1. Introduction

The Event-Based Model (EBM) is a data-driven framework designed to reconstruct the temporal ordering of biomarker changes in progressive diseases, without relying on predefined clinical staging. Rather than assuming synchronous onset across individuals, the EBM infers a probabilistic sequence of discrete events—such as regional atrophy or biomarker abnormalities—by modeling how these abnormalities accumulate over time across a population.

Originally introduced by Fonteijn et al. (2011), the EBM treats disease progression as a cascade of events, where each event corresponds to a biomarker transitioning from a normal to an abnormal state. By leveraging cross-sectional or short-term longitudinal data from both patients and controls, the model estimates the most likely event ordering and the uncertainty around it using a Bayesian inference framework. A key advantage of the EBM is its ability to disentangle inter-individual variability in disease stage from intra-individual variability in biomarker expression, making it especially powerful in heterogeneous neurodegenerative conditions such as Alzheimer’s disease or Parkinson’s disease.

Unlike traditional approaches that rely on diagnostic labels or time-from-onset information, EBM does not require a priori classification. This makes it ideal for studying early, preclinical stages of disease and for integrating multiple types of biomarkers, including clinical, structural imaging, and functional measures.

2. Theory

2.1 Set-up

2.1.1 Notations

• $E_1,…,E_N$, a set of event
• $S=(s(1),…,s(N))$, an ordering, a permutation of the integers $1, …, N$ determining the event ordering $E_{S(1)}, ..., E_{S(N)}$

( since we aim to estimate $S$ from a dataset, say $X$)

• $X_j={x_{1j},x_{2j},…,x_{Nj} },$ a set of measurements from each patient $j=1,…,J$
• $X_l ={x_{1l},x_{2l},…,x_{Nl} }$, a set of measurements from each control $l =1,…,L$

2.1.2 Assumptions

1. measurements decrease monotonically as the disease progresses; in other words, patients for whom event $E_i$ has occurred cannot revert to an earlier state where $E_i$ has not occurred
2. the event sequence is consistent over all patients.

2.2 MCMC

We want to approximate the posterior distribution of biomarker event sequences $p(S | X)$, where:

$p(S | X) = \frac{p(S)p(X | S)}{p(X)}$

Since $p(X)$ is intractable, we use Markov Chain Monte Carlo (MCMC) to sample from $p(S | X)$ without needing to calculate it directly.

2.2.1 MCMC Sampling Procedure

1. Propose a new sequence $S’$ (e.g., by swapping two events).

2. Compute the acceptance ratio:

   $\frac{p(X | S’) \cdot p(S’)}{p(X | S) \cdot p(S)}$

   Since $p(X)$ cancels out, we only need to compute $p(S)$ and $p(X | S)$.

3. Accept or reject $S’$ using Metropolis-Hastings.

If we further assume flat prior: $p(S)$ , meaning all sequences are equally likely before observing the data, then the primary focus becomes calculating p(X | S).