# Survival Mixture Density Networks

arXiv: https://arxiv.org/abs/2208.10759 · HTML: https://mlsanity.com/paper/2208.10759/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 68 references that are in the map.
- First posted on arXiv 23 Aug 2022.

## Builds on (most cited first, 22 of 22)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network](https://mlsanity.com/paper/1606.00931/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Time-to-Event Prediction with Neural Networks and Cox Regression](https://mlsanity.com/paper/1907.00825/) (2019)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [A Scalable Discrete-Time Survival Model for Neural Networks](https://mlsanity.com/paper/1805.00917/) (2018)
- [Continuous and Discrete-Time Survival Prediction with Neural Networks](https://mlsanity.com/paper/1910.06724/) (2019)
- [Deep Survival Analysis](https://mlsanity.com/paper/1608.02158/) (2016)
- [Deep Neural Networks for Survival Analysis Based on a Multi-Task Framework](https://mlsanity.com/paper/1801.05512/) (2018)
- [Deep Cox Mixtures for Survival Regression](https://mlsanity.com/paper/2101.06536/) (2021)
- [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)
- [Deep Direct Likelihood Knockoffs](https://mlsanity.com/paper/2007.15835/) (2020)
- [Survival stacking: casting survival analysis as a classification problem](https://mlsanity.com/paper/2107.13480/) (2021)
- [Deep Exponential Families](https://mlsanity.com/paper/1411.2581/) (2014)
- [Inverse-Weighted Survival Games](https://mlsanity.com/paper/2111.08175/) (2021)
- [Geometrical Insights for Implicit Generative Modeling](https://mlsanity.com/paper/1712.07822/) (2017)
- [Survival Analysis via Ordinary Differential Equations](https://mlsanity.com/paper/2009.03449/) (2020)

## Built on by (most cited first, 9 of 9)

- [An Effective Meaningful Way to Evaluate Survival Models](https://mlsanity.com/paper/2306.01196/) (2023)
- [SCMIL: Sparse Context-aware Multiple Instance Learning for Predicting Cancer Survival Probability Distribution in Whole Slide Images](https://mlsanity.com/paper/2407.00664/) (2024)
- [Position: Stop Chasing the C-index when Evaluating Survival Analysis Models](https://mlsanity.com/paper/2506.02075/) (2025)
- [Maximum Likelihood Estimation of Flexible Survival Densities with Importance Sampling](https://mlsanity.com/paper/2311.01660/) (2023)
- [Survival Modeling from Whole Slide Images via Patch-Level Graph Clustering and Mixture Density Experts](https://mlsanity.com/paper/2507.16476/) (2025)
- [ADHAM: Additive Deep Hazard Analysis Mixtures for Interpretable Survival Regression](https://mlsanity.com/paper/2509.07108/) (2025)
- [Spatially-Aware Mixture of Experts with Log-Logistic Survival Modeling for Whole-Slide Images](https://mlsanity.com/paper/2511.06266/) (2025)
- [A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature](https://mlsanity.com/paper/2605.16208/) (2026)
- [Proper Scoring Rules for Right-Censored Survival Data](https://mlsanity.com/paper/2606.06393/) (2026)

## Abstract (arXiv)

Survival analysis, the art of time-to-event modeling, plays an important role in clinical treatment decisions. Recently, continuous time models built from neural ODEs have been proposed for survival analysis. However, the training of neural ODEs is slow due to the high computational complexity of neural ODE solvers. Here, we propose an efficient alternative for flexible continuous time models, called Survival Mixture Density Networks (Survival MDNs). Survival MDN applies an invertible positive function to the output of Mixture Density Networks (MDNs). While MDNs produce flexible real-valued distributions, the invertible positive function maps the model into the time-domain while preserving a tractable density. Using four datasets, we show that Survival MDN performs better than, or similarly to continuous and discrete time baselines on concordance, integrated Brier score and integrated binomial log-likelihood. Meanwhile, Survival MDNs are also faster than ODE-based models and circumvent binning issues in discrete models.