Title: NeuralSurv: Deep Survival Analysis with Bayesian Uncertainty Quantification
Venue: NeurIPS 2025
Accepted: yes
Keywords: Survival analysis

Variational Inference

Deep Learning

Bayesian Neural Networks

Local Linearization

Polya-Gamma
Abstract: We introduce *NeuralSurv*, the first deep survival model to incorporate Bayesian uncertainty quantification. Our non‑parametric, architecture‑agnostic framework flexibly captures time‑varying covariate–risk relationships in continuous time via a novel two‑stage data‑augmentation scheme, for which we establish theoretical guarantees. For efficient posterior inference, we introduce a mean‑field variational algorithm with coordinate‑ascent updates that scale linearly in model size. By locally linearizing the Bayesian neural network, we obtain full conjugacy and derive all coordinate updates in closed form. In experiments, *NeuralSurv* delivers superior calibration compared to state-of-the-art deep survival models, while matching or exceeding their discriminative performance across both synthetic benchmarks and real-world datasets. Our results demonstrate the value of Bayesian principles in data‑scarce regimes by enhancing model calibration and providing robust, well‑calibrated uncertainty estimates for the survival function.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work introduces NeuralSurv, a deep survival analysis framework with Bayesian uncertainty quantification. To do so, it proposes a two-stage data augmentation strategy using Pólya-Gamma variables and marked Poisson processes to enable exact continuous-time inference. A local linearization technique makes Bayesian neural networks tractable via closed-form coordinate-ascent updates. Experimental results highlight the performance of the method in discrimination and calibration in data-scarce regimes.\n\nSTRENGTHS AND WEAKNESSES:\nI congratulate the authors on their work, and I am grateful for the opportunity to have reviewed this paper. The paper is well-written and a pleasure to read.\n\nStrengths:\n* The work is **very timely**: it takes steps to tackle an important problem in the in the survival analysis literature that I've noticed over recent years. As we obtain the means to statistically learn ever-more-granular survival curves (e.g., via expressive methods like SODEN [1] or mixture density networks [2]), there may or may not be support in the data for the granularity of survival curves that are being learned. As such, developing rigorous Bayesian mechanisms to credibly imbue individual survival curve estimates is an important problem. This work represents strong first steps in this direction -- e.g., the confidence intervals in Figure 1 are _exactly_ what a lot of practitioners are looking for in applied settings.\n* The experiments, as presented, are strong -- the method seems to outperform standard baselines on both concordance and calibration. I recommend the authors highlight these results not as evidence of utility (discussed more in Weaknesses), but rather as evidence that the model's design decisions (e.g., Weibull-like baseline hazard; isotropic Gaussian prior on $\\theta$; etc.) are not too restrictive for practical settings.\n* The NeuralSurv framework seems fast and scalable. By performing local linearization around the MAP and mean-field variational inference, the authors implement a closed-form coordinate ascent update with linear time complexity in model size.\n\nWeaknesses:\n* Related work and novelty: I admit that I do not have the deepest knowledge of Bayesian inference. With this in mind, the paper makes it rather difficult for me to tease apart which elements of NeuralSurv represent novel contributions, and which represent established methods in Bayesian inference. I would appreciate some clarity on this. My understanding is that the Polya-Gamma scheme in 3.1 is an established contribution; the Poisson Process Augmentation Scheme in 3.2 is novel; the Variational Mean-Field Approximation in 4.1 is an established method that is novel when applied to the survival setting; the local linearization of the BNN in 4.2 is established; and CAVI in 4.3 is established.\n  * I would appreciate if the authors added a Related Work section so that elements like this can be discussed, and so that their framework can be placed into conversation with other survival analysis frameworks.\n* There is a degree of misalignment between the experiments and the proposed contribution of the work. Whereas the experiments highlight concordance and calibration (via integrated Brier score), my understanding is that the core contribution of the work is not to implement the best ranking metric (or calibrated survival curve), but rather to (a) support the integration of prior knowledge into estimation, and (b) permit easy computation of a credible confidence interval. With that in mind, I would prefer that the authors include three additional experiments of the following form and to reframe the discussion of contributions accordingly:\n  1. Demonstrate how inference under different priors influences model performance as evaluated using classical statistics (c-index, IBS). This would inform the reader of the extent to which prior selection influences model performance on held-out data, as well as providing some guidance as to what may make for a \"good\" vs. \"bad\" prior in the survival setting.\n  2. Show several examples of survival curve estimation, both from the data-rich regime of the distribution (e.g., a patient who is similar to many other patients), and from the data-scarce regime (e.g., a patient who is dissimilar to many other patients). The aim of this experiment would be to demonstrate how the uncertainty in survival curves varies as a function of distributional support.\n  3. How do the credible intervals using this Bayesian setting compare to those computed via other methods? The authors may know the baselines here better than I do, but I'd imagine some baseline of bootstrapped confidence intervals and Monte Carlo dropout [3] would be important points of comparison.\n* I understand that the authors chose their experimental setup to highlight the performance of NeuralSurv in data-scarce regimes; but would it be possible to include additional experimental results on the performance of each method using the entire dataset (not subsampled to 100 train / 25 validation samples)? It would be good to get a sense of how the method performs as the dataset size increases. This would also allow for a more standardized comparison against the baseline methods, which are often evaluated on the entirety of COLON, METABRIC, and GBSG.\n4. The authors claim that their method supports \"time-varying covariate–risk relationships\" (Line 300); however, I don't see anything in their experimental framework supporting this. I see that $X=\\{x_i \\,:\\, i=1, ..., N\\}$ (Line 56); if the covariates were time-varying (e.g., as in [4]) I would have expected a different setup (e.g., $X=\\{x_i(t) \\,:\\, i=1, ..., N, t=0, ..., T\\}$). Could this be clarified?\n\n\n[1] Tang, Weijing, Jiaqi Ma, Qiaozhu Mei, and Ji Zhu. \"Soden: A scalable continuous-time survival model through ordinary differential equation networks.\" Journal of Machine Learning Research 23, no. 34 (2022): 1-29.\n\n[2] Han, Xintian, Mark Goldstein, and Rajesh Ranganath. \"Survival mixture density networks.\" In Machine Learning for Healthcare Conference, pp. 224-248. PMLR, 2022.\n\n[3] Gal, Yarin, and Zoubin Ghahramani. \"Dropout as a bayesian approximation: Representing model uncertainty in deep learning.\" In international conference on machine learning, pp. 1050-1059. PMLR, 2016.\n\n[4] Lee, Changhee, Jinsung Yoon, and Mihaela Van Der Schaar. \"Dynamic-deephit: A deep learning approach for dynamic survival analysis with competing risks based on longitudinal data.\" IEEE Transactions on Biomedical Engineering 67, no. 1 (2019): 122-133.\n\nQUESTIONS:\n1. Could the authors more clearly clarify which elements of the work are their own novel contribution versus drawn from other insights in the field of Bayesian inference, as well as adding a related work section providing background Bayesian inference and survival analysis methods? [I would raise my clarity score from \"2: fair\" to \"3: good\"; depending on the result of this discussion I may raise my Originality score from \"2: fair\" to \"3: good\"]\n2. Could the authors implement the following experiments, as discussed in Weaknesses? [I would raise my overall Rating from \"4: Borderline accept\" to \"5: Accept\" with these additional evaluations]\n    * Demonstrate the effect of different priors on concordance / IBS.\n    * Highlight survival curve uncertainty bounds in data-rich vs. data-scarce settings.\n    * Comparison against baseline methods of computing uncertainty intervals on survival curves.\n    * Include additional experimental results that use the whole dataset (rather than the subsampled component).\n3. Could the authors include confidence intervals on their evaluation statistics (c-index, Brier score), along with appropriate significance testing and multiple-comparisons correction? [this is another element that would contribute to me raising my overall Rating to a 5]\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces NeuralSurv, a deep survival model that offers an architecture-agnostic framework for continuous-time survival analysis that can capture complex time-varying covariate-risk relationships. The method includes a two-stage data-augmentation scheme utilizing latent marked Poisson processes and Pólya-Gamma variables. The BNN s approximated with local linearization and trained with a variational mean-field objective. The authors demonstrate good calibration and discriminative performance on synthetic and real-world datasets, particularly in data-scarce scenarios.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths \n\n- The contribution is original and significant. this paper provides a first framework for neural network-based survival analysis, with very reasonable choices and construction that are well-placed in literature. This allows following works to study survival analysis leveraging NN's expressivity to model complex time-varying covariate-risk relationships, while having relatively reliable uncertainty quantification.\n\n- Strong empirical performance on the synthetic and real world datasets. \n\n- overall very well-written paper - all the sections are clear and coherent, and the method is well-explained in a step-by-step manner. although I am not an expert on variational inference, I am still able to follow the methodology and the math in the main text. I appreciate the primers in the appendix as well. \n\nWeaknesses \n\n- Lack of discussion on sensitivity to hyperparameters and initialization. the method relies on multiple steps of data augmentation and approximation for VI. my experience with BNNs is that that they are brittle to hyperparameter choices and initializations. From the experiments section of the paper, it seems like the experiments were performed once with a single architecture. No error bars on metrics from multiple runs (which i would expect from statisticians? maybe i'm wrong). It would be great to have a discussion on the robustness of the method.\n\nQUESTIONS:\n- question about hyperparameters and initializations as elaborated in the weakness section. \n\n- is it possible to have a table to compare runtime with baseline methods as well? i understand that runtime is already discussed in discussion and appendix K, but quantitative comparison will be informative to researchers trying to using your method. (from my understanding NeuralSurv is already quite fast.)\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces NeuralSurv, a Bayesian deep learning framework for survival analysis that incorporates uncertainty quantification. The method leverages a two-stage data augmentation scheme involving Polya-Gamma variables and marked Poisson processes to enable efficient posterior estimation. The authors demonstrate superior calibration and competitive discriminative performance compared to state-of-the-art deep survival models on both synthetic and real-world datasets.\n\nSTRENGTHS AND WEAKNESSES:\nThe authors provide a Bayesian uncertainty estimate for survival functions. However, it is unknown how the proposed method relies on the prior distributions.\n\nQUESTIONS:\n1.\tThe authors should expand their discussion by referencing more literature on deep survival models, particularly focusing on key results and their limitations.  \n2.\tThe sigmoidal hazard model in (1) depends on the distribution of \\(\\theta\\). Since the random variable \\(Z(t,x)\\) in (5) is also influenced by the distribution of \\(\\theta\\), the same applies to model (1). Two critical questions arise: (a). How sensitive is the model to the prior distribution of \\(\\theta\\) when its dimensionality is high?  (b). Does the model (1), (5), and (6) converge to a fixed model as the dimension of \\(\\theta\\) approaches infinity?  \n3.\tWhat is the connection between the proposed model (1), (5), (6) and classical survival models such as the Cox proportional hazards model or the Accelerated Failure Time (AFT) model? Specifically, can the Cox or AFT model be derived as a special case of the proposed framework?  \n4.\tThe integral in the likelihood function involves only one variable, which is computationally tractable. However, the data augmentation strategy employed by the authors appears to be more computationally intensive. Could the authors provide a comparative analysis of the computational times for these two implementations?  \n5.\tCould the authors clarify which classes of models are not encompassed by their proposed framework (1), (5), (6)?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce a Bayesian neural survival analysis model. They use a neural network with sigmoid activation to modulate a parametric baseline hazard model based on observed covariates, which makes their model highly flexible and able to accommodate settings where the relationship between covariates and risk varies over time. After discussing the likelihood and placing suitable priors on neural network baseline hazard model parameters, respectively, they present a data augmentation strategy in which they represent the sigmoid function using a set of Polya-gamma random variables, then express the intractable double integral in the resulting likelihood as an expectation over a marked Poisson process. They use mean field variational inference to approximate the posterior distribution, noting as a limitation that this assumes neural network parameters are independent of the parameters of their hazard model. They conduct experiments on synthetic datasets, with the number of samples ranging from 25 to 150, followed by 8 real-world datasets, each subsampled to 125 samples. The evaluation shows that their method tends to improve on the concordance index and integrated Brier score compared to a comprehensive set of alternatives, and that the true survival function falls within their 90% credible interval in the synthetic data examples.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Addresses a true need for a fully Bayesian yet highly flexible survival analysis method\n- Method is flexible and elegant\n- Paper is well organized and generally well written\n- Experiments are compelling and illustrate the method's effectiveness and advantages for small datasets\n\nWeaknesses:\n- The method appears to be limited to very small samples. This is discussed briefly, notably in the conclusions section, and the authors note that extending e.g. to a mini-batch setup is a possible direction for future work. Still, they should expand on the details here and present runtime results for each experiment. If the method is not scalable in its current form, this is OK in my view but should be clearly stated.\n- A key benefit of the method is its flexibility in modeling the hazard function, but typically this kind of flexibility is becomes advantageous only for larger datasets, whereas their method is currently constrained to smaller datasets.\n- A substantial portion of the paper concerns the data augmentation strategy. The authors note in lines 149-150 that \"analagous data augmentation schemes have been proposed in [9, 44, 2]\". However, I think they should note this at the beginning of the section rather than the end, and they should also do more to explain how their approach is similar versus different from those already proposed. I think the authors' overall contribution is substantial even if this particular aspect is not very novel, so there is no need to obscure what is established versus new here.\n- The initial presentation of \\lambda_0 in section 2 is confusing, because it's not clear why or how \\lambda_0 depends on x. This becomes clear in 2.2, but I think it would be better to briefly comment on this when initially introducing \\lambda_0.\n\nQUESTIONS:\nMy questions are related to the weaknesses described above. Namely:\n- How well (or poorly) does the method scale to larger datasets? Please be clear about this and show runtimes.\n- How exactly does the current data augmentation strategy relate to those already proposed in [9, 44, 2]? Please be up front and detailed about this.\n\nLIMITATIONS:\nThe authors have described weaknesses but more could be done particularly around scalability & runtime, as I discuss in the \"weaknesses\" section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel survival deep learning method within a Bayesian framework. Survival analysis is a critical tool for generating personalized time-to-event predictions while effectively handling censoring distributions.\n\nThe authors propose a new variational inference algorithm that leverages an augmented data scheme. This scheme utilizes Pólya-Gamma random variables to approximate the sigmoid function and Poisson variables to tractably manage the posterior distribution of their likelihood.\n\nThe proposed method, a new Bayesian Neural Network (BNN), demonstrates superior performance on several datasets when evaluated using the Integrated Brier Score (IBS) and the C-index (a discrimination metric). It consistently outperforms other deep learning methods while maintaining scalability.\n\nSTRENGTHS AND WEAKNESSES:\nTo enhance the accessibility of the paper, it would be beneficial to include a concise introduction to key concepts in survival analysis e.g. the hazard function (\\lambda(t)), the survival function (S(t), the censoring distribution)\n\nStrengths: \n- Using Bayesian methods with a prior is not really used in the literature, and using BNN is quite a novelty. \n- Integrating this method into survival analysis offers practitioners the valuable ability to incorporate domain-specific constraints or prior beliefs, which can significantly enhance model interpretability and robustness.\n- A lot of details are provided in the manuscript on the experimental part and on the theoretical part. \n- the proofs presented in the appendix are easy to follow, contributing to the overall rigor of the paper.\n\nWeaknesses: \n- My main concern pertains to Table A5, where standard machine learning methods consistently outperform the authors' method in terms of both IBS and C-index across all datasets. While the authors claim their method may excel in high-dimensional feature settings, no experimental evidence is provided.\n- You may need to clearly write your assumptions and the different key points in your reasonning.  \n- The Integrated Brier Score (IBS) can be biased due to the unknown censoring distribution inherent in real-life datasets. Other calibration metrics, such as Kaplan-Meier (KM) calibration [1, 2] and D-calibration [1, 2], have been developed in survival analysis to provide a more accurate assessment of calibration and should be considered.\n- The authors did not compare their method against approaches that train neural networks using proper scoring rules, such as SumoNet [3] or the method proposed by Yanagisawa [4]. This comparison would provide valuable insights into the relative performance of the proposed BNN.\n\n[1]: Haider, Effective Ways to Build and Evaluate Individual Survival Distributions, 2018\n\n[2]: one implementation in Python of those calibration measures: https://github.com/shi-ang/SurvivalEVAL\n\n[3]: Rindt, Survival Regression with Proper Scoring Rules and Monotonic Neural Networks, 2021 \n\n[4]: Yanagisawa, Proper Scoring Rules for Survival Analysis, 2024\n\nQUESTIONS:\nTheoretical part: \n- Why is a sigmoidal function exclusively assumed to map between 0 and 1? If \\lambda_0 is considered the baseline hazard, does the \"personalized\" prediction only decrease? How then is \\lambda_0 interpreted? Why was a Weibull distribution chosen for \\lambda_0 instead of adopting the more common approach seen in the Cox model?\n\nExperimental part: \nI am willing to change my grade if you could give me an answer on those points: \n- Can you do one experiment to show that your method outperforms the standard ML methods on a real-life dataset? Maybe on the KKbox dataset that contains 2.5M rows and is available on Kaggle or via pycox? \n- Can you add some calibration measures to your benchmark on the real datasets? \n- Can you add SumoNet, Yanamisawa, to the benchmark to understand their calibration power, and their abilities to predict the probabilities and their discriminative power?\n- In Table A.3, some C-index are lower than 0.5, which means that those models are worse than randomness (especially DeepHit on different datasets). Could the authors provide an explanation for these results?\n- Also, on support, all the IBS are worse than it is usually seen in the literature (e.g. Table 12 in Rindt). Could the authors clarify the methodology used to compute these scores?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-23"}