The sequential MCMC filter: formulation and applications

We consider the general signal-processing problem of learning about certain attributes of interest from measurements. These attributes, which may be time-varying (dynamic) or time-invariant (static), can be anything that are relevant to the physical processes that produce the measurements. In statistical signal processing, imperfections or uncertainties in the physical processes are described using probability models, and the complete probabilistic solution to the problem is given by the distribution of the attributes conditioned on all available measurements (the posterior distribution). We describe an algorithm for computing this solution, especially in situations with many measurements or low signal-to-noise ratios. The algorithm combines sequential importance sampling (SIS) and Markov chain Monte Carlo (MCMC) so as to achieve computational efficiency and stability. MCMC is performed sequentially for batches of measurements whose sizes are determined adaptively, hence the name sequential MCMC filter. For measurements within a batch, SIS is performed. Thus, bigger batch sizes mean that MCMC is performed less frequently. SIS is computationally efficient but with a finite Monte Carlo sample size, stability is not guaranteed indefinitely. MCMC is therefore needed from time to time to "refresh" the Monte Carlo sample, eliminating any errors that may have accumulated from the SIS steps. When MCMC is performed, it does not start from scratch but uses the most recent Monte Carlo sample from SIS to construct the proposal distribution. Adaptive batch sizing is based on a Kullback-Leibler distance that is easy to compute. By extending the algorithm to multiple models, the sequential MCMC filter can deal simultaneously with the dual pillars of statistical signal processing, namely detection (more generally, model selection) and parameter estimation. We discuss general uses of the sequential MCMC filter, and demonstrate its use for simultaneous weak signal detection and parameter estimation in a real-data experiment.