Adaptive independent Metropolis–Hastings

Abstract

We propose an adaptive independent Metropolis-Hastings algorithm with the ability to learn from all previous proposals in the chain except the current location. It is an extension of the independent Metropolis-Hastings algorithm. Convergence is proved provided a strong Doeblin condition is satisfied, which essentially requires that all the proposal functions have uniformly heavier tails than the stationary distribution. The proof also holds if proposals depending on the current state are used intermittently, provided the information from these iterations is not used for adaption. The algorithm gives samples from the exact distribution within a finite number of iterations with probability arbitrarily close to 1. The algorithm is particularly useful when a large number of samples from the same distribution is necessary, like in Bayesian estimation, and in CPU intensive applications like, for example, in inverse problems and optimization.

Adaptive independent sticky MCMC algorithms

“…Theorem 1 ensures that the pdf of the states of the Markov chain becomes closer and closer to the target pdf as t increases, since 0 ≤ 1−a t ≤ 1 and thus the product in the right hand side of (9) is a decreasing function of t. This theorem is a direct consequence of Theorem 2 in [14], and ensures the ergodicity of the proposed adaptive MCMC approach. Regarding the convergence of a sticky proposal to the target, we consider the following conjecture.…”

“…The rationale behind this test is to use information from the target density in order to include in the support set only those points where the proposal pdf differs substantially from the target value at z. Note that, since z is always different from the current state (i.e., z = x t for all t), then the proposal pdf is independent from the current state according to Holden's definition [14], and thus the theoretical analysis is greatly simplified.…”

“…according to Holden's definition [14]. For the sake of simplicity, we only consider the case where a single point can be added to S t at each iteration.…”

“…The ergodicity of the adaptive sticky MCMC methods is ensured and discussed. The underlying theoretical support is based on the approach introduced in [14]. The new schemes are particularly suitable for sampling from complicated full-conditional pdfs within a Gibbs sampler [5][6][7].…”

A note on formal constructions of sequential conditional couplings

“…This is particularly the case for adaptive MCMC algorithms, a recently widely-studied subject (see e.g. [14,2,4,17,1,12,5,20,3,8,15]). For such algorithms, sometimes the coupling only holds under certain conditions about the auxiliary randomness.…”