Conditional convex orders and measurable martingale couplings

Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.Comment: 21 page

Quantitative convergence rates for sub-geometric Markov chains

We provide explicit expressions for the constants involved in the characterisation of ergodicity of sub-geometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The result is fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our result accommodates also some classes of inhomogeneous chains.Comment: 14 page

On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223--242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462--1505].Comment: Published in at http://dx.doi.org/10.1214/10-AAP682 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Convergence properties of pseudo-marginal markov chain monte carlo algorithms

We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis-Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geometric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo-marginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.Comment: Published at http://dx.doi.org/10.1214/14-AAP1022 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Increasing Role of Titin Mutations in Neuromuscular Disorders

Peer reviewe

bssm: Bayesian Inference of Non-linear and Non-Gaussian State Space Models in R

We present an R package bssm for Bayesian non-linear/non-Gaussian state space modeling. Unlike the existing packages, bssm allows for easy-to-use approximate inference based on Gaussian approximations such as the Laplace approximation and the extended Kalman filter. The package also accommodates discretely observed latent diffusion processes. The inference is based on fully automatic, adaptive Markov chain Monte Carlo (MCMC) on the hyperparameters, with optional importance sampling post-correction to eliminate any approximation bias. The package also implements a direct pseudo-marginal MCMC and a delayed acceptance pseudo-marginal MCMC using intermediate approximations. The package offers an easy-to-use interface to define models with linear-Gaussian state dynamics with non-Gaussian observation models and has an Rcpp interface for specifying custom non-linear and diffusion model