We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that...

Published nearly seventy years ago, Jeffreys' Theory of Probability (1939) has had a unique impact on the Bayesian community and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the objective Bayes school. In particular, its advances on the...

In this paper we discuss consistency of the posterior distribution in cases where the Kullback-Leibler condition is not verified. This condition is stated as : for all $\epsilon 0$ the prior probability of sets in the form $\{f ; KL(f0 , f ) \leq \epsilon\}$ where KL(f0 , f ) denotes the...

We derive rates of contraction of posterior distributions on non-parametric models resulting from sieve priors. The aim of the study was to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter is, for example,...

This chapter provides a overview of Bayesian inference, mostly emphasising that it is a universal method for summarising uncertainty and making estimates and predictions using probability statements conditional on observed data and an assumed model (Gelman 2008). The Bayesian perspective is thus...

In this paper we study the asymptotic behaviour of the posterior distribution in a mixture model when the number of components in the mixture is larger than the true number of components, a situation commonly referred to as overfitted mixture. We prove in particular that quite generally the...

This introduction to Bayesian statistics presents the main concepts as well as the principal reasons advocated in favour of a Bayesian modelling. We cover the various approaches to prior determination as well as the basis asymptotic arguments in favour of using Bayes estimators. The testing...

Consistency, asymptotic normality and e ciency of the maximum likelihood estimator for stationary Gaussian time series, were shown to hold in the short memory case by Hannan (1973) and in the long memory case by Dahlhaus (1989). In this paper, we extend these results to the entire stationarity...