Uncertainty-Calibrated Prediction of Randomly-Timed Biomarker Trajectories with Conformal Bands
Authors: Vasiliki Tassopoulou
Charis Stamouli
Haochang Shou
George J. Pappas
Christos Davatzikos
Venue: NeurIPS 2025 poster ()
Keywords: Biomarker Trajectories
Conformal Prediction
Uncertainty Quantification
Primary area: machine_learning_for_sciences

Abstract: We introduce a novel conformal prediction framework for constructing conformal prediction bands with high probability around biomarker trajectories observed at subject-specific, randomly-timed follow-up visits. Existing conformal methods typically assume fixed time grids, limiting their applicability in longitudinal clinical studies. Our approach addresses this limitation by defining a time-varying nonconformity score that normalizes prediction errors using model-derived uncertainty estimates, enabling conformal inference at arbitrary time points. We evaluate our method on two well-established brain biomarkers—hippocampal and ventricular volume—using a range of standard and state-of-the-art predictors. Across models, our conformalized predictors consistently achieve nominal coverage with tighter prediction intervals compared to baseline uncertainty estimates. To further account for population heterogeneity, we develop group-conditional conformal bands with formal coverage guarantees across clinically relevant and high-risk subgroups. Finally, we demonstrate the clinical utility of our approach in identifying subjects at risk of progression to Alzheimer’s disease. We introduce an uncertainty-aware progression metric based on the lower conformal bound and show that it enables the identification of 17.5\% more high-risk subjects compared to standard slope-based methods, highlighting the value of uncertainty calibration in real-world clinical decision making. We make the code available at \href{https://github.com/vatass/ConformalBiomarkerTrajectories}{\texttt{github.com/vatass/ConformalBiomarkerTrajectories}}.

Uncertainty-Caiibrated Prediction of Randomly-Timed Biomarker Trajectories with Conformal Bands
Vasiliki Tassopoulou 1.*
Charis Stamouli
Haochang Shou George J. Pappas"
Christos Davatzikos'
1 Center for and Data Science for Integrated Diagnostics (A12D), Perelman School of Medicine,
University of Pennsylvania, Philadelphia, PA, USA
Department of Electrical and Systems Engineering, University of Pennsylvania, PA, USA
*Department of Biostatistics and Epidemiology. University of Pennsylvania, PA, USA
E-mail: vtass @ seas. wee.ss.penn.ce.. stamouli @ seas. www..sewnn.c...e.... hshou @wwnneedciiineuuummeme www.iiiie..nnneceee pappasg @ seas. seas. massuupenn.ddu christos. datattoodavtziiosseee pewnoseeieeeiiciineeeeeeeee.ee
Abstract
We introduce novel conformal prediction framework for constructing conformal prediction bands with high probability around biomarker trajectories observed subject-specific, randomly-timed follow-up visits. Existing conformal methods typically assume fixed time grids, limiting their applicability longitudinal clin- ical studies. Our approach addresses this limitation by defining a aare-ng nonconformity score that normalizes prediction errors using model-derived uncer- tainty estimates, enabling conformal inference at arbitrary time points. We evaluate our method on two well-established brain biomarkers biomakerrs-hippoempall and ventric- a range of standard and state-of-the-ar predictors. Across
ular volume-usinn models, our conformalized predictors consistently achieve nominal coverage with tighter prediction intervals compared to baseline uncertainty estimates. To further account for population heterogeneity, we develop group-conditions conformal bands with formal coverage guarantees across clinically relevant and high-risk subgroups. Finally, we demonstrate the clinical utility of our approach in iden- tifying subjects risk of progression to Alzheimer's 's disease. We introduce uncertainty-awe progression metric based on the lower conformal bound and show that it enables the identification of 17.5% more high-risk subjects com- pared to standard slope-based methods, highlighting the value of uncertainty cal- ibration in real-world clinical decision making. We make the code available www.bb..coomeecoo waaasss/ooooooomooo Biw.aaeeeeaaeaaeeeeeeeoooooo
Introduction
Predicting biomarker trajectories is crucial for monitoring disease progression and improving patient outcomes through carly intervention. In neurodegenerative diseases such as Alzheimer's 's, volumetric biomarkers- most notably hippocampal atrophy- are closely associated with clinical decline and are widely used to track disease progression [1-3]. Given the clinical importance of biomarker trajectories, recent advances machine learning have led to growing number methods designed to forecast biomarker evolution from longitudinal data [4-8]. However, inter-individual variability in disease progression and measurement noise inherent imaging data make predictions inherently uncertain-mode cannot determine exact future values with complete confidence. If uncertainty
'qqul contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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is not properly accounted for, predictions can cause high-risk subjects to be misclassified as stable. Such potential misclassifications hinder the deployment of biomarker predictors in clinical workflows, where failure to identify high-risk individuals can delay urgently needed intervention or exclude such individuals from trials. To enable safe and reliable decision-making in healthcare, uncertainty calibration of the learned predictors is important.
popular approach for equipping predictors with reliable uncertainty estimates is given by conformal prediction [9-11]. Conformal prediction is highly versatile, accommodating any black-box predictive model (e.g.. neural networks) and any data distribution (e.g., non- Non-Gaussian). The first step in the approach is to evaluate the learned model's 's prediction error, as characterized by a so-called nonconformity score, on held-out calibration dataset. Assuming that the calibration and test data are sampled independently from the same probability distribution', the prediction error on the calibration data can be used to estimate the prediction error on the test data. Thus, model predictions can be turned into regions that are guaranteed to cover the true observations with arbitrarily high probability.
In the context of trajectory prediction, existing conformal methods have focused on fixed-time trajectories [12-16]. However, biomarker datasets typically contain randomly-timed biomarker trajectories due to missed visits and variable scheduling in clinical studies (see Figure 1). Such trajectories create a more complex setting for designing conformal prediction bands with guaranteed coverage properties, which are critical for safely delivering early diagnoses and effectively guiding clinical interventions.
In this paper, we propose novel conformal prediction method tailored to the setting of randomly-timed biomarker trajectories. Our ap- proach derives prediction bands that are guar- anteed cover the entire true biomarker tra- jectory with arbitrarily high probability. We apply our method forecasting hippocampal and ventricular-volum trajectories two well- established biomarkers monitor the onset and progression of Alzheimer's 's disease (AD). To better capture variability across clinically mean- ingful subgroups. including high-risk individu- als such as those diagnosed with Mild Cognitive Impairment (MCI), we design group-conditional conformal predictors that offer formal coverage guarantees within covariate-defined strata. We
Subjjct
Subjject
Booneeor Valae
Suub
Subbjoct
Timse (in years)
Figure 1: Illustration of randomly-timed biomarker trajectories from five subjects. Time is measured with respect to the first biomarker observation of each subject. Note the varying trajectory length and the different time points of observations across subjects.
evaluate these predictors across five key stratification--exx sex, race, diagnosis, education level, and APOE4 allele underrepresented demographic subgroups. Beyond evaluating calibration, we translate conformal uncertainty into downstream decision task: identifying MCI patients at high risk of progression to Alzheimer's s disease. We introduce a trajectory-based risk score derived from the lower conformal bounds and show that this subject selection in scenarios such as trial enrichment or early intervention planning.
by clinical risk and the need for equitable performance across
metric enables more inclusive and safety-aware
Our contributions can be summarized as follows: 1) We introduce new conformal prediction frame- work for randomly-timed biomarker trajectory forecasting. Our method defines a nonconformity score over multiple time points-simila in spirit to Yu et al. [17]. Cleaveland et al. [18] the key distinction that our time points are not fixed across individuals but randomly distributed, reflecting real-world clinical follow-up schedules. 2) We test our approach on two well-established neurodegenerative biomarkers bionatkrrs-hippoaampll volume and ventricular gaatumn--iing real clinical data. By conformalizing range of standard and state-of-the-aar predictors, we demonstrate that our method achieves the nominal coverage while maintaining tight prediction bands, outperform- ing baselines that rely solely model-specific uncertainty estimates. 3) To address population heterogeneity, we incorporate covariate-based based stratification into the calibration step and develop group-conditional conformal predictors with formal subgroup-level coverage guarantees. We empiri- cally validate these guarantees across five demographic and clinically relevant stratifications: sex, race, diagnosis, education level, and APOE4 allele status. 4) We demonstrate the clinical utility of our
"Conformal methods typically require that the calibration and test data are exchangeable milder condition than that of independent and identically distributed data (see Appendix A).

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seprtaainty-alibratte conformal prediction bands in a downstream decision task, where the goal is to identify MCI subjects at high risk of progression Alzheimer's disease. To this end, we introduce a novel uncertainty-awart progression metric- rate of change bound-which leverages conformal uncertainty to provide an estimate of the biomarker's rate of change over time. This metric enables the identification of 17.5% more high-risk MCI subjects who progress to Alzheimer's disease compared to standard approaches.
2 Related Work
Uncertainty-Calibratee Prediction of Biomarker Trajectories. Uncertainty-calibaaee predictors predicccooo for biomarker trajectories integrate uncertainty quantification into their architecture. Bayesian methods achieve this by employing probabilistic frameworks to model uncertainty. For instance, Gaussian Processes (GPs) [19] utilize kernel-defined priors over functions to encode structural assumptions, such as smoothness. Their posterior distributions quantify prediction uncertainty through covariance-based inference under Gaussian likelihoods. Several studies [8, 20-24] involve predictors with inherent uncertainty measures based on Gaussian distribution assumptions. In contrast, Lin et al. [25] introduce conformal variant for fixed-time time biomarker trajectories, ensuring asymptotic longitudinal coverage without any distributional assumptions.
Conformal Prediction for Fixed-Time Trajectories. Conformal methods for predicting fixed-time trajectories have been developed for settings with single time series data [26-28) and multiple time series data [13, 17, 18, 25, 29, 30]. Our setting differs from the above, involving data with generally non-temporal inputs (e.g., sex) and randomly-timed trajectory outputs (e.g., hippocampal-volurg measurements taken at random time points for each subject). We focus on the latter line of works, whose setting is special case of ours.
Lin et al. [25] present conformal prediction method with asymptotic longitudinal coverage guar- antees. Stankeviciute et al. [13] and Lindemann et al. [30] design conformal prediction bands with finite-sample coverage guarantees, by leveraging union bound argument. Sun and Yu [29] propose using copulas to model the uncertainty of predictions at multiple time points. Their approach ensures coverage with narrower prediction bands, but is limited to situations with ample calibration data, as noted by the authors. Yu et al. [17] and Cleaveland et al. [18] overcome this limitation by defining a normalized nonconformity score jointly over multiple time points. Their framework provides narrow prediction bands with simultaneous coverage guarantees over the entire trajectory length. In the next section, we extend their approach to our setting of randomly-timed trajectories, where observations are collected at varying time points.
3 Conformal Prediction for Randomly-Timed Trajectories
In this section, we introduce a novel conformal prediction method tailored to settings with randomly- timed trajectories. In Subsection 3. we present the problem of conformal prediction for data with vector-valued inputs and randomly-timed trajectory outputs. In Subsection 3.2, we derive conformal prediction bands with coverage guarantees for our setting, inspired by the approach of Yu et al. [17] and Cleaveland et al. [18] for fixed-time trajectories.
3.1 Problem Formulation
Our setting of randomly-timee -imeed trajectories is characterized by triplet of random variables (X, YY, where X E IRd denotes the input, T C N the set of time points at which observations Y, E IR are collected, and Y := (Y; t T the corresponding trajectory output. In the context of biomarkers, the triplet (X, T, Y' can be interpreted, for instance, in the following way: i) the input X may involve demographic covariates (e.g., sex, race) as well as the biomarker observation on the patient's 'ts first clinical visit, ii) the set T may include the times of the patient's 's subsequent visits, and iii) the output Y the corresponding biomarker observations at these times. We note the implicit dependence of the trajectory output Y on the set of time points T.
Suppose we have a dataset D = ((x(), T(9, Y() 1-- such that for each test example (X, TYY). the random variables (X) T(), Y())..., (X((), ((N), (()), and (X, T, Y) are exchange- able'. We note that exchangeability is weaker than the standard assumption of independent and identically distributed data, imposing a reasonable condition for biomarker datasets, where each triplet (x ((), (i) corresponds to different subject. Our goal is to leverage the dataset D and
Exxchaneaability implies that the joint probability distribution of a sequence of random variables remains unchanged under any permutation of the variables (see definition in Appendix A).
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We focus on the standard tractable variant of conformal prediction, referred to as split conformal pre- diction [11]. Specifically, we start by splitting the dataset 1) into training set D Dtrain and calibration set Deal- Let Zerain == {i (X(1), T((), y()) } and Teal == {i (x("), T()), y()) E Dea1) denote the corresponding sets of indices. Employing the dataset Derain we train an arbitrary model that maps the input X to predictions Y; of the future observations Y2 (see Appendix E for examples of predictive models). Moreover, let 8:(0) denote the estimate of Y: returned by the learned predictor. the imput X to design a high-confidene prediction band for the unknown trajectory Y (see Figure 3). Our problem is formalized in the following statement.
given dataset D, consisting of N data and a test example (X,7.Y), such that all (x(() (0,),(000 y()) and (x, Y) are exchangeable. Given a failure probability E (0, 1), design
Problem 3.1. Consider
(X(") T(), y(D)..... ..... (X(N), (TMM,,
conformal intervals C₁ (X) S I for the unknown observations Yt, such that:
P (vt E T Y2 E (x))) 2 - a.
The intervals (x) may also depend on the dataset D as well the parameters N and a.
3.2 Conformal Prediction Bands for Randomly-Time Trajectories
In this subsection, we introduce conformal prediction for randomly-time timmed trajectories, under the setting of Subsection 3.1. Inspired by the conformal methods in [17, 18] for fixed-time trajectories, design a prediction band that guarantees covering the unknown trajectory y with arbitrary confidence (see (1)).
The idea is to leverage the data in Deal and the corresponding predictions P,(0) to transform the estimates Y, into intervals that ensure the guarantee (1). To this end, we define the normalized nonconformity scores:
R = max
(2)
BET
where (-): R + R_ is an arbitrary normalizing function, for all i E Teal- A natural way of defining the function (-) is by leveraging some notion of predictive standard deviation (std). In cases of
predictors with inherent predictive stds, the function (-) can be defined accordingly (see Appendix E for examples). Otherwise, the factors o ( can be computed from the given data, employing
ideas from [17, 18]. In the following theorem, we use the scores in (2) to design prediction band
composed of intervals Cx(X) C R that ensure the guarantee (1). The proof the theorem given in Appendix B.
Theorem 3.2 (Conformal Prediction Bands for Randomly- Timed Trajectories). Fix a failure prob- ability E (0, 1). Let Ye be the prediction of the future observation Y, at time point t. Consider the nonconformity scores R(O) defined as in (2). for any normalizing function af)). Then, if R is the a(] + () 4-1h smallest value of the set {R(6 i
E Ical] U }. the guarantee (1) holds with intervals (X) = (Y) Y, Y, + (Y()))) Vt.
Notice in Theorem 3.2 that the normalizing factors a (Y:) are leveraged to scale the score R into Ro (Yi) across different time points t. For the intervals (X) to be bounded, we need to have T([Daail + 1)(1 a]] < D We note that, while the set of time points T is priori unknown, can design a prediction band for Y by deriving the intervals Cx for all time points / & (1...., .T}. for a user-defined maximum time T E N4, and interpret the intervals at time points outside T as interpolations among the true time points (i.e., those contained in T). For example, in Figure 3, the conformal prediction band is guaranteed cover the true observations (black bullets) at the corresponding (a priori unknown) time points.
Remark 3.3. The nonconformity scores in (2) are inspired by the conformal methods in [17, 18), which guarantee coverage for fixed-time trajectories, where the sets of time points T and (1), i 1, ,. N. are identical and known priori. However, in the context of biomarker trajectory prediction, observations are collected at random time points for each subject, depicted in Figure 1. Therefore, the application of the approach from [17, 18] is hindered by the lack of prior knowledge of the time points at which observations are collected for the test subject. More details on why
'The notation [-] represents the ceiling function.

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Figure 2: (a) We compare the mean coverage and mean interval width of our baseline and conformal prediction bands for hippocampal- and ventricular-volume trajectories. Error bars denote the 95th percentile of the metrics across 10 data splits. The baseline predictors (solid bars) either exceed the desired coverage by noticeable margin (DKPG, DME) or fall short of it (DQR, DRMC, Bootstrap). In contrast, all conformalized predictors (striped bars) achieve the nominal coverage while also maintaining relatively tight intervals. (b) We show the temporal evolution of the mean interval width of the 90% conformal prediction bands for hippocampal- (left) and ventricular-volume (right) trajectories. The curve corresponds to the CP-DKGP predictor and yields relatively tight intervals early in time, with bands steadily widening as the prediction horizon extends, reflecting the expected growth in uncertainty time. Reported interval widths are the standardized scale.
these previous approaches cannot be used in our setting are given in F. Our method removes this assumption and guarantees coverage of randomly-timed trajectories (see Theorem 3.2).
Beyond the coverage guarantee (1), our conformal prediction method provides with distribution- agnostic framework for obtaining uncertainty-awar variants of arbitrary trajectory predictors. In the following section, demonstrate the applicability of our approach to several standard and state-of-the-art predictive models.
4 Prediction of Brain Biomarker Trajectories
In this section, we apply our conformal prediction method in case study involving two well- established brain biomarkers. Particularly, we focus on predicting hippocampal- and ventricular- volume trajectories, both of which are critical for diagnosing and monitoring the progression of neurodegenerative diseases, such as Alzheimer's disease (AD) [31]. For each biomarker, we use
dataset 2, 200 samples with (X(+), ( ())) from subject i: i) (*) are input features consisting of demographic and clinical covariates, such sex, race, and clinical diagnosis, well volume measures from 145 brain regions, collected the subject's first visit; ii) ( includes the set of time points of the subject's 's subsequent visits, relative to their first visit; iii) (())) is the output trajectory involving the corresponding biomarker values at the observed time points. For our experiment, we combine the preprocessed and harmonized neuroimaging measures from well-known longitudinal studies- Alzheimer's s Discase Neuroimaging Initiative (ADNI) ADNI [32] and Baltimore Longitudi- nal Study of Aging (BLSA) w3ic-which focus on AD and Brain Aging, respectively. Details on the studies and our preprocessing pipelines can be found Appendix D.
To demonstrate the model-agnostic nature of our framework, we apply it to various standard and state-of-thed aht predictors: i) Deep Kernel Gaussian Process (DKGP) model [24], ii) Deep Mixed
Effects (DME) model [20], iii) Deep Quantile Regression (DQR) model [34], iv) Deep Regression with Monte Carlo (DRMC) dropout model [35], and v) Bootstrap Deep Regression model [36].
Details the architecture and training of each model can be referred to Appendix E. We note that
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Hippocamppa Volume (standardized,
2
DKGP
CP-DKGP CP-DRMC
True Trajectory
DRMC
0
-1
-2
0
10
20
30
40
50
60
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Figure 3: We compare trajectory bounds for DKGP and DRMC before and after conformal calibration. The baseline DKGP model produces overly wide bounds, while DRMC yields narrow, overconfident intervals that fail to capture later observations. After conformalization, CP-DKGP and CP-DRMC adjust their bounds to achieve the nominal coverage, correcting over- and under-confidence respec- tively.
existing conformal prediction methods are designed for fixed-time settings and cannot be applied to randomly-timed trajectories, which precludes direct comparisons. We elaborate further in Appendix
For each predictor, we present comparative case study between its baseline variant, which is trained on the entire given dataset, and conformalized variant, which is trained on subset of the given dataset, described in Subsection 3.2. To design prediction bands for the baseline variants, we employ the inherent uncertainty measure provided by their predictive standard deviation (see Appendix E.2 for details). By leveraging the same uncertainty measure, we define the normalizing function (-) employed the design of our conformal prediction bands (see Theorem 3.2). We note that while baseline bands provide notion of uncertainty- prediction, the desired coverage might not be achieved when model assumptions are violated.
In all experiments, we evaluate the performance of the baseline and conformal prediction bands in terms of mean coverage (i.e., the proportion of test trajectories entirely contained within the prediction bands) and mean interval width (across all test trajectories and corresponding time points). Specifically, we compute these metrics by averaging over 10 random splits of the dataset into given data and test data, with the test set comprising 10% of the total dataset. For the design of our conformal prediction bands, we perform an additional splitting of the given data into training and calibration sets. In all cases, the calibration set size is selected as 20% of the given dataset (see Appendix for details about this choice). Moreover, in this section, we consider confidence level of 0.9 (i.e., 0.1), while additional confidence levels are explored in Appendix G.2.
In Figure 2a), we illustrate the mean coverage and mean interval width attained by all the baseline and conformalized predictors for the two biomarkers of interest. We observe that in the case of the DKGP and DME predictors, both the baseline and conformalized variants achieve the desired coverage. However, the mean coverage of the conformalized predictors is closer to 0.9, and the corresponding prediction bands are tighter, as indicated by their reduced mean interval width compared to the baselines. We also see that unlike the corresponding baseline variants, the conformalized DQR, DRMC, and Bootstrap predictors are able to achieve the desired coverage, confirming our coverage guarantee in (1). Figure 2b shows the temporal evolution of conformal interval width for CP-DKGP. The intervals remain stable over time and increase only gradually with prediction horizon. The mean interval width with time for the remaining predictors provided in Appendix G.3. To illustrate the bands qualitatively, we visualize baseline and conformal prediction intervals for hippocampal-voluus trajectories from DKGP and DRMC example subject in Figure 3.

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5 Group-Conditional Application across Covariate Subpopulations
So far, we have developed and tested conformalized predictors of biomarker trajectories that guar- antee coverage across the overall patient population. However, these guarantees may not hold uniformly when applied to subpopulations defined by demographic or clinical covariates. In par- ticular, high-risk groups (e.g., individuals with MCI or genetic risk factors) and underrepresented demographic subgroups may be inadequately captured by population- based conformal bands, leading to miscalibrated uncertainty estimates. To address this, employ group-conditional variant of our approach, directly derived from the conformal framework in [9, 37]. We describe this method in Subsection 5.1 and evaluate it in our case study on hippocampal- and ventricular-volun trajectories in Subsection 5.2.
5.1 Group-Conditional Conformal Prediction for Candomly-Times Trajectories
In this subsection, we present a group-conditional variant of the conformal prediction method described in Subsection 3.2. Specifically, we provide group-conditional conformal prediction bands for randomly-tiin tidddmly--imeed trajectories, following the Mondrian conformal prediction framework from 37].
Mondrian conformal prediction a general approach for deriving group-conditional variants of conformalized predictors. To apply it to our setting of randomly-timed trajectories for distinct covariate subpopulations, consider a grouping function G : R - g, where 9 a predefined set of covariate groups. The idea is to stratify the calibration data by group, and apply the conformal method from Subsection 3.2 to each of the groups. In this way, we can derive group-conditional conformal bands that guarantee coverage within each covariate subpopulation. We formalize our result in the following corollary, adopting the notation from Section 3. The proof is omitted, as directly follows from the application of Theorem 3.2 across all subpopulations, similar to the proof of [37, Proposition 3].
Corollary 5.1 Condiiionniii Conformal Prediction Bands for Randomly-Ting Timed Trajectories). Fix a failure probability a E (0, 1). Let Y, be the prediction of the future observation Y, at time point t.
Consider the nonconformity scores R() defined as in (2), for any normalizing function a-)). For each
9 E G. let Deal.g == {(x(), (1) T(i) (() E Deal : G(X()) = 9} be the corresponding subset of Deal and Leat, == E Ical G
()) = 9} the corresponding set of indices. Moreover, for each g E G. let Rg denote the [(]Deal + +1 - -th smallest value of the set (R()) i E Ical.g) U {co}.
Then, for every E G, we have:
P P(Y Y, E C((X) G(X) =9) 20211 a
with =
+ +Y, Yr + Ra (P)). Vt.
Simply put, the above corollary guarantees that, given test example from known covariate group we can derive conformal prediction intervals with guaranteed coverage of a for the unknown trajectory Y. for any (0, 1).
5.2 Application across Heterogeneous Demographic and Clinical Covariate Groups
In this subsection, we test our group-conditional conformal prediction method in the case study of hippocampal- and ventricular-volun group-conding conditionl conformal prediction bands for a total of five population stratifications, each based on distinct demographic or clinical covariate. Among demographic factors, we consider sex, race, and education level, with the following respective subpopulations: i) females and males, ii) Asians, Blacks, and Whites, and iii) subjects with less than 16 years of education (Edu < 16) and more than 16 years of education (Edu > 16). As for clinical covariates, we consider diagnosis and APOEA allele status, a genetic risk factor associated with Alzheimer's s disease. As noted in Section 4, our diagnostic composition divides subjects into: Cognitively Normal (CN), Mildly Cognitively Impaired (MCI), and subjects diagnosed with Alzheimer's 's disease (AD). Moreover, subjects are classified based on their APOE4 status as non-carriers, heterozygotes and homozygotes depending whether they have zero, one, or two copies of the APOE4 allele, respectively.
trajectories, described in Section 4. Specifically, we design
To demonstrate the impact of our group-conditions. conformal method, we compare it with the approach from Subsection 3.2 for the five population stratifications. Herein, we refer to the latter method as population conformal prediction, owing to its coverage guarantees across the overall population. Our experiments focus on the setup of Section 4, employing the state-of-the-ar Deep Kernel Gaussian Process model from [24], as well as the combined cohort of the ADNI and BLSA
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Figure 4: We compare the mean coverage of population and group-conditional conformal prediction bands for hippocampal- and ventricular-volume trajectories. Error bars denote the 95th percentile of the metries across 10 data splits. Results are presented across five population stratifications based on individual covariates. For most covariate groups, the population conformalized predictors fail to achieve the desired coverage, while the group-conditional conformalized predictors consistently attain the desired confidence level.
studies. For the composition of the dataset across all five population strata, we refer the readers to Table 3 in Appendix D. We conduct experiments for 10 distinct splits of the data into training, calibration, and test sets, as detailed in Section For each of the splits, we first learn a model from the corresponding training data, and then design: i) single population conformalized predictor, and ii) five group-conditional conformalized predictors for the respective population stratifications. For the group-conditional conformal prediction bands, the test data stratified by group. The mean coverage and mean interval width are computed by averaging over all 10 data splits. Results are presented for a confidence level of 0.9, while additional values are in Appendix H.
Figure shows the mean coverage achieved by population- and group-conditional conformal pre- dictors across five covariate stratifications for peroocaapal-volume trajectories. We observe that the population conformalized predictors (in blue) fail to achieve the nominal coverage across most covariate groups in several cases-inee inceusiin clinically high-risk or underrepresented groups such as individuals diagnosed with Mild Cognitive Impairment (MCI), APOE4 homozygotes, and Asian subjects. On the other hand, while group-conditional predictors (red) consistently attain the nominal 90% coverage across all subgroups. These results confirm that group-conditional conformal bands provide more reliable coverage across diverse patient subpopulations and underscore the translational potential of our approach in clinical settings where early risk identification is essential. In the next section, we illustrate how conformal uncertainty can be directly leveraged in downstream clinical task, enabling more informed decision decision-making making and improving individual-level risk stratification.
6 Clinical Utility of Conformal Bands: Identifying High-Risk Subjects
Having established coverage guarantees, we now focus on clinical utility for identifying high- risk individuals. Specifically, we apply our conformal bounds to stratify MCI patients by risk of Alzheimer's s progression, enabling early detection and targeted intervention. In clinical practice, the rate of change (RoC) of longitudinal biomarker between the subject's 's first clinical visit and subsequent clinical visit has been widely used as progression marker for early detection and cohort enrichment [2, 38-40J. the case of predicted biomarker trajectories, we can define the predicted rate of change RoC of subject i as the slope of the predicted trajectory between an initial and future time point, denoted by to and f.N, respectively, as follows:
(1)
(1)
RoC (
IN
iN t
While the predicted RoC is widely used as a progression marker, relying on trajectory predictions alone underestimate risk in uncertain cases. We therefore introduce the Rate of Change Bound
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(RoCB) as an uncertainty-awan progression metric that quantifies the worst-case rate of change with high probability. Let L ( and U/(²) denote the lower and upper bounds of an uncertainty interval obtained for subject at time point with a given method (e.g., a baseline method or our conformal prediction method). To account for the different behaviors of distinct biomarkers, the RoCB of subject i is defined as follows:
(tN) - (t)
if biomarker decreases with progression,
tN to U (tN) - (to) tN to
(oCB()
(4)
if biomarker increases with progression.
Note that the RoCB provides notion of uncertainty-aware slope. In particular, for decreasing biomarkers, such as hippocampal volume, the worst-case decline is captured by the lower bound of the interval, whereas for increasing biomarkers, such as ventricular volume, the upper bound of the interval is used. In the experiments below, we focus on hippocampal volume and therefore compute RoCB using the lower bound.
In this experiment, we focus on trajec-
the hippocampal-volume tory as a key neurodegenerative biomarker. We evaluate each sub- ject's predicted rate of change as baseline progression metric and compare it to the proposed rate of change bound. The RoC is de- fined as the slope between the baseline prediction and the fu- ture predicted mean value, while the RoCB is computed using the lower bound the prediction in- terval the final timepoint.
Table I: Youden-optimise discrimination on z-standardised pre- dieted rate of change RoC and rate of change bound RoCB for MCI converters, with 95% bootstrap confidence intervals. For hip- pocampal volume (a decreasing biomarker), RoCB corresponds to the lower bound of the interval.
Method
Metric
Precision
Recall
Fi
DRMC
RoC
-0.006
0.436 + 0,022
0,671 + 0.058
0.528 +: 0.023
RoCB
-0.012
0.403 1 0.022
0.884 1 0.058
0.553 1. 0.023
CF-DRMC
RoC
-0.006
0.432 + 0.022
0,740 + 0.095
0,546 + 0.024
RoCB
-0.020
0.395
0,915 + 0.095
0.552
We analyze subjects diagnosed with MCI at their first clinical visit. Those remaining MCI at all follow-ups are labeled MCI-Stable; those converting to Alzheimer's are MCI-Progressors, A threshold T applied to RoC or RoCB determines trial eligibility: subjects below T are classified as high-risk. The cohort includes 462 MCI-Stable participants (mean follow-up 22.5 months; range 0-153) and 258 MCI-MMMCI-see PrCl-Progressors (mean 23.5 months; range 0-142).
To evaluate the clinical value of our conformal bands, we compare the discriminative performance of DRMC (with its baseline uncertainty measures see Section 4) and its conformalized counterpart (CP-DRMC) using both the predicted rate of change and rate of change bound. For each metric (RoC or RoCB), we determine the optimal threshold -" via Youden's index. As shown in Table 1, the RoCB metric substantially improves recall within both models. For DRMC, recall increases
from 67.1 1% (RoC) to 88.4% (RoCB), and for CP-DRMC, from 74.0% 91.5%. These increases demonstrate the benefit of incorporating predictive uncertainty into the slope-basee progression metric. Importantly, even under the same RoCB metric, conformalization yields an additional improvement: CP-DRMC achieves a recall of 91.5% compared to 88.4% for DRMC. This increase highlights the impact of conformal calibration itself: not only produces well-calibrated prediction intervals, but also systematically improves risk identification by lowering uncertainty bounds. Although precision decreases modestly when using RoCB instead of RoC, the F1 score remains similar, confirming that the increased sensitivity does not come at the cost of overall discrimination. Appendix Section I (Table 4) presents the quantitative results for the remaining predictors.
To further illustrate how conformal prediction improves clinical decision-making Figure 5 presents two representative hippocampal-voliie volume trajectories comparing DRMC and CP CRDMC On the left, an MCI subject who converts to AD is misclassified by DRMC: the model's 's overconfident prediction band fails to capture the true trajectory, and the resulting RoCB (-0.01) SD month lies above the Youden- threshold (7*), leading to missed detection. In contrast, CP-DRMC produces wider and better-calibrated prediction band that lowers the RoCB to (-0.020 SD month -1), correctly classifying the subject as an AD progressor. On the right, we show stable MCI subject incorrectly

---

CP-DRMC True Trajectory
DRMC Threshold (f--0.012)
CF-DRMC Threshold (r--0.02)
Hippocampal Volumee (slendariized
CP- DRMC--0.011
Hippocampal Volume (standardizz)) (rrdzzdd)
RoCB: CR-DRMC=-00016 DRMC=-0.012
0.5
Hole MCI
0.0
-0.5
-0.5
-1.0
-1.0
-1.5
Ab
-1.5
MC1
-2.0
40
-2.0
0 20
40
60
80
Time months)
Time (in months)
Figure 5: We visualize hippocampal-volume trajectories for two MCI participants. On the left, converter from MCI to AD. The over-con over-confident DRMC band (green) fails to cover the true trajectory and yields a lower-bound rate of change of -0.011.mo -1 above the Youden-optimal threshold so the subject would be missed. Conformal calibration (CP- (CP-DRMC, lilac) lowers the band, produces RoCB 0.020,mo I -", and correctly flags the converter. On the right, a non-converter who remains MCI. DRMC erroneously predicts conversion, whereas the wider CP-DRMC band raises the RoCB above the threshold and prevents false enrolment. Together these examples demonstrate how conformalisation rescues under-scored converters while simultaneously reducing false positives among stable cases.
identified as high-risk by DRMC due an overly steep RoC. The conformal predictor expands the interval, raising the RoCB to larger value and avoiding a false inclusion.
7 Discussion on Clinical Impact
Our framework brings distribution-free, uncertainty-calirrat calibrated forecasting to biomarker trajectories observed at randomly-timed visits-a pervasive reality in longitudinal clinical studies. By confor- malizing trajectory predictions, we equip downstream decisions (e.g., early intervention, enrichment for MCI or APOE4 subgroups) with formal coverage guarantees at subject-specific time points. Im- portantly, our approach is aggell-g and can conformalize any biomarker predictor to enhance reliability in precision trial design [41], where slope-based metrics guide inclusion the clinical trial [2]. Our rate change bound (RoCB) uses the conformal band to favor conservative, safety-oriente screening without parametric uncertainty assumptions. Both population- and group-conditional calibration improve trustworthiness in high-risk or underrepresented subgroups.
Limitations and Future Work
Our contribution adapts split conformal and group-conditional conformal prediction randomly-ting settings via a time-varying normalizing function. We focus on single biomarkers; joint guarantees across coupled trajectories (e.g., hippocampus, ventricles, cognitive measures) are not yet addressed, though multivariate conformal prediction that provides joint coverage is a natural extension of our approach to be pursued next. Furthermore, the group-conditional conformal prediction faces challenges: as we create intersectional subgroups (e.g.. APOE4 status, race, sex, age), the resulting ones become too small for effective calibration, yielding wide or uninformative bands. Future work will explore alternative approaches for multi-covariate conditioning that maintain coverage guarantees while avoiding the sample size limitations of fully stratified calibration. Additionally, our coverage guarantees assume the test and calibration data are drawn from the same underlying distribution (i.e., they are exchangeable). When this assumption is violated such as when applying the method to new cohorts from different sites, imaging protocols, or demographic compositions the coverage guarantees may no longer hold. Addressing distribution shift remains an open challenge. Furthermore, online variants that update prediction bands as new patient visits arrive could support continual adaptation, though the theoretical properties of such approaches in our setting remain be established.
10

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Acknowledgement
This research is supported by the NIH U241 RF1 AG054409 grant, the NIA contract ZIA- AG000191 and ASSET (AI-Enabled Systems: Safe, Explainable and Trustworthy) Center.
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24
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Technical Appendices
A
Exchangeability
Exchangeability of a sequence of random variables implies that their joint probability distribution remains unchanged under any permutation of the variables. This condition is milder than the standard one of independent and identically distributed random variables. Below we provide the formal definition of exchangeability from [10].
Definition A.1. The variables z y.... Z(N+1) exchangeable if for every permutation (-)) of the integers ,,...,,. + 1, the variables W ....., W(N-1), where W Z(a(0), have the same joint probability distribution as ( 2(N+1)
B Proof of Theorem 3.2
For simplicity of presentation, we introduce the notation (X(++1), T(++1) y to represent the test sample (X, T, Y) and the notation (N++1) to represent the random variable R defined in the theorem statement. Since the random variables (X((),T))) Y()), i e [I...., + 1}, are exchangeable by assumption, the random variables (xx(*) T Y()), i Teal U + 1}, are also exchangeable. Somewhat abusing notation, we define the nonconformity score function:
T((,,
Y P/l a(P)) S
R(X,T,Y) = max KET
that applied to each datapoint (X()) T()) ((), Teal U (N + 1}, yields the corresponding score R(, Conditioned on the dataset Derains the function R(.) is deterministic. Therefore, we conclude that conditioned on Drain the random variables R(), Teal U UNN + 1}, are exchangeable. Hence,
from [42, Lemma obtain the conditional property:
Y Pel
max IET
<
R Devain 21-a,
(S)
(Y.)
where R is defined as in the theorem statement. By marginalizing over D (5) yields the unconditional property:
Y Pil
max FET
SR 2 a.
(6)
o (Yi)
The guarantee (1) directly follows from rewriting (6) as:
P (vL T |Y, - Prl S (Y))) 2
and setting C(XX) [-Ro (1) + P,, P. Ro (P.)), for all l.
C Discussion Clinical Impact
Our conformal prediction framework provides prediction bands that enable clinicians to perform informed prognosis and decision-making with greater reliability. This is important as predictive models are increasingly applied in healthcare for both patient management and drug development. In the latter case, Cummings et al. [41] highlighted the need for Al-informed clinical trials, referred to as precision trial design. Along these lines, Maheux et al. [2] evaluates a predictive model for biomarker trajectories, in the context AD, where derived measures- such as biomarker rate of change serve as quantitative indicators of whether a subject is likely to progress to the disease during the clinical trial. This assessment ultimately informs decisions on subject inclusion in drug administration process. By conformalizing such predictors, either population level or calibrated covariate groups, such as MCI subjects, we provide confidence bands alongside predictions, thus increasing the reliability of these critical clinical decisions.
Sernne
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Table 2: Demographic and clinical characteristics of the clinical studies. The joint cohort of ADNI and BLSA comprises our base population, whereas the remaining cohorts represent external populations. For the total time of observation and the age, we report the mean and standard deviation over all subjects contained in each study. CN: Cognitively Normal, MCT: Mild Cognitive Impairment, AD: Alzheimer's Disease.
Study
Subjects
Obs. Time (mo)
Males (%)
Age
Diagnosis (%)
CN
MCI
AD
ADNI+BLSA
2200
65 39
53.2
75.4 + 8.6
49.3
34.8
15.9
Table 3: Composition of the joint cohort the ADNI and BLSA studies across five demographic and clinical covariates.
Covariate
Groups
Percentage Subjects
Number of Subjects
Sex Race
Male
52.89
1164
White
87.46
1924
Black
8.60
189
Asian
3.94
87
Education Years
Less than 16
54.56
1200
More than 16 Cognitively Normal
45.44
1000
Diagnosis
49.12
1081
Mild Cognitive Impairment Alzheimer's s Disease Heterozygous Homozygous None
34.76
765
16.01
352
APOE4 Status
32.16
707
7.31
161
60.54
1332
D Clinical Datasets and Preprocessing
Our data consists of neuroimaging and demographic measures taken from subjects in the iSTAGING consortium [43]. Specifically, the neuroimaging measures are the 145 anatomical brain Regions of Interest (ROI) volumes (119 ROIS in gray matter, 20 ROIS white matter and ROIs in ventricles) extracted using multi-atlas label fusion method [44]. Phase-level harmonization is applied these 145 ROI volumes to remove site effects [45]. To train, calibrate and test the conformalized predictors data from two cohorts: the Alzheimers Disease Neuroimaging Initiative [32], which is public-private collaborative longitudinal cohort study and has recruited participants categorized as Cognitively Normal (CN), Mildly Cognitively Impaired (MCI) and diagnosed with Alzheimer's Disease (AD) through 4 phases (ADNI1, ADNIGO and ADNI2) [46]. We also use Baltimore Longitudinal Study of Aging (BLSA) follows participants who cognitively normal enrollment with imaging and cognitive exams since 1993 [33].
We also extract from iSTAGING cohort: the OASIS [47], The Wisconsin Registry for Alzheimer's Prevention (WRAP) study [48], the Australian Imaging, Biomarker, and Lifestyle (AIBL) study [49], the Coronary Artery Risk Development in Young Adults (CARDIA) [50], the PreventAD [51] and PENN.
For the clinical variables, we utilize Age at Baseline, Sex. Years of Education, and APOE4 Allele status, the latter being known risk factor for Alzheimer's Disease. Diagnostic categories were designated as Cognitively Normal (CN), Mild Cognitive Impairment (MCI) and Alzheimer's 's Disease (AD). Subjects diagnosed with alternative forms of dementia, such as Lewy Body Dementia and Frontotemporal Dementia, were excluded from the study. These exclusions are minimal (less than 10 subjects) and did not impact the overall sample size. After filtering, our dataset consists of 2200 subjects. Furthermore, Years of Education was dichotomized: subjects with more than 16 years of education were coded as '1', while those with years or fewer were coded as 'O'. Also, duplicate acquisitions within the same month are discarded.
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E
Predictors
E.1
Architectural Design and Training
Our input data consists of multivariate features, including volumetric imaging features, demographic information, and clinical variables. All our predictors generate biomarker trajectories by using a learned model with an additional time input variable E N+, that outputs an estimate of the corresponding future biomarker value Y,. Below we present five examples of such models, along with corresponding learning algorithms.
Deep Kernel Regression. A fully connected feedforward neural network is used to linearly transform input data into a low-dimensional latent space. The transformed input is then passed to a Gaussian Process with trained using exact inference by minimizing the negative marginal log- ligeliilihood. For details on exact GP inference, refer [19]. DKGP approach builds upon the deep kernel learning paradigm presented in [52] and applied for trajectory prediction in [24]. The DKGP is trained for 100 epochs, using the Adam optimizer [53] with weight decay 0.02 and a learning rate of 0.01 for the deep network parameters and 0.2 for the hyperparameters of the Gaussian Process.
mean function and radial basis function (RBF) kernel. The GP component is
Deep Mixed Effects. The DME model leverages an embedding network and deep mean function to capture both global trends and local variations in structured regression tasks. The embedding network projects inputs into 64-dimensional latent space, while the mean function, implemented as an MLP, maps the latent features to a scalar regression output. The Gaussian Process (GP) with an RBF kernel operates the latent space to model residuals, with warping applied when an embedding function is used. The model is trained end-to-end using variational inference, minimizing the negative Evidence Lower Bound (ELBO). Separate Adam optimizers are employed for the mean function and GP kernel parameters, each using a learning rate of 10 and an L2-penalty of 10-3 For each training iteration, the GP parameters are adapted over Radaptt 10 steps with an inner learning rate of 10 The training process spans 50 epochs, alternating between optimizing the mean function and GP parameters. Details on the DME model can be found in [20].
Deep Quantile Regression. The model is a fully connected feedforward neural network designed for quantile regression, predicting multiple quantiles simultaneously (e.g., 0.1, 0.5, 0.9). includes a 128-unit hidden layer and a 64-unit hidden layer, both followed by ReLU activations and dropout with 0.2 rate for regularization. The output layer has one unit per quantile to estimate. The model trained using quantile loss function [54]. DQR is using Adam for the optimization, with learning rate of 0.01 for 200 epochs.
Deep Regression with Monte Carlo Dropout. The model is fully connected feedforward neural network with Monte Carlo Dropout for uncertainty estimation in regression tasks. It consists of a 128-unit hidden layer, a 64-unit hidden layer, and single-unit output layer for continuous scalar prediction. ReLU activations are applied the hidden layers, and dropout layer with a fixed rate 0.2 is applied during both training and inference to approximate Bayesian uncertainty. DRMC is using Adam for the optimization with learning rate of 0.01 for 200 epochs.
The model is fully connected feedforward neural network for regression tasks, compris- ing a 128-unit hidden layer, a 64-unit hidden layer, and a single-unit output layer. ReLU activations are applied to hidden layers, and the output layer produces a raw scalar value for regression. The model deployed as part of a bootstrap ensemble, where 10 instances are trained on different bootstrap- resampled subsets of the train data. Each instance is trained with consistent hyperparameters (epochs and learning rate) to enhance robustness and reduce variance by aggregating predictions.
Bootstrap.
E.2 Predictive Standard Deviation
We outline the models and their uncertainty quantification mechanisms. Predictions are denoted as Y, with standard deviation (Y.).
Deep Kernel Regression (DKGP) and Deep Mixed Effects (DME). Both models estimate the posterior predictive distribution using exact inference. The posterior predictive distribution is gaussian distribution, and from that we extract the predictive mean that corresponds to the point estimates Y, and the predictive variance that corresponds to requested confidence level, i. 0.90.
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From that we calculate the standard deviation (Y)). Details on the mathematical formula of the predictive mean and predictive variance can be found in [19]
Deep Quantile Regression (DQR). Deep Quantile Regression predicts the desired quantiles- lower, mean, and upper-for specific confidence level. From these predictions, we calculate the quan- tiles corresponding to different percentiles, such as the 10th, 50th, and 90th percentiles, enabling uncertainty quantification. The confidence level represents the range between the lower and upper percentiles. The predicted variance is estimated using the spread between the upper and lower quantiles, adjusted by z-score corresponding to the desired confidence level (e.g.. 1.645 for a 90% confidence interval). This calculation assumes that the predictive distribution is approximately Gaussian, allowing the z-score to be used as a scaling factor for the quantile spread.
Deep Regression with Monte Carlo (DRMC) dropout. Monte Carlo dropout approximates Bayesian inference via multiple stochastic forward passes with dropout. The model generates multiple predictions, and from these we extract the standard deviation of the sample. We run inference 100 times.
Bootstrap Deep Regression (Bootstrap). Bootstrap Deep Regression leverages an ensemble of 20 deep regression models, each trained on different bootstrap sample, estimate uncertainty. During inference, the ensemble produces multiple predictions for the same input, and the sample standard deviation of these predictions quantifies predictive uncertainty.
to
F Relation to the Broad Conformal Prediction Literature
Our conformal prediction method provides an extension to the setting of randomly timed trajectories, which necessary for applying conformal prediction using real-world biomarker data. None of the previous methods, including [25] can handle randomly-timed trajectories. Below, we elaborate on why prior CP methods for fixed-time trajectories cannot be applied in our setting of randomly timed biomarker trajectories. Our simple yet effective normalization function is able to accommodate conformal inference on randomly-time trajectories. The normalizing function enables us to compute normalized prediction error across all time steps, ensuring that component within the maximum dominates the others in terms of scale. This function leverages the model's 's predictive standard deviation to measure prediction uncertainty, effectively normalizing errors in standard way. Our conformal prediction intervals at each time point in a test trajectory are not influenced by the trajectory's 's length. The bound for the prediction error | - -Yl at each time is derived by scaling the score R by the time-specific uncertainty estimate or (Y))), yielding the value R (Yi)). This time-varying scaling ensures that the conformal bands adapt to the uncertainty at each time point. Additionally, a true "worst-case" scenario would arise only if (Y,) = 1 at all time points, which does not hold here.
Our extension of conformal prediction to randomly timed trajectories is necessary for real-world biomarker data. None of the previous methods [13, 18, 25, 30]can handle randomly- trajectories. Closest to our work are the approaches by [18] and [17], which are designed under the assumption of fixed-time trajectories-is. the entire population. In contrast, our setting involves trajectories with observations collected at random times due to missed visits and variable scheduling typical in clinical studies. To adapt the method from [17, 18] to our setting, one could select a subset of the data corresponding to a fixed set of time points (e.g.. 3, 9, and 12 months) and extract only the corresponding observations from each subject. However, this leads to substantial dataset reduction. In a simple experiment following this approach, we found that only 37 trajectories with observations at all four selected time points remained out of the total dataset of 2200 trajectories. Beyond dataset reduction, this method limits the applicability of the conformal prediction bands. More specifically, since [18] and [17] define normalization factors only at the pre-specifted time points (e.g.. 3, 7,9, 12 months), their method can produce prediction intervals only at those times during inference. Thus, if test subject visits the clinic at months 4, 5, and 14, the available intervals at months 3, 7, 9, and 12, provided by [18] and [17], do not allow valid conformal inference at months 4, 5, and 14. In contrast, we introduce a normalizing function ot.) that produces predictive uncertainty estimates o(Y.) for any time point 1, even if no observations at that time are available in the calibration data. This allows our method
observations are made at the same pre-defined time points across
to produce valid conformal intervals at arbitrary test-time points, in contrast to existing conformal prediction methods designed for fixed-time trajectories.
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G
Case Study: Details and Extended Results
G.1
Calibration Set Size Selection
In this section, we outline the procedure for determining the suitable calibration set size for conformal prediction. Specifically, we vary the fraction of the training set used for calibration from 0.01 to 0.30, while the remaining portion is reserved for predictor training. We then evaluate the resulting conformal intervals on held-out validation set terms of two key metries: (i) mean coverage and (ii) mean interval width.
This procedure is repeated for both hippocampus and ventricular volume biomarkers, confidence levels of 0.90, 0.95, and 0.99, and for all five conformal predictors. Our goal is to pick the size that achieves the desired coverage with the smallest possible interval width.
Our findings, visualized in figure indicate that as the calibration set fraction increases, the empirical coverage exceeds the nominal coverage level. We identify a calibration set fraction of 0.20 practical choice that achieves coverage while avoiding wide prediction intervals. This calibration set fraction corresponds to calibration set with 414 subjects.
CF DKDD
CP-DME
CP UF-DQR
CP-DRMC
CP CP-Bootstrgg
Hippocampal Volume
Nominal Oooergg 11:0
Nominal Coorage 0 95
Nocumall Coverage
Mean Coverage
Commrage
Metrs Coveragg
0.35
0:50
1.00
2.00
0.00
0.88
0.80
Callantion
0.10
0.35
0.92
0.5
Plodkon
2.00
0.30
0.50
Calbmick Set Hiatoo
2.10
020
Moa Intervl Wadts
2:30
Moan lateral Walts
0.50
0.55
Calibeatin: Sei. Fraction
0.33
0.35
0.20
0.25
Callirstion Sot Proxtoon
0.00
0.55
Caillmitinn Sot. Frackir
0.33
0.55
6.30
5.00
3300
Ventricular Volume
Nominal Coverage 0.30
Coverage
Nocninal Coourage 0.90
1,000
1.00
Mean Conarag
Chrestgge
0.973
Mou Contrage
0.08
0.55
0.300
0.36
0.50
0.34
0.803
0.85
0.50
Calibeation: Sec Fractien
0.10
0.16
0.20
0:25
0.30
0.303
0.00
0.06
0.15
0.25
0:30
0.00
0.05
0.20
0.15
0.20
0.25
0.30
Calitration
Collbotion Sel Fraction
Men loooa W332
Wiehh
3.8
0.2
0.00
Califcation Set Tractoo
0.30
5.20
Callintion Paztion
0.25
Chiliaaiin
Figure 6: Mean coverage and mean interval width for hippocampal- and ventricular-volume trajec- tories at nominal coverage levels of 0.90, 0.95, and 0.99. Error bars denote the 95th percentile of the metrics across 10 data splits. Each column corresponds to a different nominal coverage level, and each curve represents one of five conformalized predictors (CP-DKGP, CP-DME, CP-DQR, CP-DRMC, CP-toootrr. Dashed red lines indicate the nominal coverage. The horizontal axis shows the fraction of the training set used for calibration, ranging from 0.01 0.30
26

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G.2 Comparison between Baseline and Conformalized Predictors for Other Confidence Levels
In this section, provide additional results comparing the conformalized predictors with their baseline counterparts at confidence levels of 0.95 and 0.99. For both hippocampal- and ventricular- volume trajectories,
observe trends similar to those reported at the 0.90 confidence level in Section of the main paper. Specifically, conformal adjustments effectively boost empirical coverage the desired nominal level. Gaussian Process based methods, such the DKGP and DME already achieve the nominal coverage. Conformalized-DGGe coverage with tighter bounds, which corresponds less conservative uncertainty quantification. On the contrary, for the methods of DQR, DRMC and Boostrap, that provide undercoverage, their conformalized versions achive nominal coverage by widening the intervals.
and conformalized-DMr. attain nominal
These findings confirm the effectiveness of our conformal prediction across varying confidence levels.
Euetlne Predittor
Conformiiied Prodicser
Nocuinal Coverage 0.95
Hippocampus Volume
Ventricular Volume
1a
1.0
2.6
Xoan Internal Wolll
Soon lncerval Whlti
Marn Crerney
Choeregge
B
MRGP
DNB
Bostery
CP DPR
CP BEMi
DQR
OP WE
CP Exji
0.0
TAGOP
DMB
OF LMM
on FROA
CP-Dbstemay
DMB
EQj
FROe
oP SIGE
CPNNI
coP OP-D0D
CP dhib-taa
Daseliee Predictcr
Ckusormalized Predictor
Nomunall Corerage (1.99
Maz Inieaal Walt
Moan
Moan
D:3
oe Beraii
EMEE
CR-D-3R OF DONO
ch boootao
D247
BRMC
(P-DROF
CE OIE
Figure 7: Mean coverage and mean interval width of baseline and conformalized predictors for trajectories at nominal confidence levels of 0.95 (top) and 0.99
hippocampal- and ventricular-volie (bottom). The dashed red line indicates the nominal coverage. Error bars denote the 95th percentile of the metrics across 10 data splits. Overall, the conformalized methods exhibit coverage closer the nominal level, by adjusting accordingly the interval width.
Next, we demonstrate how uncertainty changes over time starting from the initial acquisition, which, in our study, corresponds to the test subject's first hospital visit.
G.3 Mean Interval Width of the Conformalized Predictors Time
Having established in the previous section that each conformalized baseline achieves desired coverage levels across different calibration set sizes and confidence levels, we now focus to the evolution of the conformal prediction intervals over time. In practical scenarios, the uncertainty in predicting future measurements naturally increases as we move further away from a known data acquisition, and understanding this growth in interval width is crucial for longitudinal analyses.
27

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Figure 8 shows how the conformal prediction intervals of our selected methods vary with the time elapsed from the most recent (known) acquisition. Each row corresponds different conformalized predictor, while the horizontal axis indicates increasing time from the known data point, and the vertical axis shows the corresponding mean interval width. As expected, most methods exhibit relatively tight intervals when predicting only short time ahead, but the intervals expand as the temporal distance grows. This behavior reflects the increasing uncertainty associated with predicting further into the future.
CP-DXGP
CP-DMB
CP-DQR
CP DRMC
CP Booksica
Hippocanpol Volume
Ventriculkr Volumo
0.975
1.240
0.350
Interval Wizth
Mom [nterval Widtt
1,800
1.600
1,600
4,00
2,000
2,500
5,000
2,000
2,500
3.000
3,000
2,000
2,000
0.0
5.0
10.0
12.5
0.0
5.0
10.0
12.5
Time (itl years)
Time
years]
Figure Temporal evolution of the mean interval width (averaged per year) for the five conformalized predictors. We observe that as time increases, the average conformal predictive intervals increase.
28

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H
Croup-Conditoonal Application across Covariate Subpopulations: Extended Results
Figure 10 depicts the mean interval widths for hippocampal- and ventricular- trajectories across subgroups defined by demographic (Sex, Race, Education) and clinical (Diagnosis, APOE4 alleles) covariates. The population conformal prediction generates narrower intervals for all the subgroups, leading to undercoverage, particularly for high-risk or underrepresented groups such as Black and Asian participants, MCI patients, and APOE4 homozygotes. In contrast, group-conditional conformalized predictors (in red) produces wider intervals order ensure that empirical coverage aligns with nominal levels within the specific subpopulations. While for the hippoampal volume all subpopulations were not covered, for the ventricular volume several subpopulations were already reaching the nominal coverage of 0.9, specifically for females, Asian, Black, subjects with less than 16 years of education as well as the Non-carriers. From the Figure 10 we observe that these are the ones that already reach the nominal level of coverage exhibit the lower increase in the interval width withing the subpopulation. For example, females have lower mean interval width than males.
Pogdltio Codican Prodetio
Neniiaal Chorage
Taze
Eduratio Yoees
Eingnais
APOS4 Allaa
Pipercampal
22
0.56
Moan
0.94
L77
Freal
Mide
Juim
Bled.
Wans
Ede sia
AD
Mas
Prodd
Ma
Adaa
Blaak
Wile
Eata sin
MCT
Shebens
CearpeCrnittteed
Necrinal Doveress 0.55
1.00
4.0
0.02
Maee Interal Watth
3.50
3.20
3.00
2:0
6.96
Aman
Whoo
AD
Noo
Herooggo
Male
Wale
Edo
Elc
Figure 9: We compare the mean coverage of population and group-conditional conformal prediction bands for ventricular-volume trajectories. Error bars denote the 95th percentile of the metrics across 10 data splits. Results presented across five population stratifications based on individual covariates (sex, race, education level, cognitive diagnosis, and APOE4 alleles status). We observe that group-conditional intervals are adjusted per covariate group in order reach the nominal coverage. On the contrary, the population conformal prediction provides consistent intervals across all groups, that cause the coverage disparities.
29

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Popsiation Conformal Prodiction:
Grou--Corfiikeul Conformal Prodiction
Nominal Concage
Sex
Race
Education Years
Diagnosis
APOE4 Alleles
Hippocauppa Volume
Ventricnlar Volume
1.7
133
18-
Monn Interval Wahta
Mown Intersal Whhts
1.2
15-
14
1.0
Fonnle
Male
Asion
Black
Whate
Elo
Edu =10
CN
MC1
AD
Nono
Holerotygora Honnaygoos
Punole
Male
Adian
Eshokk
White
Edu E
Eis -1.
CN
MC1
AD
Noue
Eatssorygou
Hontosyous
Figure 10: We compare the mean interval width of population and group-conditioal conformal prediction bands for hippocampal- and ventricular-volus trajectories. Error bars denote the 95th per- centile of the metrics across 10 data splits. Results are presented across five population stratifications based on individual covariates (sex, race, education level, cognitive diagnosis, and APOE4 alleles status). We observe that voup-cooniitinal intervals are adjusted covariate group in order to reach the nominal coverage. the contrary, the population conformal prediction provides consistent intervals across all groups, that cause the coverage disparities.
Figure 9 presents the empirical coverage and mean interval widths for hippocampal-volumee trajecto- ries across subgroups at varying confidence levels 0.95 and 0.99. Again, we observe the same trend as in the Figure 4, where population conformal prediction bands show noticeable undercoverage for underrepresented subgroups, such as Black and Asian participants, APOE4 homozygotes, and MCI patients. For the nominal level of 0.95 only the Homozygotes appear to be covered by the population conformal predictor. Again, the group-conditional conformal prediction adapts the intervals each subpopulation in order to achieve the nominal coverage withing subpopulation. For the nominal level of 0.99 we observe that less subpopulations are miscovered and the majority of them align closely to the nominal level. This is expected the confidence bands wide enough in order capture any subpopulation variability.
Figure 12 presents the empirical coverage and mean interval widths for ventricular-volume trajectories across subgroups varying confidence levels 0.95 and 0.99. For ventricular volume conformal prediction bands, similar pattern is observed. Population conformal prediction underperform for the subgroups defined by Diagnosis and APOE4 status. Specifically, at the nominal level of 0.95, MCI and AD subjects well the heterozygotes and homozygotes not covered by the population conformal prediction. However, the group-conditional conformal prediction tackles this by widening the intervals across those subgroups and thus achieving nominal coverage.
30

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Pepsittion Contomaal Producton
Groug Oukkuaal
Nettinal Contage
Set
Race
Educsiom Yoors
Dragooss
APGB4 Alddes
2.-
016
0.66
Noxa Saturval
0.9
12
Roale
Ma
Witt
New
Fress
AMAC
Bi
Whit
3011
Non:
Popuktio Shalama Prediton
Contl-oa Outtonal Predeio
Nominal Contage 6.0
1.00
6.38
2.6
8:97
0.35
0.35
6.01
Baak
ON
NO1
AD
Stoe
Mak
Adea
Blok
MCS
Eio
Bis
Ed
Figure 12: We compare the mean coverage and mean interval width of population and group- conditional conformal prediction bands for ventricular-volume trajectories for the nominal coverage of 0.95 and 0.99. Error bars denote the 95th percentile of the metrics across 10 data splits. Results presented across five population stratifications based individual covariates (sex, race, education level, cognitive diagnosis, and APOE4 alleles status).
I Clinical Utility of Conformal Bands: Identifying High-Risk Subjects
To complement the clinical analysis in Section 6, we further investigate the behavior of predictors that produce overconservative uncertainty estimates. Such models generate wide prediction intervals that capture most future outcomes, resulting in high recall but limited precision. While this behavior may be considered safe in clinical contexts, can lead to over-inclusion low-risk subjects and reduced specificity. Here, we examine whether conformalization can help refine these models by tightening the prediction bounds, thereby improving the balance between sensitivity and precision. We focus on two representative predictors: DME and DKGP. The total quantitative results are presented in Table
The DME model exhibits strongly conservative behavior. Using RoC, it identifies 81.4% of MCI converters, and recall increases 98.4% under RoCB, indicating near-complete inclusion pro- gressors. However, this recall comes at the cost of low precision-311 33.8% for RoC and 36.7% for resulnn in many false positives. Conformalization (CP-DME) slightly tightens the intervals, leading to modest drop in RoCB recall to 94.2%, while improving precision to 36.9%.
The DKGP model also demonstrates overconservative behavior, producing wide prediction intervals that result in strong recall but imprecise discrimination. Under the RoC metric, DKGP identifies 80.2% of MCI converters, and RoCB pushes recall even higher to 92.2%. However, this gain comes with reduced precision-faae falling from 50.6% to 37.4%. After conformalization. CP-DKGP produces slightly narrower intervals. While this modestly lowers recall under RoC (to 72.1 1%). the RoCB-based recall remains high at 90.3%, and precision increases slightly (to 37.6%). These results highlight the role of conformalization in refining overly broad uncertainty estimates: by reducing
31

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excess conservativeness, CP-DKGP achieves a more balanced trade-off between identifying true progressors and limiting false positives, while still preserving the safety benefits of RoCB-basse b-aseed risk stratification.
Table 4: Youden-optimised discrimination on z-standardised rate of change (RoC) and lower-bound rate of change (RoCB) for MCI converters, with 95% bootstrap confidence intervals (Cls).
Method
Metric
Precision
DRMC
RoC
-0.006
0.436 (0.367. 0.455]
0.671 [0.693, 0.9191
0.528 (0.504, 0.593]
RoCB
-0.012
0.403 (0.367, 0.455]
0.884 [0.693, 0.919]
0.553 (0.504, 0.593]
CP-DRMC
RoC
-0.006
0.432 (0.360, 0.446]
0.740 [0.667. 0.956]
0.546 (0.498. 0.590
RoCB
0.395 (0.360, 0.446]
(0.667, 0.956]
(0.498, 0.590]
DKGP
RoC -0.007
0.506 (0.340, 0.414]
0,802 [0.8 807 0.9931
0.621 [0.494, 0.573]
RoCB
-0.019
0.374 [0.340, 0.414]
0.922 (0.807, 0.993]
0.532 [0.494, 0.573]
CP-DKGP
RoC
-0.007
0.507 [0.338, 0.414]
0.721 [0 809, 0.996]
0.595 [0.491, 0.573]
RoCB
-0015
(0.338, 0.414]
(0.809, 0.996]
(0.491, 0.573]
DME
RoC
-0.000
0.378 [0.333. 0.406]
0.814 [0.813. 1.000]
0.516 [0,489, 0.571]
RoCB
-0.011
0.367 (0.333, 0.406]
0,984 [0.813, 1.000]
0.535 [0.489, 0.571]
CP-DME
RoC
-0.000
0.378 (0.335. 0.408]
0.814 [0.836, 1.000]
0,516 (0.492, 0.569]
RoCB
-0.012
(0.335, 0.408]
(0.836, 1.000]
[0.492, 0.569]
Bootstrap
RoC
-0,008
0.501 [0,363, 0.450)
0.698 [0,754, 0.9301
0.583
[0.503. 0.5881
RoCB
-0011
0.407 (0.363, 0.450]
0.837 (0.754, 0.930)
0.548 00.503.0.588
CP-Bootstrap
RoC
-0.007
0.454 (0.352. 0.432]
0.733 (0. 804, 0.956]
0.561 (0.50g (0.504. 0.581)
RoCB
-0.024
0.387 (0.352, 0.432]
804, 0.956]
0.539 [0.504, 0.581]
32

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I.1 Threshold-Frie Free Evaluation of RoCB vs RoC Metrics
To address concerns regarding the reliance on threshold-speciffe performance (e.g., Youden's J), we conducted comprehensive threshold-free evaluation of our uncertainty-avvnn biomarker, RoCB, compared against the predicted rate of change (RoC). This analysis spans ten trajectory prediction methods, including DKGP, CP-DRMC, Bootstrap, and others.
For each method, we computed ROC-AUC, PR-AUC, Fl score, precision, recall, and balanced accuracy across 720 test subjects. These metrics were chosen to reflect not only the discriminative ability (AUCs) but also the clinical decision trade-offs in high-risk detection (recall vs. precision).
RoCB consistently improves recall across all models, often by large margins (c.g.. +62% CP- DQR). This demonstrates its strength in identifying high-risk individuals under worst-case prediction scenarios. However, this gain often comes at the cost reduced precision and AUC metrics, reflecting conservative, sensitivity-focusee models, with the largest increase seen CP-DQR (+9.8%), indicating that in certain settings RoCB enhances both sensitivity and overall prediction balance.
behavior. Despite this trade-off, F1-score improves in 6 out of 10
These results confirm that RoCB is a replacement for RoC, but complementary metric that prioritizes reliable early detection. This behavior is particularly beneficial in clinical contexts such as trial enrichment or preclinical screening, where high recall is often more critical than specificity.
Table 5: Threshold-fres evaluation RoC vs. RoCB using ROC-AUC, PR-AUC, Recall, and F1-score. Values shown as (RoC/ / RoCB). LRoC improves recall consistently and maintains or improves of 10 models.
Model DRMC
AUC
PR
Rec
F1
0.597 0.544
0.421 0.355
0.671 0.884
0.528 0.553
0.555
CP-DRMC
0.608 / 0.526
0.421
0.344
0.740 0.911
0.546 0.552
DKGP
0.716/ 0.435
0.523 0.300
0.802/0.922
0.621
CP-DKGP
0.697 0.4338
0.502/0.300
0.721/0.903
0.595 0.531
DQR
0.637 / 0.508
0.441 /0.334
0.791
0.583 /
CP-DQR
0.594/0.483
0.431 0.325
4519/0.844
0.482/0.529
DME
0.495 / 0.412
0.370 0.299
0.814 0.984 0.984
0.516 0.535
CP-DME
0.495 / 0.411
0.370 0.293
0.814 / 0.942
0.516 0.533
Bootstrap
0.686 0.520
0.495 0.343
0.698 0.837
0.583 0.548
CP- Pe-Bootstrap
0.624 / 0.464
0.432 0.314
0.733 0.888
0.561 0.539
33


Reviews:
[{"review_id": "wIVDeLtZGi", "reviewer": "Reviewer_W7CN", "summary": "This paper aims to construct conformal prediction bands for randomly timed trajectories, where outcomes are observed only at random time points and may neither be consecutive nor follow any fixed time grid. The authors propose to utilize normalized nonconformity scores originally introduced in [Yu et al.], in conjunction with the standard split conformal prediction pipeline. Furthermore, they extend their study to group-conditional conformal prediction, leveraging the method from [Romano et al.]. The proposed methodology is applied to two well-established longitudinal clinical datasets. Additionally, the authors introduce a new metric based on conformal prediction bounds to assess the risk of progression from Mild Cognitive Impairment (MCI) to Alzheimer’s disease.\n\n[Yu et al.] Xinyi Yu, Yiqi Zhao, Xiang Yin, and Lars Lindemann. Signal temporal logic control synthesis among uncontrollable dynamic agents with conformal prediction.\n[Romano et al.] Yaniv Romano, Rina Foygel Barber, Chiara Sabatti, and Emmanuel Candès. With malice toward none: Assessing uncertainty via equalized coverage", "questions": "1. Compared to [Yu et al.], what unique challenges arise in extending the method to randomly-timed observations, aside from substituting the predictive model with one capable of handling such data?\n2. In Figure 2, could you please report the standard deviations or confidence intervals associated with the coverage estimates?", "limitations": "No, please refer to the weaknesses section for further details.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: \nThe paper is clearly written, well-structured, presents a clear problem statement, and is easy to follow.\n\nWeaknesses: \n1. Novelty: My main concern lies with the paper's technical novelty. The proposed method essentially applies existing conformal prediction frameworks (as introduced in [Yu et al.]) to the setting where outcomes are observed at irregular, random time points. The normalized nonconformity score presented in Equation (2) appears to be identical to that of [Yu et al.], and the overall methodology follows the standard split conformal prediction paradigm. Similarly, the group-conditional conformal procedure mirrors the approach in [Romano et al.] without apparent modification. That being said, while the setting of randomly-timed observations is practically relevant and underexplored, the methodological contribution appears to be more an adaptation rather than an innovation. If there are subtle but important theoretical or algorithmic contributions to handling the randomness in time stamps, I encourage the authors to clearly articulate and highlight them.\n2. Empirical Evaluation: Some aspects of the experimental results are counterintuitive. If I have misunderstood, I would appreciate clarification. In particular, regarding Figure 4: when reporting stratified coverage by features such as Race, did you partition the test set into mutually exclusive and collectively exhaustive groups (e.g., Asian, Black, and White)? If so, it is puzzling that the conformal coverage appears to fall below the nominal level (e.g., 90%) for all subgroups. If each subgroup undercovers, then by the law of total probability, the overall marginal coverage would also be below target - contradicting the guarantees of marginal conformal prediction. A more detailed discussion or clarification of this point would be helpful.\n3. Limitations: The paper does not provide a dedicated discussion on its limitations.", "was_revised": true, "final_justification": "The authors addressed my concerns regarding novelty during the discussion period and provided helpful answers to my questions. Accordingly, I have raised my score to 3. That said, I still have some reservations, as in my view the contribution—while valuable—offers only a modest advance over the method proposed in [Yu et al.] in the context of the broader conformal prediction literature, which makes me hesitant to fully recommend acceptance at NeurIPS.", "rebuttal": "We would like to thank the reviewer for acknowledging that our paper is **well structured** and **clearly written**.\nIn summary, our response addresses the following **key points**:  \n1. **Major concern:** We elaborate on the contributions of our conformal method based on our detailed comparison in Appendix F. We explain that our method: i) is applicable to randomly-timed trajectories via the **novel use of a normalizing function**, ii) provides **provable coverage guarantees in the context of randomly-timed trajectories**, and iii) is very **impactful for practical applications with randomly-timed trajectories**, as approaches for fixed-time trajectories cannot be applied to such settings. \n2. We provide updated results that  address the issue pointed out by the reviewer regarding performance disparities across groups in population conformal prediction. \n3. As requested, we discuss the limitations of our approach, and we have already revised the manuscript accordingly. \n\nIn our response below, we aim to address in detail each of these concerns, along with several additional questions related to the experimental results.\n\n---\n\n**Weaknesses** \n\n- Our core technical contribution is the introduction of a **normalizing function** that enables conformal prediction for randomly timed trajectories. This adaptation allows nonconformity scores to be evaluated at arbitrary, subject-specific timepoints—supporting valid coverage even under irregular and asynchronous sampling.   The normalizing function enables us to compute a normalized prediction error across all time steps, ensuring that no component within the maximum dominates the others in terms of scale. The above are explained in detail in Appendix F, but in the revised manuscript we will provide intuition for the reader earlier in Section 3. Thanks to this methodological advance, our conformal method is the **first to provide formal coverage guarantees for the setting of randomly-timed trajectories**.\nBeyond its theoretical contributions, our approach addresses a critical practical gap: it is, to our knowledge, the first to implement conformal prediction in the setting of **randomly-timed trajectories**. Prior conformal methods (e.g., Stankeviciute 2021, Yu 2023, Cleaveland 2024) assume fixed, synchronized time grids across subjects, making them unsuitable for real-world clinical data where observation schedules vary widely. Our framework removes this constraint, unlocking conformal prediction for a broad range of applications in longitudinal healthcare modeling.\n\n- We thank the reviewer for pointing out the discrepancy in stratified coverage. Upon review, we identified that when we evaluated the population conformal prediction per race-group we had excluded several subjects due to fallacious filtering. \n We sincerely thank the reviewer for catching this issue. We revised the figure accordingly and below we attach the results for the Race. The rest of the results are unaffected. \nWe add the corrected results too as a table. \n\n*Table: Comparison of coverage under population vs. group-conditional conformal prediction, stratified by race (White, Black, Asian)*\n\n| Race   | Population Conformal Prediction (%) | Group-Conditional Conformal Prediction (%) |\n|--------|--------------------------------------|--------------------------------------------|\n| White  | 92.0                                 | 92.0                                       |\n| Black  | 88.0                                 | 92.0                                       |\n| Asian  | 86.0                                 | 96.0                                       |\n\n- Indeed, we do not explicitly discuss limitations of the current approach (only our section on future work hints to open problems). We will include a dedicated paragraph on limitations in the final version of the paper. One key limitation of our current group-conditional approach is its **lack of scalability** with respect to the number of covariates. While stratifying by a single covariate (e.g., sex, race, diagnosis) allows for interpretable subgroup analysis, it becomes impractical to condition on **combinations of multiple covariates** as this may lead to very small subgroup sample sizes (e.g., Asian, Homogenous for APOE4). Even though our method can be applied to such subgroups, in practice we expect that small sample sizes corresponding to underrepresented subgroups will yield uninformative conformal bands. This restricts the practical applicability of the analysis and may overlook intersectional disparities in model calibration. As discussed in our future work, we aim to extend the group-conditional procedure to **intersectional subgroups** (e.g., Black females aged 55–65), which would provide more clinically relevant and equitable coverage guarantees. Another limitation is that our current framework cannot be directly applied to external datasets while preserving the original coverage guarantees, unless a new calibration step is performed on the target study. When applied to external studies with potentially different demographic distributions, data acquisition protocols, or diagnostic criteria, the assumptions for valid CP are violated. Therefore, without prior recalibration, applying our method to unseen cohorts may lead to invalid or misleading prediction intervals. \n\n**Questions**\n\n-  We would like to clarify that our contribution goes beyond simply changing the predictive model. The key methodological novelty lies in the introduction of a **time-aware normalizing function** used in the nonconformity score, which enables conformal prediction in the context of **randomly timed longitudinal trajectories**. Existing methods, such as *Yu et al.* (2023), assume that all subjects share a common set of fixed observation times across the training, calibration, and test sets. These approaches define normalization terms `σ_t` only at pre-specified time points and therefore provide valid conformal coverage **only at those points**, limiting their applicability to real-world data where visits occur at variable and often irregular times. Attempting to apply such methods to real biomarker trajectories would necessitate discarding a large portion of the dataset — e.g., in our experiments, selecting four fixed time points (3, 7, 9, 12 months) yielded only 37 usable trajectories out of 2200.\nTo address this, we introduce a **normalizing function `σ(·)`** that produces predictive uncertainty estimates `σ(Ŷ_t)` at **any time point `t`**, even if no observations exist at that time in the calibration set. This allows us to compute normalized nonconformity scores `|Ŷ_t - Y_t| / σ(Ŷ_t)` at **arbitrary, subject-specific test-time points**, thus supporting valid coverage under asynchronous and irregular sampling schedules. Crucially, this normalization ensures that no individual time point dominates the score due to scale differences, and that prediction intervals adapt to time-varying uncertainty. We explain the full technical details of this construction in **Appendix F**, but in the revised manuscript, we will provide **earlier intuition in Section 3** to help guide the reader through this key innovation.\n\n\n\n- The confidence intervals are also visualized in the Figure 2a. This is the table with the 95% CI of the coverage metric across the folds: \n\n*Table: 95% confidence intervals for marginal coverage and average interval width across methods, before and after conformalization.*\n\n| Method         | Coverage CI           | Interval Width CI      |\n|----------------|------------------------|-------------------------|\n| DKGP           | [0.9628, 0.9980]       | [1.8508, 1.8517]        |\n| DME            | [0.9070, 0.9687]       | [1.9554, 1.9568]        |\n| DQR            | [0.4099, 0.5380]       | [0.6132, 0.6510]        |\n| DRMC           | [0.4227, 0.5513]       | [0.6524, 0.7467]        |\n| Bootstrap      | [0.3675, 0.4073]       | [0.4771, 0.4915]        |\n| CP-DKGP        | [0.8758, 0.9486]       | [1.2311, 1.2317]        |\n| CP-DME         | [0.8567, 0.9346]       | [1.5976, 1.5987]        |\n| CP-DQR         | [0.8584, 0.9364]       | [1.3722, 1.4677]        |\n| CP-DRMC        | [0.8584, 0.9364]       | [1.6931, 1.9319]        |\n| CP-Bootstrap   | [0.8697, 0.9442]       | [1.9360, 2.1190]        |"}, {"review_id": "XlZAIJTsFm", "reviewer": "Reviewer_Ragn", "summary": "This work proposes a conformal prediction method for longitudinal biomarker trajectories with irregular follow-up times. The proposed time-adaptive nonconformity scores leverage uncertainty estimates to enable conformal predictions of trajectory values at arbitrary time points. The method is demonstrated to outperform baseline approaches in terms of coverage and interval width on brain biomarker trajectories. \nTo address population heterogeneity (e.g., gender, race, and existing risk factors), the authors propose stratifying the calibration data by relevant groups. This approach is demonstrated to attain nominal coverage, unlike their unconditional (non-stratified)  counterparts.\nTo further demonstrate the clinical utility, the authors propose a *lower-bound rate of change* (LRoC) derived from the lower bound of the conformal prediction band, as a tool for identifying patients at high risk of Alzheimer’s disease.", "questions": "1. Figure 2, what the mean interval width is averaging over, the subject index $i$, the time index $t$, or both? If averaging over $t$ is involved, does the irregularly sampled $t$ affect the result?\n\n2. Figure 4, while coverage is attained, why is it further away for the nominal level compared to those in figure 2? Is this due to the baseline prediction model $\\hat Y$ and $\\sigma$ not using the grouping $G$? If so, why the baseline models not using them?\n\n3. Line 349, how is the prediction horizon $t_N$ specified? Further, RoC's current definition in line 361 -- 364 is ambiguous. It is beneficial to add formula for RoC as well, preferably side-by-side to that of LRoC in (3).\n\n4. Line 370, trial eligibility? Isn't this $\\tau$ is just used for identifying high-risk or not?\n\n5. While Table 4 in the appendix includes all results utilizing different base predictors, why Table 1 only shows DRMC? \n\n6. Is it possible to also show threshold $\\tau^*$ in certain way in figure 5 for better illustration?", "limitations": "The manuscript lacks a dedicated discussion of limitations, despite Section 8 providing some directions for future work. While these future directions may implicitly highlight constraints of the current approach, a more explicit and constructive elaboration of limitations, particularly in light of the discussed weaknesses, would be beneficial.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strength:**\n\n1. This work presents a novel extension of the conformal prediction framework to handle longitudinal data with irregularly spaced time points, a critical advantage in clinical and biomedical applications where sampling times are often inconsistent across patients. The reviewer agrees that this addresses an important limitation of existing approaches, which typically require uniformly sampled time points for all trajectories.\n\n2. The proposed method achieves nominal coverage with tight prediction bands compared to baseline approaches that rely on model-specific uncertainty estimates. This is demonstrated on a brain biomarker example, where the necessity of uncertainty calibration—particularly in the presence of population heterogeneity—is clearly illustrated through comparisons with baseline and/or non-stratified methods.\n \n**Weaknesses:**\n\n1. The method relies on time-wise uncertainty estimates to construct normalize nonconformity scores, yet it does not explicitly account for registration problem (e.g., misalignment due to irregular or asynchronous sampling schedules). Such misalignment could distort both the estimated mean trajectory and the uncertainty estimates, which assume a consistent temporal structure. It is unclear whether the authors considered how registration (or lack thereof) might affect the validity of these uncertainty estimates in their data analysis. This omission raises questions about whether the current approach adequately captures the true variability in trajectory dynamics or whether the results could be sensitive to unaccounted temporal misalignment. Future work could explore the impact of registration problem on the proposed uncertainty-aware predictions.\n\n2. The specification of groups (e.g., line 263) is not thoroughly discussed in the manuscript, despite being a critical modeling choice that could influence results. See also my later Question 2. The authors could benefit from a deeper discussion in Section 8.\n\n3. The contrast between RoC (rate of change) and LRoC could have been more effectively leveraged to highlight the clinical value of the proposed uncertainty-aware metrics. However, the current presentation (e.g., Table 1) may not fully clarify such advantage, as the gain in recall could stem from either reduced precision or improved discriminative power. Alternative summaries, such as ROC-AUC or PR-AUC, would provide a more comprehensive evaluation of performance across thresholds, rather than focusing solely on a single Youden-optimized $\\tau^*$ point.", "was_revised": true, "final_justification": "The authors addressed my initial concern regarding the registration problem during the rebuttal. While acknowledging that this issue lies beyond the scope of the current manuscript, they agreed that future investigation in this area could be valuable. Additionally, the authors provided supplementary results employing threshold-free evaluation to further demonstrate the advantages of the proposed LRoC method. They also clarified several implementation details and notations in response to my initial questions. I thank the authors for their prompt and thorough replies. Overall, my assessment of this manuscript is positive.", "rebuttal": "We thank the reviewer for their feedback and for highlighting the key strengths of our work. We are delighted that reviewer recognizes that we **address a critical limitation of prior methods**, while achieving nominal coverage with tight prediction bands compared to baseline approaches. We are also grateful for the reviewer’s appreciation of our empirical demonstration on brain biomarkers and the importance of uncertainty calibration in heterogeneous clinical populations.\n\nIn summary, our response addresses the following **key points**:  \n1. We explain that our conformal method is designed so that it can directly handle irregular trajectories without the need of temporal alignment.\n2. We expand and enhance the comparison between the RoC and LRoC with the requested metrics. \n\n---\n\n**Weaknesses**\n\n- In our setting, each subject's time axis is defined relative to their baseline visit, and our predictors are trained to forecast future biomarker values based on this subject-specific temporal reference frame. Also, we would like to clarify that our method is specifically designed to handle trajectories that are not temporally aligned across subjects. In our formulation, the set $\\(\\mathcal{T}\\)$ of time points at which observations are collected for a given subject is treated as a **random variable**, and our coverage guarantee in **Theorem 3.2** is stated with respect to the joint distribution over the triplet $\\((X, \\mathcal{T}, Y)\\)$. This formulation explicitly incorporates the randomness of observation times **without relying on any assumption of a shared or aligned time axis across subjects**. Furthermore, our nonconformity scores are computed at subject-specific timepoints relative to each individual's baseline, and we do not perform any form of temporal registration or warping. The validity of our coverage guarantee holds under this irregular and asynchronous observation setting and is rigorously proven in **Appendix B**.\n- We thank the reviewer for raising this point regarding group specification. To clarify, the grouping function $\\( G: \\mathbb{R}^d \\rightarrow \\mathcal{G} \\)$, introduced in line 263, is instantiated using single categorical covariates in our current experiments. As described in Section 5.2, we evaluate group-conditional conformal prediction across five clinically relevant stratifications: sex, race, education level, diagnosis, and APOE4 allele status. These covariates are well associated with Alzheimer’s Disease and Brain Ageing  [1,2]. We will elaborate about this in the revised manuscript.\n-  We thank the reviewer for the insightful suggestion to contrast RoC and LRoC using threshold-free evaluation metrics. In response, we conducted a comprehensive analysis across ten trajectory models (e.g., DKGP, CP-DRMC, Bootstrap), using **partial AUC** (PrAUC) computed at varying sensitivity thresholds. Rather than relying on full ROC-AUC, which can obscure performance at clinically relevant operating points, we report **partial AUC at 0.8 sensitivity**, a threshold that offers a strong balance between recall and precision. A sensitivity of 0.8 is clinically appropriate for high-risk screening applications such as early MCI-to-AD conversion detection—prioritizing early identification of converters without overwhelming clinicians with false positives. The difference in partial AUC between RoC and LRoC is negligible for DKGP (Δ = −0.003), reverses in favor of LRoC for DQR (Δ = +0.007), and even improves substantially for DRMC (Δ = +0.032). This supports our claim that **LRoC can enhance conservative prediction under uncertainty without meaningfully degrading AUC-based performance**. These findings clarify that LRoC does not aim to outperform RoC across all metrics, but rather serves a complementary role—offering robust sensitivity under worst-case assumptions. We have included these comparisons in Appendix Table 4 and updated the manuscript to emphasize the 0.8-sensitivity PrAUC as a clinically justified and empirically stable operating point. We will include the additional metrics in Table 4 of Appendix I.\n\n**Questions**\n\n- The mean interval width reported in Figure 2 is computed over both indices $i$ and $t$, as briefly mentioned in line 227 of the manuscript. The averaging over $t$ is necessary for a fair empirical evaluation of our method, since the probability distribution in our coverage guarantee is over triplets (X,$\\mathcal{T}$,Y), where $\\mathcal{T}$ is the set of random points corresponding to a test subject. Our method is specifically designed to handle irregularly sampled trajectories, and the mean interval width we compute provides the empirical average over random subjects with different sets of random time points.\n\n- The coverage differences observed in Figure 4, relative to Figure 2, are primarily due to the use of a **common calibration set** across the population and group-conditional conformal prediction experiments. Specifically, we used a fixed calibration set comprising 20% of the subjects, which was selected based on the calibration set size exploration for the Population CP (see Appendix G.2, and applied the same set when evaluating Group-Conditional CP. This decision was made to ensure a **fair comparison** between population and group-conditional methods under consistent data conditions. However, we would not expect the calibration set size that we selected for the Population CP to also provide tight intervals for the Group-Conditional CP. **In practice, if we were to apply Group-Conditional CP, we would make another exploration (similar to that in Appendix G.2) for the calibration set size that would result in tight bands for the individual groups.**   In particular, we would select the calibration set size **separately for each group** (e.g., CN, MCI, AD) to achieve tighter adherence to the nominal level. \n- The prediction horizon is defined individually for each subject as the time span between their baseline and last available observation. On average, this corresponds to **43 months** (standard deviation: **32 months**), but the horizon varies across subjects due to irregular follow-up schedules. We will clarify this point in the revised manuscript and will also **update the notation** throughout the paper to make it explicit that both $t_0$ and $t_N$ are **subject-specific** (i.e., indexed by subject $i$).\n\n    We agree that the mathematical definition of Rate of Change (RoC) in lines 361–364 should be included in the manuscript. In the revision, we will provide an explicit formula for RoC and place it side-by-side with the LRoC definition in Equation (3). Specifically, we define RoC for subject $i$ as:\n$$\n\\operatorname{RoC}_i = \\frac{\\hat{y}_i(t_N^{(i)}) - y_i(t_0^{(i)})}{t_N^{(i)} - t_0^{(i)}} \n$$\nWhere:\n- $t_0^{(i)}$ and $t_N^{(i)}$ denote the subject-specific baseline and final timepoints,\n- $\\hat{y}_i(t)$ is the predicted biomarker value at time $t$.\n\n- In our framework, the term “trial eligibility” refers to identifying high-risk subjects likely to experience rapid decline toward Alzheimer’s disease. Such subjects are eligible to be included in a clinical trial and therefore the terms “trial-eligible” and “high-risk” subjects are used interchangeably in the manuscript. In the revised manuscript, we will make sure to clarify the above.\n\n- Table 1 focuses on DRMC as a representative baseline due to space constraints in the main manuscript. However, as shown in Appendix I Table 4, the same conclusion holds across all base predictors—namely, that LRoC improves recall relative to RoC, often with comparable or improved F1 scores. We will include the full set of results in the main paper in the final version where space permits it.\n\n- We will update the corresponding plot to include a decision boundary line with slope equal to the Youden-optimized threshold. This line will clarify how the threshold corresponds to the separation criterion in the LRoC and CP-LRoC. \n\n**Limitations**\n\nIndeed, we do not explicitly discuss limitations of the current approach (only our section on future work hints to open problems). We will include a dedicated paragraph on limitations in the final version of the paper. \nOne key limitation of our current group-conditional approach is its **lack of scalability** with respect to the number of covariates. While stratifying by a single covariate (e.g., sex, race, diagnosis) allows for interpretable subgroup analysis, it becomes impractical to condition on **combinations of multiple covariates** as this may lead to very small subgroup sample sizes (e.g., Asian, Homogenous for APOE4). Even though our method can be applied to such subgroups, in practice we expect that small sample sizes corresponding to underrepresented subgroups will yield uninformative conformal bands. This restricts the practical applicability of the analysis and may overlook intersectional disparities in model calibration. As discussed in our future work, we aim to extend the group-conditional procedure to **intersectional subgroups** (e.g., Black females aged 55–65), which would provide more clinically relevant and equitable coverage guarantees. Another limitation is that our current framework cannot be directly applied to external datasets while preserving the original coverage guarantees, unless a new calibration step is performed on the target study. When applied to external studies with potentially different demographic distributions, data acquisition protocols, or diagnostic criteria, the assumptions for valid CP are violated. Therefore, without prior recalibration, applying our method to unseen cohorts may lead to invalid or misleading prediction intervals. \n\n[1] Corder, E. H., et al. (1993). Gene dose of apolipoprotein E type 4 allele and the risk of Alzheimer's disease in late onset families\n\n[2] Meng, X., et al. (2012). Education and dementia in the context of the cognitive reserve hypothesis: a systematic review with meta-analyses and qualitative analyses"}, {"review_id": "DPrL2THP7k", "reviewer": "Reviewer_FjGd", "summary": "This paper introduces a method to estimate continuous biomarker trajectories with uncertainty bands from sparsely and irregularly sampled data, such as patient visits during a chronic disease journey.  The technical novelty is to use conformal methods, which previously rely on fixed time points, but to extend the formulation to accommodate irregular time points.  The authors demonstrate the method on an Alzheimer’s data set using standard image-derive biomarkers, hippocampal and ventricular volume.  Results show that a method that uses the lower bounds of the uncertainty bands more reliably detects progressors from MCI to AD than a standard method.", "questions": "What advantages does the conformal method offer over alternatives that can estimate uncertainty bands such as the Gaussian process model mentioned above?", "limitations": "Fine", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nThe formulation seems nice, although I am not particular familiar with conformal methods.\n\nThe problem is real – both the reconstruction of biomarker trajectories from irregular time points and the estimation of uncertainty bands – and has genuine utility in clinical and application and not doubt other application areas.\n\nResults show some potential benefit of the approach.\n\nWeaknesses\n\nThere’s no comparison against other methods that can estimate uncertainty bands.  The obvious choice would be Gaussian process models.\n\nFurther to the point above, one limitation of the proposed method is that it only considers biomarkers individually.  Other work on disease progression modelling – see https://pubmed.ncbi.nlm.nih.gov/38191721/ for an overview – typically creates models of multiple biomarker trajectories on the same time axis, which is more powerful.  In particular, Lorenzi et al’s Gaussian Process disease progression model https://www.sciencedirect.com/science/article/abs/pii/S1053811917307061, considers multiple biomarkers with uncertainty bands so seems on the face it more advanced than the proposed method.", "was_revised": true, "final_justification": "It's early stage.  The extension to multiple biomarkers is essential for this to be relevant.  Still feel the paper lacks a bit the wider context of work on disease progression modelling - as mentioned in my original review - but I do think this paper is acceptable so increasing score one notch.", "rebuttal": "We thank the reviewer for their feedback and their thoughtful review. We are happy that the reviewer found our problem—reconstructing biomarker trajectories from **irregular time points and quantifying uncertainty** through prediction intervals—**genuinely useful for clinical and other application areas**, and saw the **potential benefit** of our approach. \n\nIn summary, our response addresses the following **key points**:  \n1. We explain how our conformal method **compares with five baseline uncertainty quantification methods** (e.g., Gaussian-process based bands) based on our comparative case from Section 4.\n2. We elaborate on how our conformal algorithm can be **extended to account for multiple biomarkers**.\n\n---\n\n**Weaknesses** \n\n- Our current comparison in Section 4 consists of a total of five baselines, including Gaussian Process-based baselines—see also Appendix for comparative experiments for Sections 5 and 6. Specifically, both **DKGP** (Deep Kernel Gaussian Process) and **DME** (Deep Mixed Effects model) are GP-based methods capable of estimating uncertainty bands. In figure 2 you can notice that we compare the DKGP and DME with their conformalized counterparts, where our approach is able to achieve the nominal coverage with tighter bands.\n\n- While our current implementation focuses on single-biomarker prediction bands, our conformal algorithm can be directly extended to multivariate predictors that jointly model multiple biomarker trajectories. As noted in the future work section, we are actively pursuing this direction. A natural generalization involves defining the nonconformity score as the maximum normalized residual across all biomarkers and timepoints, allowing for multivariate trajectory-wise conformal coverage. More specifically, this means that in the current expression of the nonconformity scores we would add an additional index $j=1,...,B$, ranging over all $B$ biomarkers we care about and the same index would be added in the true and predicted biomarker values, which will now be denoted by $Y_{t,j}^{(i)}$ and $\\hat{Y}_{t,j}^{(i)}$, respectively.\n\nNote also that our method is not intended to substitute specific predictive models, but rather to provide a conformalization procedure that can be applied to **any base predictor**—including multivariate models such as the Gaussian Process disease progression framework proposed by Lorenzi et al. While GP-based models are powerful, they come with limiting assumptions (e.g., Gaussianity, stationarity) that may not hold in clinical data. Our conformal approach complements these models by providing **distribution-free coverage guarantees** in the presence of model misspecification.\n\n**Questions**\n\nGP-based methods estimate uncertainty based on modeling assumptions, but often produce overly conservative intervals, as shown in Figure 2 for DKGP and DME. Frequentist methods like Bootstrap and Monte Carlo Dropout can underestimate uncertainty and lead to narrow, miscalibrated bands. Our conformal method improves both cases: it can be applied on top of any of these predictors to produce **narrower or better-calibrated intervals** that achieve valid coverage guarantees. Unlike model-based or sampling-based methods, our conformal prediction method makes no distributional assumptions and provides formal coverage guarantees that hold by construction."}, {"review_id": "Q7hoF8QACX", "reviewer": "Reviewer_uuzK", "summary": "The paper presents a nice application of conformal prediction, for the prediction of biotracker trajectories with random observation times.\nThe task is as follows: in an initial consultation (at time step t_0 = 0) we collect covariates $X$, potentially with some biomarker reading $Y_0$.\nWe then want to predict 'confidence bands' $C_t(X)$ for the trajectory $Y_t$ (for $t \\in [0, T]$ or some discretization thereof) such that, given some random times $\\mathcal{T} = \\{T_1, ..., T_k\\}$, it holds that\n$\\mathbb{P}[\\forall t \\in \\mathcal{T},\\ Y_t \\in C_t(X)] \\geq 1 - \\alpha$ for some $\\alpha$ chosen a priori.\nThe authors approach this via split conformal prediction techniques -- namely (vanilla) split conformal and group-conditional split conformal prediction -- with a conformity score designed for their setting.\nThey then go on to show the applicability of their approach in a real-world setting involving the prediction of brain biomarkers, highlighting the increased reliability when using their method, as well as the advantage of doing group-conditional CP with well-chosen covariates.\nThey conclude by showing how their uncertainty quantification can be used to confidently identify high-risk individuals, by building upon a metric that is reasonably well-established in the existing practice.", "questions": "My main concern is as stated above, and led me to give a borderline accept rather than a straight accept.\nCould the authors perhaps elaborate more on either (i) how their methodology differs from existing CP methods besides the conformity score, and/or (ii) why the problem being solved would be of particular interest to the NeurIPS community, in contrast to the many other problems straight-forwardly solvable by CP? Are there any additional properties enjoyed by CP in the specific setting being considered?", "limitations": "Limitations were appropriately discussed.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The application presented in the paper is very nice, and the writing is easy to follow.\nThe methodology is sound, well-motivated and achieves good results.\nMy understanding of the specific example of brain biomarkers is limited, but the larger problem of biomarker prediction with random observation times is broadly applicable and relevant.\n\nMy main concern is that, at the end of the day, the proposed method reduces to simple split conformal and group-conditional split conformal prediction, treating the biomarkers at the random times as the labels. The only significant difference is a novel conformity score (which is a straightforward modification from existing ones, involving only adding a maximum over the time).\n\nThat said, the methodology and experimental validation are quite sound.\nThe section on group-conditional conformal prediction also cleanly presents the advantage of conditional guarantees. It's worth mentioning that the covariates used for the groups are some of the more comprehensive I have seen in applied conformal prediction (i.e., more than the usual applied CP paper), and clearly were carefully thought through.\n\nI think it's also worth noting that there is some more recent work on CP that might be of interest to the authors. For example, stronger conditional guarantees than simple group-conditional CP can be attained via the work of https://arxiv.org/abs/2305.12616.\nAlso, similar ideas of involving maxima/suprema in the computation of the conformity scores have been considered in other contexts, e.g. in strategic CP (https://arxiv.org/abs/2411.01596) and in CP with adversarial robustness (e.g. https://arxiv.org/abs/2503.05239).\n\nFinally, some minor nitpicks and suggestions:\n- line 153-154: $\\sigma$ might not be a function of (just) $\\hat{Y}_t^{(i)}$, right?\n- It would be best to make it clear in Problem 3.1 (line ~135) that though the guarantee cares only for $t \\in \\mathcal{T}$, the conformal bands are defined for all $t \\in [0, T]$ (or some discretization thereof).\n- Regarding the checklist:\n    - Question 2 is marked as [TODO], but was answered with [YES] (judging by the justification).\n    - Data&code: while I don't think it is essential that the code for this paper be reviewed, I strongly recommend that it be indeed published with the paper.\n    - CIs on metrics: the authors claim in the checklist to provide CIs of all metrics, however I cannot find them for the tables with Precision, Recall & F1 (e.g. Table 1, and in the supplementary material). These can be easily computed e.g. with bootstrap, right?\n    - Declaration of LLM usage is missing?", "was_revised": true, "final_justification": "One one hand, I think this is a nice applied paper; I particularly appreciate that rather than just doing conformal, the authors care about what benefits conformal guarantees actually give in practice. On the other hand, the method introduced has relatively little novelty, as noted during the discussion period. While the solid application makes me lean towards acceptance, this limited methodological novelty keeps me from giving the paper a score of 5.", "rebuttal": "We thank the reviewer for their extensive and constructive feedback. We are very glad that the reviewer found the application to be **well-motivated**, the methodology and experimental validation **sound**, and the writing **clear and accessible**. We also appreciate your positive comments on the **broad relevance** of predicting biomarkers under randomly timed observations, as well as our **carefully designed group-conditional covariates**, which you noted as **among the most comprehensive seen in applied conformal prediction** and go beyond what is typically explored in the literature.\n\nIn summary, our response addresses the following **key points**:  \n1. We address all the suggestions (include the CIs in Table 1 and Table 4 of the paper) \n2. We elaborate on the novelty of our work as well as the impact on the NeuRIPS community.\n\n---\n\n\n**Weaknesses** \n\n- We agree with the reviewer that the normalizing function $\\sigma$ does not depend only on $\\hat{Y_t}^{(i)}$. A more complete notation would be $\\sigma(\\hat{Y_t}^{i},X_t;f)$, with $f$ denoting the predictor and $X_t$ the input at time $t$. This emphasizes that the uncertainty estimate depends not only on the predicted value $\\hat{Y_t}^{(i)}$ but also on the covariates as well as properties of the predictor $f$ used to generate it. In the manuscript, we used the simplified notation $\\sigma(\\hat{Y}_{t}^{(i)})$ for readability. We will clarify in the final version that this is a simplified form and that the normalizing function is indeed dependent on other quantities.\n\n  To make this dependence concrete:\n    - For Gaussian Process-based predictors (e.g., DKGP, DME), $\\sigma$  returns the posterior predictive standard deviation from the model.\n    - For frequentist ensemble methods (e.g., Bootstrap, Monte Carlo Dropout), $\\sigma$ corresponds to the empirical standard deviation  across multiple forward passes or resampled models at time $t$ .\n\n\n- Thank you for the helpful comment. We will make the clarification in Problem 3.1 as you suggested. \n- Regarding your question about the CIs, in the revised manuscript  we have included the CIs in both tables. Here we present for reference the updated results for the Table 1 of the main manuscript: \n\n*Table: Youden-optimized discrimination on $z$-standardized rate of change (RoC) and lower-bound rate of change (LRoC) for MCI converters, with 95% bootstrap confidence intervals.*\n\n| Method       | Metric | $\\tau^*$   | Precision        | Recall           | F$_1$             |\n|--------------|--------|------------|------------------|------------------|-------------------|\n| DRMC         | RoC    | $-0.006$   | **0.436 ± 0.022** | 0.671 ± 0.058    | 0.528 ± 0.023     |\n|              | LRoC   | $-0.012$   | 0.403 ± 0.022     | **0.884 ± 0.058** | **0.553 ± 0.023** |\n| CP--DRMC     | RoC    | $-0.006$   | **0.432 ± 0.022** | 0.740 ± 0.095    | 0.546 ± 0.024     |\n|              | LRoC   | $-0.020$   | 0.395 ± 0.022     | **0.915 ± 0.095** | **0.552 ± 0.024** |\n\n   We have also updated the Table 4 of Appendix accordingly. \n\n- Also we plan to release the code upon acceptance and we have made the modifications in the checklist. \n\n**Questions**\n\nWe appreciate the opportunity to clarify the broader significance of our work. To the best of our knowledge, this is the **first conformal prediction framework tailored to randomly timed biomarker trajectories**, a setting that arises in clinical studies. Existing CP methods assume fixed or synchronized timepoints across subjects, which makes them incompatible with the asynchronous and irregular observation patterns present in real-world clinical data. \nWhile our nonconformity score is a key adaptation, our contribution goes beyond that: we formally extend split conformal prediction to the setting where time index \\(\\mathcal{T}\\) is random. This includes a **normalization mechanism** that enables inference at arbitrary, unseen timepoints and supports guarantees even under irregular follow-up—a critical requirement for clinical deployment **with real data from clinical studies**. In Appendix F, we explain in detail how **methods designed for fixed-time trajectories cannot be applied in our setting**, which highlights the **importance of our results mainly from a practical (rather than a technical) perspective**.\nThis problem setting is of particular interest to the **NeurIPS AI for Health and Medicine community**. Our approach not only enables **distribution-free guarantees** in this setting but also highlights real-world challenges (e.g., misaligned visit times, subgroup disparities, calibration under irregular sampling) that are underexplored in the CP literature.\nAdditionally, our paper outlines several **methodological and applied directions** for future research—such as multivariate conformal prediction, adaptive methods for evolving trajectories, and group-conditional guarantees for intersectional subgroups—which we believe will help **bridge the gap between theory and practice** in clinical ML applications."}]

Author rebuttal:

Meta-review: This paper presents a method for constructing conformal prediction bands for longitudinal biomarker trajectories observed at irregular and random time points, addressing a common challenge in clinical data where visits do not follow a fixed schedule. The approach builds on normalized nonconformity scores introduced in Yu et al. and employs standard split conformal prediction, further extending to group-conditional conformal prediction following Romano et al. The methodology is demonstrated on Alzheimer’s disease biomarkers (e.g., hippocampal and ventricular volume) using two well-established clinical datasets, with a proposed lower-bound rate of change (LRoC) metric to identify patients at higher risk of progression from MCI to Alzheimer’s.

Strengths noted across reviews include the paper’s clarity, structure, sound methodology, and practical relevance for clinical applications. The approach is recognized for adapting conformal prediction to irregular time points, achieving nominal coverage with relatively tight prediction bands, and carefully considering covariates in group-conditional analysis. Reviewers highlight the method’s potential utility in reliably detecting disease progression, its applicability to heterogeneous populations, and its straightforward yet effective application of conformal principles. However, multiple weaknesses are also raised. Several reviewers question the level of technical novelty, noting that the method largely adapts existing frameworks (split conformal, group-conditional conformal, normalized scores from Yu et al.) with only minor modifications. Others flag omissions: the lack of discussion on trajectory registration/misalignment problems, the absence of comparison to established alternatives, and the restriction to single biomarkers rather than joint modeling across multiple trajectories. Empirical concerns are also raised, including seemingly inconsistent subgroup coverage results (potentially contradicting marginal guarantees), insufficient discussion of how groups were specified, and underdeveloped analysis of the added value of LRoC relative to other metrics. Minor issues include missing confidence intervals on reported metrics, absent code release, and missing declarations in the checklist. Overall, reviewers find the application relevant and promising but encourage stronger articulation of novelty, deeper evaluation, and broader comparisons to strengthen the contribution.

On balance, I concur with the reviewers that the paper’s technical novelty is limited, as it largely repurposes existing conformal prediction frameworks with only modest extensions. Nonetheless, it addresses a practically important and underexplored problem i.e. uncertainty quantification for irregularly sampled biomarker trajectories, and demonstrates that conformal prediction can be applied effectively in this clinically significant setting. In that sense, while the contribution may not be theoretically groundbreaking, it is a well-executed and meaningful application that offers real value.
Decision: Accept (poster)