JUMP MARKOV CHAIN MONTE CARLO ALGORITHMS FOR

     BAYESIAN INFERENCE IN HIDDEN MARKOV MODELS

     Tobias Rydén, Matematisk Statistik i Lund


A hidden Markov model (HMM) is a bivariate stochastic process
{(X_k,Y_k)} such that (i) {X_k} is a finite state Markov chain
and (ii) given {X_k}, the process {Y_k} is a sequence of
conditionally independent random variables with the conditional
distribution of Y_n depending on X_n only. The chain {X_k} is
generally not observable, hence the word `hidden', so that
inference has to be based on {Y_k} alone. 

HMMs have during the last decade become widely spread for
modelling sequences of weakly dependent random variables with
applications in areas like speech processing, communication
networks, biochemistry, biology, medicine, econometrics,
environmetrics, etc. Sometimes the hidden Markov chain {X_k}
does indeed exist, so that the physical nature of the problem
suggests the use of an HMM, in other cases HMMs just provide a
good fit to data.

One of the most difficult problems in HMM inference is to
estimate the number of states, d say, of {X_k}. Classical
approaches to this problem include likelihood ratio tests
and penalized likelihoods (AIC/BIC). In this talk we present
a Bayesian approach: by placing a prior on the unknown d
we obtain a posterior distribution for d and the other
parameters of the model. This distribution is analytically
untractable but can be explored using jump Markov chain
Monte Carlo algorithms.