Exact posterior distributions over the segmentation space and model selection for multiple change-point detection problems

In segmentation problems, inference on change-point position and model selection are two difficult issues due to the discrete nature of change-points. In a Bayesian context, we derive exact, non-asymptotic, explicit and tractable formulae for the posterior distribution of variables such as the number of change-points or their positions. We also derive a new selection criterion that accounts for the reliability of the results. All these results are based on an efficient strategy to explore the whole segmentation space, which is very large. We illustrate our methodology on both simulated data and a comparative genomic hybridisation profile.

💡 Research Summary

This paper tackles two long‑standing challenges in multiple change‑point detection: (1) obtaining exact posterior distributions for discrete change‑point locations and numbers, and (2) selecting an appropriate model while accounting for the reliability of the inferred segmentation. Working within a Bayesian framework, the authors first formalize the segmentation space as the union of all possible partitions of a data sequence of length n into K+1 contiguous segments, for K ranging from 1 to a user‑defined maximum. For each segment they assume a parametric likelihood (e.g., Gaussian with unknown mean and variance) and a conjugate prior (normal‑inverse‑Gamma), which yields closed‑form marginal likelihoods for any candidate segment.

The central theoretical contribution is an explicit, non‑asymptotic expression for the joint posterior (p(K,\tau|y)), where (\tau) denotes the ordered set of change‑point positions. By pre‑computing sufficient statistics for all possible sub‑intervals (