On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions

We exhibit examples of high-dimensional unimodal posterior distributions arising in nonlinear regression models with Gaussian process priors for which Markov chain Monte Carlo (MCMC) methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialized ('cold start') algorithms that are local in the sense that their step sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis-Hastings adjusted methods such as preconditioned Crank-Nicolson and Metropolis-adjusted Langevin algorithm. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Non-stationary phase of the MALA algorithm

The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo method which creates a Markov chain reversible with respect to a given target distribution, π N , with Lebesgue density on R N ; it can hence be used to approximately sample the target distribution. When the dimension N is large a key question is to determine the computational cost of the algorithm as a function of N. The measure of efficiency that we consider in this paper is the expected squared jumping distance (ESJD), introduced in Roberts et al. (Ann Appl Probab 7(1):110-120, 1997). To determine how the cost of the algorithm (in terms of ESJD) increases with dimension N, we adopt the widely used approach of deriving a diffusion limit for the Markov chain produced by the MALA algorithm. We study this problem for a class of target measures which is not in product form and we address the situation of practical relevance in which the algorithm is started out of stationarity. We thereby significantly extend previous works which consider either measures of product form, when the Markov chain is started out of stationarity, or non-product measures (defined via a density with respect to a Gaussian), when the Markov chain is started in stationarity. In order to work in this non-stationary and non-product setting, significant new analysis is required. In particular, our diffusion limit comprises a stochastic PDE coupled to a scalar ordinary differential equation which gives a measure of how far from stationarity the process is. The family of non-product target measures that we consider in this paper are found from discretization of a measure on an infinite dimensional Hilbert space; the discretised measure is defined by its density with respect to a Gaussian random field. The results of this paper demonstrate that, in the non-stationary regime, the cost of the algorithm is of O ( N 1 / 2 ) in contrast to the stationary regime, where it is of O ( N 1 / 3 ) .

Error Bounds and Normalising Constants for Sequential Monte Carlo Samplers in High Dimensions

In this paper we develop a collection of results associated to the analysis of the sequential Monte Carlo (SMC) samplers algorithm, in the context of high-dimensional independent and identically distributed target probabilities. The SMC samplers algorithm can be designed to sample from a single probability distribution, using Monte Carlo to approximate expectations with respect to this law. Given a target density in d dimensions our results are concerned with d → ∞, while the number of Monte Carlo samples, N, remains fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative -error of the estimate of the normalising constant associated to the target. We also establish marginal propagation of chaos properties of the algorithm. These results are deduced when the cost of the algorithm is O(Nd2).

Error Bounds and Normalising Constants for Sequential Monte Carlo Samplers in High Dimensions

In this paper we develop a collection of results associated to the analysis of the sequential Monte Carlo (SMC) samplers algorithm, in the context of high-dimensional independent and identically distributed target probabilities. The SMC samplers algorithm can be designed to sample from a single probability distribution, using Monte Carlo to approximate expectations with respect to this law. Given a target density inddimensions our results are concerned withd→ ∞, while the number of Monte Carlo samples,N, remains fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative-error of the estimate of the normalising constant associated to the target. We also establish marginal propagation of chaos properties of the algorithm. These results are deduced when the cost of the algorithm isO(Nd2).