Abstract
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, pi ^N, with Lebesgue density on {mathbb {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 {{mathcal {O}}}(N^{1/2}) in contrast to the stationary regime, where it is of {{mathcal {O}}}(N^{1/3}).
Highlights
Metropolis–Hastings algorithms are Markov Chain Monte Carlo (MCMC) methods used to sample from a given probability measure, referred to as the target measure
The measure of computational cost considered in this paper is the expected squared jumping distance, introduced in [19] and related works
If the proposal variance scales with N like N −ζ, for some ζ > 0, we will say that the cost of the algorithm, in terms of expected squared jumping distance (ESJD), is of the order N ζ
Summary
Metropolis–Hastings algorithms are Markov Chain Monte Carlo (MCMC) methods used to sample from a given probability measure, referred to as the target measure. Scaling for MALA, in the stationary regime, was later obtained in the setting of non-product measures defined via density with respect to a Gaussian random field [17]. In this paper we contribute further understanding of the MALA algorithm when initialized out of stationarity by considering non-product measures defined via density with respect to a Gaussian random field. Considering such a class of measures has proved fruitful, see e.g. In this paper our primary contribution is the study of diffusion limits for the the MALA algorithm, out of stationarity, in the setting of general non-product measures, defined via density with respect to a Gaussian random field. An interesting starting point of such work might be the study of non i.i.d. product measures as pioneered by Bédard [2,3]
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