In this paper, we shall optimize the efficiency of Metropolis algorithms for multidimensional target distributions with scaling terms possibly depending on the dimension. We propose a method for determining the appropriate form for the scaling of the proposal distribution as a function of the dimension, which leads to the proof of an asymptotic diffusion theorem. We show that when there does not exist any component with a scaling term significantly smaller than the others, the asymptotically optimal acceptance rate is the well-known 0.234.formance of the algorithm. It was also shown that the correct proposal scaling is of the form ℓ 2 /d for some constant ℓ as d → ∞. The simplicity of the obtained asymptotically optimal acceptance rate (AOAR) makes these theoretical results extremely useful in practice. Optimal scaling issues have been explored by other authors, namely [5,6,10,12,13].In this paper, we carry out a similar study for d-dimensional target distributions with independent components. The particularity of our model is that the scaling term of each component is allowed to depend on the dimension of the target distribution, which constitutes a critical distinction with the i.i.d. case. We provide a condition under which the algorithm admits the same limiting diffusion process and the same AOAR as those found in [11]. This is achieved, in the first place, by determining the appropriate form for the proposal scaling as a function of d, which is now different from the i.i.d. case. Then, by verifying L 1 convergence of generators, we prove that the sequence of stochastic processes formed by, say, the i * th component of each Markov chain (appropriately rescaled) converges to a Langevin diffusion process with a certain speed measure. Obtaining the AOAR is thus a simple matter of optimizing the speed measure of the diffusion.The paper is structured as follows. In Section 2, we describe the Metropolis algorithm and introduce the target distribution setting. The main results are presented in Section 3, along with a discussion concerning inhomogeneous proposal distributions and some extensions. We prove the theorems in Section 4 using lemmas proved in Sections 5 and 6, finally concluding the paper with a discussion.2. Sampling from the target distribution.2.1. The Metropolis algorithm. The idea behind the Metropolis algorithm is to generate a Markov chain X 0 , X 1 , . . . having the target distribution as a stationary distribution. In particular, suppose that π is a d-dimensional probability density of interest with respect to Lebesgue measure. Also, let the proposed moves be normally distributed around x, that is, N (x, σ 2 I d ) for some σ 2 and where I d is the d-dimensional identity matrix. The Metropolis algorithm thus proceeds as follows. Given X t , the state of the chain at time t, a value Y t+1 is generated from the normal density q(X t , y) dy. The probability of accepting the proposed value Y t+1 as the new value for the chain is α(X t , Y t+1 ), where α(x, y) =    min 1, π(y) π(x) , π(x)q(x, y) > 0,...

Recent optimal scaling theory has produced a condition for the asymptotically optimal acceptance rate of Metropolis algorithms to be the well-known 0.234 when applied to certain multidimensional target distributions. These d-dimensional target distributions are formed of independent components, each of which is scaled according to its own function of d. We show that when the condition is not met the limiting process of the algorithm is altered, yielding an asymptotically optimal acceptance rate which might drastically differ from the usual 0.234. Specifically, we prove that as d → ∞ the sequence of stochastic processes formed by say the i * th component of each Markov chain usually converges to a Langevin diffusion process with a new speed measure υ, except in particular cases where it converges to a one-dimensional Metropolis algorithm with acceptance rule α *. We also discuss the use of inhomogeneous proposals, which might reveal essential in specific cases.

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Linear regression is ubiquitous in statistical analysis. It is well understood that conflicting sources of information may contaminate the inference when the classical normality of errors is assumed. The contamination caused by the light normal tails follows from an undesirable effect: the posterior concentrates in an area in between the different sources with a large enough scaling to incorporate them all. The theory of conflict resolution in Bayesian statistics (O'Hagan and Pericchi (2012)) recommends to address this problem by limiting the impact of outliers to obtain conclusions consistent with the bulk of the data. In this paper, we propose a model with super heavy-tailed errors to achieve this. We prove that it is wholly robust, meaning that the impact of outliers gradually vanishes as they move further and further away form the general trend. The super heavy-tailed density is similar to the normal outside of the tails, which gives rise to an efficient estimation procedure. In addition, estimates are easily computed. This is highlighted via a detailed user guide, where all steps are explained through a case study. The performance is shown using simulation. All required code is given.MSC 2010 subject classifications: Primary 62F35; secondary 62J05.

Abstract:The authors provide an overview of optimal scaling results for the Metropolis algorithm with Gaussian proposal distribution. They address in more depth the case of high-dimensional target distributions formed of independent, but not identically distributed components. They attempt to give an intuitive explanation as to why the well-known optimal acceptance rate of 0.234 is not always suitable. They show how to find the asymptotically optimal acceptance rate when needed, and they explain why it is sometimes necessary to turn to inhomogeneous proposal distributions. Their results are illustrated with a simple example. Echelonnage optimal de l'algorithme Metropolis : en route vers des lois cibles généralesRésumé : Les auteurs font un survol des résultats sur l'échelonnage optimal de l'algorithme Metropolis avec loi instrumentale gaussienne. Ils s'intéressent de plus près au cas des lois cibles multivariéesà composantes indépendantes mais non identiquement distribuées. Ils tentent d'expliquer en termes intuitifs pourquoi le taux d'acceptation optimal bien connu de 0.234 ne convient pas toujours. Ils montrent comment déterminer au besoin le taux d'acceptation asymptotiquement optimal et ils expliquent pourquoi il est parfois indispensable d'avoir recoursà des lois instrumentales non homogènes. Leurs résultats sont illustrés dans un cas simple.