The research effort in this dissertation is targeted to investigate theoretical properties of some key statistics used in the sequential Monte Carlo (SMC) sampling, and to extend SMC to model checking, prior smoothing, and constrained state estimation. A novel application of SMC estimation to population pharmacokinetic models is also introduced.
Asymptotic properties of two key statistics in the SMC sampling, importance weights and empirical effective samples size, are discussed in the dissertation. The sum-normalized nature of importance weights makes it extremely difficult, if not impossible, to analytically investigate their properties. By using expectation-normalized importance weights, we are able to show the theoretical estimate of empirical effective sample size under various situations. In addition, the superiority of optimal importance function over prior importance function is verified based on the expectation-normalized weights.
The usage of SMC is also demonstrated for checking incompatibility between the prior and the data, using observation's predictive density value. When the prior is detected to be incompatible with the data, prior smoothing is proposed with a popular numerical method, Moving Horizon Estimation (MHE), to obtain a better estimate of the initial state value. Specifically, the incorporation of MHE smoothing into SMC estimation is among the first efforts to integrate these two powerful tools.
Convergence of constrained SMC (Chen, 2004) is verified and its performance is further illustrated with a more complex model.
SMC estimation is applied to a multi-dimensional population pharmacokinetic (PK) model. It is shown that the SMC sampling is faster than Markov Chain Monte Carlo (MCMC), and it doesn't suffer from the lack of convergence concern for MCMC.