A tutorial derivation of the reversible jump Markov chain Monte Carlo (MCMC) algorithm is given. Various examples illustrate how reversible jump MCMC is a general framework for Metropolis-Hastings algorithms where the proposal and the target distribution may have densities on spaces of varying dimension. It is nally discussed how reversible jump MCMC can be applied in genetics to compute the posterior distribution of the number, locations, eects, and genotypes of putative quantitative trait loci.

this paper) reversibility holds, that is f P(x, A)(,x) = f PC, B A for all A, B , whereby r is clearly invariant

Abstract. I show how any reversible Markov chain on a finite state space that is irreducible, and hence suitable for estimating expectations with respect to its invariant distribution, can be used to construct a non-reversible Markov chain on a related state space that can also be used to estimate these expectations, with asymptotic variance at least as small as that using the reversible chain (typically smaller). The non-reversible chain achieves this improvement by avoiding (to the extent possible) transitions that backtrack to the state from which the chain just came. The proof that this modification cannot increase the asymptotic variance of an MCMC estimator uses a new technique that can also be used to prove Peskun’s (1973) theorem that modifying a reversible chain to reduce the probability of staying in the same state cannot increase asymptotic variance. A non-reversible chain that avoids backtracking will often take little or no more computation time per transition than the original reversible chain, and can sometime produce a large reduction in asymptotic variance, though for other chains the improvement is slight. In addition to being of some practical interest, this construction demonstrates that non-reversible chains have a fundamental advantage over reversible chains for MCMC estimation. Research into better MCMC methods may therefore best be focused on non-reversible chains. 1

Bayesian methods have proven to be powerful tools for computed tomographic reconstruction in realistic physical problems. However, Bayesian methods require that a number of modeling and computational problems be addressed. This paper summarizes a coherent system of statistical modeling and optimization techniques designed to facilitate efficient, unsupervised Bayesian emission and transmission tomographic reconstruction. New results are included on improved convergence behavior of these methods. 1. INTRODUCTION For the past 10 years, Bayesian estimation techniques have been studied for application in computed tomography, providing a framework for accurately modeling physical measurements and incorporating prior knowledge[1, 2, 3]. These model based techniques can improve the quality of reconstructions by incorporating more information about the data and measurement process. In particular, Bayesian reconstruction methods commonly employ a forward model which accounts for e#ects such ...

A revised version of this article is published in Communications in Statistics, Simulation and

In this thesis, we study the optimal scaling problem for sampling from a target distribution of interest using a random walk Metropolis (RWM) algorithm. In or-der to implement this method, the selection of a proposal distribution is required, which is assumed to be a multivariate normal distribution with independent com-ponents. We investigate how the proposal scaling (i.e. the variance of the normal distribution) should be selected for best performance of the algorithm. The d-dimensional target distribution we consider is formed of independent components, each of which has its own scaling term θ −2 j (d) (j = 1,..., d). This constitutes an extension of the d-dimensional iid target considered by Roberts, Gelman & Gilks (1997) who showed that for large d, the acceptance rate should be tuned to 0.234 for optimal performance of the algorithm. In a similar fashion, we show that for the aforementioned framework, the relative efficiency of the algorithm can be characterized by its overall acceptance rate.

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General methods for obtaining maximum likelihood estimates in exponential families are demonstrated using a constrained autologistic model for estimating relatedness from DNA fingerprint data. The novel features are the use of constrained optimization and two new algorithms for maximum likelihood estimation. The first, the "phase I " algorithm determines the support of the MLE in the closure of the exponential family (a distribution in the family conditioned on a face of the convex support of the natural statistic) when the MLE does not exist in the traditional sense (a point in the natural parameter space). The second, the maximum Monte Carlo likelihood algorithm uses the Metropolis algorithm or the Gibbs sampler to obtain estimates when exact calculation of the likelihood is not possible. Separate papers on each algorithm accompany

Abstract. In this paper we study the Metropolis algorithm in connection with two mean–field spin systems, the so called mean–field Ising model and the Blume–Emery–Griffiths model. In both this examples the naive choice of proposal chain gives rise, for some parameters, to a slowly mixing Metropolis chain, that is a chain whose spectral gap decreases exponentially fast (in the dimension N of the problem). Here we show how a slight variant in the proposal chain can avoid this problem, keeping the mean computational cost similar to the cost of the usual Metropolis. More precisely we prove that, with a suitable variant in the proposal, the Metropolis chain has a spectral gap which decreases polynomially in 1/N. Using some symmetry structure of the energy, the method rests on allowing appropriate jumps within the energy level of the starting state, and it is strictly connected to both the small world Markov chains of [15, 16] and to the equi-energy sampling of [22] and [26]. 1. Introduction. The Metropolis algorithm, introduced in [29] and later generalized in [18], is

Peskun’s theorem shows that the asymptotic variance of an MCMC estimator based on a reversible Markov chain will not increase if the matrix of transition probabilities for the chain is modified so as to increase the off-diagonal terms. I present a new proof of this result, which is more intuitive than Peskun’s original proof, and which may provide hints for how to prove other results of this nature.