Learning Survival Distributions with the Asymmetric Laplace Distribution

TL;DR

This paper introduces a parametric survival model based on the Asymmetric Laplace Distribution ($ALD$) to learn covariate-conditioned time-to-event distributions with closed-form summaries. It trains via maximum likelihood by combining the $f_{ALD}$ density for observed events and the $S_{ALD}$ survival for censored data, enabling direct computation of mean, median, quantiles, and other summaries. The architecture uses a shared encoder with three heads predicting $(\theta,\sigma,\kappa)$ (or $q$) to specify $ALD(\theta,\sigma,\kappa)$, and the loss naturally handles censoring without discretization. Across synthetic and real-world datasets, the ALD approach demonstrates competitive or superior discrimination and calibration compared to parametric, semiparametric, and nonparametric baselines, while highlighting calibration challenges in highly skewed regimes and noting distributional assumptions as a limitation. Overall, the method provides interpretable, distributional survival predictions with practical advantages for risk assessment and decision-making in domains with censoring.

Abstract

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

Figures (11)

• Figure 1: The proposed neural network architecture for predicting the parameters of the Asymmetric Laplace Distribution $\mathcal{AL}(\theta, \sigma, \kappa)$.
• Figure 2: Performance on discrimination and calibration metrics. (a) concordance and (b) calibration. Reported are test averages with standard deviations over 10 runs.
• Figure 3: Examples of best and worst calibration curves. Slope and intercept of the linear fit are shown in the legend.
• Figure 4: Performance on calibration metrics.
• Figure 5: Calibration Linear Fit. The blue and orange lines represent the curves for Cal$[S(t \mid \mathbf{x})]$ and Cal$[f(t \mid \mathbf{x})]$, respectively. The gray dashed line represents the idealized result where the slope is one and the intercept is zero.

Theorems & Definitions (2)

• Definition 3.1: kotz2012laplace
• Corollary 3.2