Learning Survival Distributions with the Asymmetric Laplace Distribution

Conference: ICML2025
arXiv: 2505.03712
Code: The paper includes code in supplementary materials
Area: Survival Analysis / Probabilistic Modeling
Keywords: Survival Analysis, Asymmetric Laplace Distribution, Parametric Models, Quantile Regression, Maximum Likelihood Estimation

TL;DR

This paper proposes a parametric survival analysis method based on the asymmetric Laplace distribution (ALD). By using a neural network to learn the three parameters of the ALD (location, scale, and asymmetry), it achieves continuous, closed-form estimation of the survival distribution, comprehensively outperforming existing parametric and non-parametric approaches in both discriminative and calibration performance.

Background & Motivation

Survival analysis (also known as time-to-event analysis) aims to predict the event time distribution given covariates, and is widely applied in fields such as healthcare, finance, and engineering. The core challenge lies in right-censoring: for some samples, the event has not occurred by the end of the observation period.

Existing methods are categorized into three classes, each with limitations:

• Parametric methods (Exponential, Weibull, Log-normal): Assume a fixed distributional form, lacking flexibility and suffering performance degradation when real data deviates from the assumptions.
• Semi-parametric methods (Cox proportional hazards model, DeepSurv): Rely on the proportional hazards assumption, with reduced reliability under high censoring rates.
• Non-parametric methods (DeepHit, CQRNN): Flexible but produce discrete or piecewise-constant estimates, making it difficult to extract continuous distribution summaries (such as mean, median, and quantiles) and incurring high computational overhead.

The core motivation of this paper is: Is it possible to find a distribution that offers both closed-form solutions (the advantage of parametric models) and sufficient flexibility (the advantage of non-parametric models)? The asymmetric Laplace distribution (ALD) perfectly meets this demand.

Method

Asymmetric Laplace Distribution (ALD) Definition

The ALD is controlled by three parameters \((\theta, \sigma, \kappa)\), corresponding to location, scale, and asymmetry, respectively:

\[f_{\text{ALD}}(y;\theta,\sigma,\kappa) = \frac{\sqrt{2}}{\sigma}\frac{\kappa}{1+\kappa^2} \begin{cases} \exp\left(\frac{\sqrt{2}\kappa}{\sigma}(\theta-y)\right), & y \geq \theta \\ \exp\left(\frac{\sqrt{2}}{\sigma\kappa}(y-\theta)\right), & y < \theta \end{cases}\]

Its CDF also has a closed-form expression, allowing the mean, median, mode, variance, and arbitrary quantiles to be calculated analytically.

Key property: Through the reparameterization \(q = \kappa^2/(\kappa^2+1)\), the ALD can be naturally linked to quantile regression.

Neural Network Architecture

The model adopts an architecture consisting of a shared encoder and three independent prediction heads:

• Shared Encoder: Fully connected layers with ReLU activation to extract covariate features.
• Three Output Heads: Predict \(\theta\), \(\sigma\), and \(\kappa\) respectively, all utilizing an Exp activation to ensure non-negativity.
• Residual Connections: Enhance gradient flow and training stability.

Maximum Likelihood Learning

The likelihood is constructed based on whether the event is observed:

\[-\mathcal{L}_{\text{ALD}} = \sum_{n \in \mathcal{D}_O} \log f_{\text{ALD}}(y_n|\mathbf{x}_n) + \sum_{n \in \mathcal{D}_C} \log S_{\text{ALD}}(y_n|\mathbf{x}_n)\]

where \(\mathcal{D}_O\) represents the set of observed event times (\(e=1\)), \(\mathcal{D}_C\) represents the set of censored times (\(e=0\)), and \(S_{\text{ALD}} = 1 - F_{\text{ALD}}\) is the survival function.

Key advantages over CQRNN: - CQRNN requires setting pseudo-values \(y^* = 1.2 \max_i y_i\) (which are data-sensitive) and defining an approximate censoring quantile \(q_c\) (where precision is limited by grid granularity). - The proposed method directly maximizes the survival probability without requiring extra hyperparameters. - It yields a continuous parametric distribution rather than discrete quantile points.

Comparison with CQRNN

CQRNN is based on the pinball loss of the Portnoy estimator, which requires predefining a quantile grid \(q=\{0.1, 0.2, \ldots, 0.9\}\) and estimating \(\theta_q\) individually. In contrast, the proposed method only needs to learn three parameters to obtain the complete distribution, bypassing the difficulties of quantile grid selection and pseudo-value tuning.

Key Experimental Results

Dataset Configurations

The model is evaluated on 14 synthetic datasets and 7 real-world datasets, covering different censoring ratios (0.20 to 0.80) and feature dimensions (1 to 14). The real-world datasets span multiple domains such as oncology and cardiology:

Dataset	No. of Features	Train Set	Test Set	Censoring Ratio
METABRIC	9	1523	381	0.42
WHAS	6	1310	328	0.57
SUPPORT	14	7098	1775	0.32
GBSG	7	1785	447	0.42
TMBImmuno	3	1328	332	0.49
BreastMSK	5	1467	367	0.77
LGGGBM	5	510	128	0.60

Overall Performance Summary (21 datasets × 9 metrics = 189 comparisons)

Baseline	Significantly Win	Significantly Lose	Draw
vs. DeepHit	113	22	54
vs. LogNorm MLE	113	6	70
vs. DeepSurv	77	27	85
vs. CQRNN	43	22	124

Key Findings

• IBS Metric: ALD wins in all cases against LogNorm, DeepSurv, and DeepHit across the 21 datasets, and achieves 19 wins and 1 loss against CQRNN.
• Calibration Metrics: In terms of slope/intercept calibration for \(S(t|\mathbf{x})\) and \(f(t|\mathbf{x})\), ALD comprehensively outperforms traditional parametric and non-parametric methods.
• Discriminative Metrics (C-Index): The primary advantage of ALD is demonstrated in comparison with DeepHit (15 wins, 0 losses), while showing comparable performance to CQRNN.
• All experiments were replicated 10 times, with significance determined using the Wilcoxon signed-rank test.

Highlights & Insights

1. Elegant Mathematical Framework: By leveraging the closed-form PDF/CDF of the ALD, the proposed work returns survival analysis to the clean form of classic parametric models while maintaining sufficient flexibility.
2. Practical Value of Continuous Distribution: Unlike DeepHit (which discretizes) and CQRNN (which outputs quantile points), ALD yields a continuous distribution, allowing direct calculation of the mean, median, variance, and arbitrary quantiles.
3. Minimal Hyperparameters: Eliminates the need to tune extra parameters such as quantile grids or pseudo-values, significantly reducing tuning difficulty.
4. Outstanding Calibration: Performance is particularly robust on distribution-level calibration metrics, indicating that ALD indeed fits the shape of real survival distributions more accurately.
5. Unified Treatment of Censoring Types: Can be adapted to other censoring types, such as left-censoring, by simply modifying the likelihood function.

Limitations & Future Work

1. Inherent Distribution Assumption: Although ALD is more flexible than Normal or Weibull distributions, it still belongs to the family of unimodal distributions, which may feel inadequate when facing multimodal survival distributions.
2. ALD Support Range Includes Negative Numbers: \(t < 0\) is meaningless in survival analysis; although the authors claim this rarely occurs in practice, it lacks theoretical guarantees.
3. Evaluation Limited to Structured Data: Has not been applied to high-dimensional unstructured covariates such as imaging or time series.
4. Unstable MAE Metric: Performance is mixed in terms of MAE compared to CQRNN/DeepSurv, indicating that the advantage in point prediction is less pronounced.
5. Extension to Mixture Distributions: Modeling a mixture of ALDs could count as a direction to handle more complex survival distributions.
6. Competing Risks Scenario: The current framework only processes single-event scenarios and has not been extended to multi-event competing risks settings.

Rating

• Novelty: ⭐⭐⭐⭐ — Applying ALD to survival analysis is a novel entry point, though the link between ALD itself and quantile regression has been extensively studied.
• Experimental Thoroughness: ⭐⭐⭐⭐⭐ — Solid experimental design with 21 datasets, 9 metrics, 10 replications, and statistical testing.
• Writing Quality: ⭐⭐⭐⭐ — Clear mathematical derivations and an in-depth, thorough comparison with CQRNN.
• Value: ⭐⭐⭐⭐ — Provides a practical and elegant new parametric option for survival analysis.

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