MENSA: A Multi-Event Network for Survival Analysis with Trajectory-based Likelihood Estimation

TL;DR

MENSA addresses the challenge of multi_event survival analysis by formulating a multi_state framework where each event is a state transition. It jointly learns time_to_event distributions via a shared representation and event_specific Weibull mixtures, augmented with a trajectory_based likelihood to encode temporal ordering. Across four multi_event datasets, MENSA demonstrates strong discrimination and calibration, with ablations confirming gains from joint training and the trajectory term, and computes favorable efficiency relative to deep baselines. The method offers practical value in settings where events can occur in sequence or concurrently, and where local event ordering is clinically informative.

Abstract

Most existing time-to-event methods focus on either single-event or competing-risks settings, leaving multi-event scenarios relatively underexplored. In many healthcare applications, for example, a patient may experience multiple clinical events, that can be non-exclusive and semi-competing. A common workaround is to train independent single-event models for such multi-event problems, but this approach fails to exploit dependencies and shared structures across events. To overcome these limitations, we propose MENSA (Multi-Event Network for Survival Analysis), a deep learning model that jointly learns flexible time-to-event distributions for multiple events, whether competing or co-occurring. In addition, we introduce a novel trajectory-based likelihood term that captures the temporal ordering between events. Across four multi-event datasets, MENSA improves predictive performance over many state-of-the-art baselines. Source code is available at https://github.com/thecml/mensa.

Figures (10)

• Figure 1: Example of a multi-event prediction.
• Figure 2: Overview of the MENSA architecture. The model first maps input covariates through a shared multilayer perceptron (MLP) layer, producing a common representation $\tilde{\bm{x}}$. This is passed through event-specific covariate layers, each producing $\tilde{\bm{x}}_{p}$ for state $p \in \mathcal{P}$. Each adapted representation then parameterizes a mixture of Weibull distributions to predict a time-to-event density function, $f_p(t)$, capturing the time of transition from the transient state (state 0) to each event state. The model is trained using a combined loss consisting of a multi-event likelihood, $\mathcal{L}_{\text{ME}}$, which captures how well the model fits event times across all events, and a trajectory likelihood, $\mathcal{L}_{\text{trajectory}}$, which enforces known temporal ordering constraints between pairs of events.
• Figure 3: Population-level summaries of predicted survival functions across the PRO-ACT test set for each event. Each panel shows the median expected survival $\mathbb{E}[S_E(t)]$ with narrow (40–60th percentile) variability bands. Different colors correspond to MENSA models trained with varying numbers of Weibull distributions $\Psi \in \{1, 3, 5, 7\}$. The inset boxes report event-specific performance metrics (Global C-index, IBS, mMAE).
• Figure 4: A single-event model with a single transition function.
• Figure 5: A competing-risks model for mortality from two causes.