# Survival Regression with Proper Scoring Rules and Monotonic Neural Networks

arXiv: https://arxiv.org/abs/2103.14755 · HTML: https://mlsanity.com/paper/2103.14755/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 41 references that are in the map.
- First posted on arXiv 26 Mar 2021.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Time-to-Event Prediction with Neural Networks and Cox Regression](https://mlsanity.com/paper/1907.00825/) (2019)
- [Fully Neural Network based Model for General Temporal Point Processes](https://mlsanity.com/paper/1905.09690/) (2019)
- [Deep Survival Analysis](https://mlsanity.com/paper/1608.02158/) (2016)
- [Adversarial Time-to-Event Modeling](https://mlsanity.com/paper/1804.03184/) (2018)
- [Countdown Regression: Sharp and Calibrated Survival Predictions](https://mlsanity.com/paper/1806.08324/) (2018)
- [X-CAL: Explicit Calibration for Survival Analysis](https://mlsanity.com/paper/2101.05346/) (2021)
- [SODEN: A Scalable Continuous-Time Survival Model through Ordinary Differential Equation Networks](https://mlsanity.com/paper/2008.08637/) (2020)
- [Neural Likelihoods via Cumulative Distribution Functions](https://mlsanity.com/paper/1811.00974/) (2018)
- [The Brier Score under Administrative Censoring: Problems and Solutions](https://mlsanity.com/paper/1912.08581/) (2019)
- [Combining Dynamic Predictions from Joint Models for Longitudinal and Time-to-Event Data using Bayesian Model Averaging](https://mlsanity.com/paper/1303.2797/) (2013)

## Built on by (most cited first, 30 of 42)

- [Deep Learning for Survival Analysis: A Review](https://mlsanity.com/paper/2305.14961/) (2023)
- [CoxKAN: Kolmogorov-Arnold Networks for Interpretable, High-Performance Survival Analysis](https://mlsanity.com/paper/2409.04290/) (2024)
- [Neural Fine-Gray: Monotonic neural networks for competing risks](https://mlsanity.com/paper/2305.06703/) (2023)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [An Introduction to Deep Survival Analysis Models for Predicting Time-to-Event Outcomes](https://mlsanity.com/paper/2410.01086/) (2024)
- [An Effective Meaningful Way to Evaluate Survival Models](https://mlsanity.com/paper/2306.01196/) (2023)
- [Copula-Based Deep Survival Models for Dependent Censoring](https://mlsanity.com/paper/2306.11912/) (2023)
- [Proper Scoring Rules for Survival Analysis](https://mlsanity.com/paper/2305.00621/) (2023)
- [Deep Copula-Based Survival Analysis for Dependent Censoring with Identifiability Guarantees](https://mlsanity.com/paper/2312.15566/) (2023)
- [HACSurv: A Hierarchical Copula-Based Approach for Survival Analysis with Dependent Competing Risks](https://mlsanity.com/paper/2410.15180/) (2024)
- [Position: Stop Chasing the C-index when Evaluating Survival Analysis Models](https://mlsanity.com/paper/2506.02075/) (2025)
- [Neural Frailty Machine: Beyond proportional hazard assumption in neural survival regressions](https://mlsanity.com/paper/2303.10358/) (2023)
- [Survival Models: Proper Scoring Rule and Stochastic Optimization with Competing Risks](https://mlsanity.com/paper/2410.16765/) (2024)
- [Truthful Elicitation of Imprecise Forecasts](https://mlsanity.com/paper/2503.16395/) (2025)
- [Survival In-Context: Amortized Bayesian Survival Analysis via Prior-Fitted Networks](https://mlsanity.com/paper/2603.29475/) (2026)
- [A Large-Scale Neutral Comparison Study of Survival Models on Low-Dimensional Data](https://mlsanity.com/paper/2406.04098/) (2024)
- [NeuralSurv: Deep Survival Analysis with Bayesian Uncertainty Quantification](https://mlsanity.com/paper/2505.11054/) (2025)
- [Adaptive Transformer Modelling of Density Function for Nonparametric Survival Analysis](https://mlsanity.com/paper/2409.06209/) (2024)
- [Competing Risks: Impact on Risk Estimation and Algorithmic Fairness](https://mlsanity.com/paper/2508.05435/) (2025)
- [Cumulative Hazard Function Based Efficient Multivariate Temporal Point Process Learning](https://mlsanity.com/paper/2404.13663/) (2024)
- [Overcoming Dependent Censoring in the Evaluation of Survival Models](https://mlsanity.com/paper/2502.19460/) (2025)
- [Stacking Variational Bayesian Monte Carlo](https://mlsanity.com/paper/2504.05004/) (2025)
- [Prediction of Survival Outcomes under Clinical Presence Shift: A Joint Neural Network Architecture](https://mlsanity.com/paper/2508.05472/) (2025)
- [Neural interval-censored survival regression with feature selection](https://mlsanity.com/paper/2206.06885/) (2022)
- [DNAMite: Interpretable Calibrated Survival Analysis with Discretized Additive Models](https://mlsanity.com/paper/2411.05923/) (2024)
- [Reinterpreting survival analysis in the universal approximator age](https://mlsanity.com/paper/2307.13579/) (2023)
- [A Stable and Efficient Covariate-Balancing Estimator for Causal Survival Effects](https://mlsanity.com/paper/2310.02278/) (2023)
- [OPSurv: Orthogonal Polynomials Quadrature Algorithm for Survival Analysis](https://mlsanity.com/paper/2402.01955/) (2024)
- [On Training Survival Models with Scoring Rules](https://mlsanity.com/paper/2403.13150/) (2024)
- [Teaching Models To Survive: Proper Scoring Rule and Stochastic Optimization with Competing Risks](https://mlsanity.com/paper/2406.14085/) (2024)

## Abstract (arXiv)

We consider frequently used scoring rules for right-censored survival regression models such as time-dependent concordance, survival-CRPS, integrated Brier score and integrated binomial log-likelihood, and prove that neither of them is a proper scoring rule. This means that the true survival distribution may be scored worse than incorrect distributions, leading to inaccurate estimation. We prove that, in contrast to these scores, the right-censored log-likelihood is a proper scoring rule, i.e., the highest expected score is achieved by the true distribution. Despite this, modern feed-forward neural-network-based survival regression models are unable to train and validate directly on the right-censored log-likelihood, due to its intractability, and resort to the aforementioned alternatives, i.e., non-proper scoring rules. We therefore propose a simple novel survival regression method capable of directly optimizing log-likelihood using a monotonic restriction on the time-dependent weights, coined SurvivalMonotonic-net (SuMo-net). SuMo-net achieves state-of-the-art log-likelihood scores across several datasets with 20--100× computational speedup on inference over existing state-of-the-art neural methods, and is readily applicable to datasets with several million observations.