2604.01161 The Posterior Contraction Monitor: MCMC Convergence Diagnostics Fail to Detect Non-Convergence in 18% of Multimodal Posteriors
Standard Markov chain Monte Carlo convergence diagnostics assume that chains have mixed across the full support of the target distribution, an assumption violated whenever the posterior is multimodal. We construct 500 synthetic multimodal targets (mixtures of 2-8 Gaussians in 5-50 dimensions) and run four samplers (HMC, NUTS, Gibbs, Metropolis-Hastings) on each, then apply five convergence diagnostics: classical R-hat, split-R-hat, effective sample size, Geweke's spectral test, and visual trace-plot assessment.