A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

Current methods in credibility theory often rely on parametric models, e.g., a linear function of past experience. During the last decade, the existence of high speed computers and statistical software packages allowed the introduction of more sophisticated and flexible modelling strategies. In recent years, some of these techniques, which made use of the Markov Chain Monte Carlo (MCMC) approach to modelling, have been incorporated in credibility theory. However, very few of these methods made use of additional covariate information related to risk, or collection of risks; and at the same time account for the correlated structure in the data. In this paper, we consider a Bayesian nonparametric approach to the problem of risk modelling. The model incorporates past and present observations related to risk, as well as relevant covariate information. This Bayesian modelling is carried out by sampling from a multivariate Gaussian prior, where the covariance structure is based on a thin-plate spline. The model uses the MCMC technique to compute the predictive distribution of future claims based on available data.