This paper investigates Bayesian estimation for Gaussian Markov random elds. In particular, a new class of inhomogeneous model is proposed. This inhomogeneous model uses a Markov random eld to describe spatial variation of the smoothing parameter in a second random eld which describes the spatial variation in the observed intensity image. The coupled Markov random elds will be used as prior dis...

In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large–scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel es...

Abstract Under a compactness assumption, we show that a φ-irreducible and aperiodic MetropolisHastings chain is geometrically ergodic if and only if its rejection probability is bounded away from unity. In the particular case of the Independence Metropolis-Hastings algorithm, we obtain that the whole spectrum of the induced operator is contained in (and in many cases equal to) the essential ran...

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This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen [8], for target distributions with a non-compact support. The conditions ensuring a strong law of large numbers and a central limit theorem require that the tails of the target density decay super-exponentially, and have regular enough convex con...

The Metropolis-Hastings algorithm transforms a given stochastic matrix into a reversible stochastic matrix with a prescribed stationary distribution. We show that this transformation gives the minimum distance solution in an L1 metric.

Consider a d-ary rooted tree (d≥ 3) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum S n of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to infinity. We analyze the Metropolis algori...

This paper considers high-dimensional Metropolis and Langevin algorithms in their initial transient phase. In stationarity, these algorithms are well-understood and it is now well-known how to scale their proposal distribution variances. For the random walk Metropolis algorithm, convergence during the transient phase is extremely regular to the extent that the algorithm’s sample path actually r...