architecture with a shared encoder and independent heads for estimating the ALD parameters, the paper aligns with recent deep learning efforts in survival analysis (e.g., DeepSurv, DSM, Deep Cox, Neural Frality Machine, DeepHit, and CQRNN) that aim to model complex, non-linear relationships and to address limitations of traditional models like the Cox proportional hazards model.", "essential_references_not_discussed": "* Deep Survival Machine (DSM) —which models survival times using a learned mixture of parametric distributions—shares the goal of providing a full survival distribution without relying on the proportional hazards assumption. However, while DSM flexibly combines several parametric components (such as Weibull or log-normal distributions) to capture heterogeneous survival patterns, the proposed ALD-based method offers a closed-form formulation for key event summaries (mean, median, quantiles) via maximum likelihood estimation. This paper was not discussed and compared.\n* Similarly the infamous DeepCox model in this field was also not compared.", "other_strengths_and_weaknesses": "Strengths:\n\n* The paper offers detailed derivations, theoretical insights, and even proofs (presented in the appendix) that explain the loss formulation and the properties of the ALD. This depth of analysis underpins the methodological contributions.\n* The use of the ALD provides a neat way to capture survival distributions in closed form. This results in efficient computation of various summary statistics, which enhances interpretability compared to discretized methods.\n\nWeaknesses:\n\n* The model’s performance is contingent upon the assumption that survival times are well modeled by an ALD. In cases such as the SUPPORT dataset, where the event distribution is highly skewed, this assumption may not hold, potentially affecting calibration.\n* Experiments on highly censoring scenarios (e.g., refer to the experiments by DSM) are encouraged to more comprehensively evaluate the model at tails.", "other_comments_or_suggestions": "N.A.", "questions_for_authors": "* How sensitive is the proposed ALD-based method when the true underlying survival distribution deviates from the ALD assumption? Can you provide results from ablation studies or additional experiments that test this robustness?\n* In datasets with very high censoring rates (e.g., the experiment settings of DSM), what modifications or additional regularizations might be necessary to maintain reliable calibration and discrimination? Have you considered how the method might be adapted to such scenarios?", "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:

Meta-review: The reviewers overall leaned toward acceptance after the discussion period, where the authors provided detailed comments including running additional experiments. Overall, the proposed ALD method is a valuable contribution to the survival analysis community and achieves reasonably good experimental results. There remain concerns that it is not always better than other survival methods, which I think is a reasonable concern although I wouldn't expect such a method (or just about any other method) to be uniformly better/the "best" at every dataset.
Decision: Accept (poster)