Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2406.04653

Variational Markov chain mixtures with automatic component selection

Christopher E. Miles·Robert J. Webber
View original source ↗Read source document ↗Export metadata ↓

Abstract

Markov state modeling has gained popularity in various scientific fields since it reduces complex time-series data sets into transitions between a few states. Yet common Markov state modeling frameworks assume a single Markov chain describes the data, so they suffer from an inability to discern heterogeneities. As an alternative, this paper models time-series data using a mixture of Markov chains, and it automatically determines the number of mixture components using the variational expectation-maximization algorithm.Variational EM simultaneously identifies the number of Markov chains and the dynamics of each chain without expensive model comparisons or posterior sampling. As a theoretical contribution, this paper identifies the natural limits of Markov state mixture modeling by proving a lower bound on the classification error. It then presents numerical experiments where variational EM achieves performance consistent with the theoretically optimal error scaling. The experiments are based on synthetic and observational data sets including Last.fm music listening, ultramarathon running, and gene expression. In each of the three data sets, variational EM leads to the identification of meaningful heterogeneities.

Explore related subjects

stat.MEcs.NAmath.NAstat.ML

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Christopher E. Miles, Robert J. Webber. 2025-12-21. Variational Markov chain mixtures with automatic component selection. https://doi.org/10.1007/s11222-026-10842-8

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bias-Correction for Privacy-Protected Spatial Autoregressive Models with Application to Restaurant Network Analysis

Spatial autoregressive (SAR) models and their extensions are important tools for studying network effects. However, with an increasing emphasis on data privacy, data providers often implement protection measures that render standard SAR models inapplicable. In this study, we introduce a privacy-protected SAR model that incorporates noise into both the response and covariates to meet privacy requirements. With noise present in both components, the traditional quasi-maximum likelihood estimator becomes difficult to compute because the likelihood function cannot be directly formulated. To bypass this hurdle, we begin with a pseudo-likelihood approach, initially omitting the noise in the covariates. A Newton-Raphson algorithm is then applied to compute the estimator; however, the estimator is biased. To address this, we propose a bias-corrected Newton-Raphson-type algorithm that simultaneously accounts for noise in both the response and covariates. We further show, under appropriate regularity conditions, that the resulting estimator is consistent and asymptotically normal. To further enhance computational efficiency, we also develop a bias-corrected least squares estimator. Several extensions are discussed, and the finite-sample performance of the proposed methods is evaluated through extensive simulations. We apply the proposed methodology to restaurant transaction data from a third-party payment platform. Our method identifies a statistically significant competitive network effect among restaurants and further reveals meaningful restaurant-customer interaction patterns.

stat.ME↗
arXiv2026

A variational framework for modal estimation

Multivariate mode estimation arises in many statistical problems such as inverse problems, multimodal sampling, and density-based clustering, but becomes challenging in moderate to high dimensions, especially when the underlying density is not directly evaluable. We introduce GERVE (Gibbs-measure Entropy-Regularized Variational Estimation), a sample-based method for estimating multivariate modes by approximating Gibbs distributions directly from samples, without estimating or evaluating the density. GERVE uses Gaussian-mixture variational annealing and natural-gradient optimization, producing a mixture concentrated in high-density regions whose component responsibilities also provide a clustering of the observations. We prove theoretical guarantees in two regimes: as the Gibbs temperature goes to zero, the optimal variational mixture concentrates around the global modes of the population density; at fixed positive temperature, we prove existence, consistency, and asymptotic normality of empirical maximizers and propose a bootstrap procedure for uncertainty quantification. Simulations and a real-data experiment show that GERVE accurately recovers modes and produces meaningful clusters.

stat.ME↗
arXiv2026

Objective Model Prior Probabilities in Variable Selection

For many years it was routine to use equal model prior probabilities in Bayesian model uncertainty analysis. At least twenty years ago it became clear that this was problematic, leading to support of much too large models in the increasingly huge model spaces being considered in genomics and other fields. A popular replacement was to adopt a suggestion of Harold Jeffreys for the variable selection problem in which a total of $k$ possible variables are being considered for inclusion in the model: give the collection of all models containing $d$ variables ($d = 0, . . . , k$) prior probability $1/(k + 1)$ and then divide this prior probability equally among the models in the collection. Many other choices of model prior probabilities that impose severe parsimony have also been introduced. We begin by reviewing the problems with using equal model prior probabilities and then discuss some serious problems with the Jeffreys choice. Finally, we introduce and study a number of objective alternative choices of model prior probabilities, from both numerical and theoretical perspectives.

stat.ME↗