State-space models have been used in a number of fields such as epidemiology, finance, and systems biology. In the Bayesian context, simultaneous fixed parameter and state estimation in these models is performed either using Markov chain Monte Carlo for batch analyses or sequential Monte Carlo for real-time analyses. We discuss computational developments in these areas that exploit the use of mixtures of distributions to overcome common difficulties.
One sweep in the standard MCMC approach for these models requires sampling from the full conditional distribution for each individual state. This approach can lead to arbitrarily poor mixing rates and therefore extremely long run times are necessary to reach convergence and for reasonable parameter estimates. An alternative approach samples from the entire sequence of latent states jointly, but, in general, this full conditional is unavailable in closed form. We propose a method utilizing mixtures to provide highly accurate approximations to the filtering and smoothing distributions. This approximation provides an easily sampled independent proposal distribution for use in a Metropolis-Hastings algorithm. We provide an example based on systems biology to illustrate the methodology.
In real-time analyses, sequential Monte Carlo methods, often called particle filters, are computationally efficient. The standard approach includes fixed parameters in the state vector with degenerate evolutions. This, in turn, causes degeneracy in the particle components representing the fixed parameters. Two techniques for combating this phenomenon are available: the first relies on approximating the filtered distribution for the parameters by a mixture and the second relies on a sufficient statistic representation. Unfortunately, the first technique still suffers from degeneracy issues and the second is usable only on a small subclass of models. We propose an approach that combines these ideas and therefore reduces degeneracy but is usable on a much larger model class.