Towards smooth particle filters for likelihood estimation with multivariate latent variables

In parametrized continuous state-space models, one can obtain estimates of the likelihood of the data for fixed parameters via the Sequential Monte Carlo methodology. Unfortunately, even if the likelihood is continuous in the parameters, the estimates produced by practical particle filters are not, even when common random numbers are used for each filter. This is because the same resampling step which drastically reduces the variance of the estimates also introduces discontinuities in the particles that are selected across filters when the parameters change. When the state variables are univariate, the methodology of [23] gives an estimator of the loglikelihood that is continuous in the parameters. We present a non-trivial generalization of this method using tree-based o(N) (and as low as O(N logN)) resampling schemes that induce significant correlation amongst the selected particles across filters. In turn, this reduces the variance of the difference between the likelihood evaluated for different values of the parameters and the resulting estimator is considerably smoother than naively running the filters with common random numbers. Importantly, in practice our methods require only a change to the resample operation in the SMC framework without the addition of any extra parameters and can therefore be used for any application in which particle filters are already used. In addition, excepting the optional use of interpolation in the schemes, there are no regularity conditions for their use although certain conditions make them more advantageous. In this thesis, we first introduce the relevant aspects of the SMC methodology to the task of likelihood estimation in continuous state-space models and present an overview of work related to the task of smooth likelihood estimation. Following this, we introduce theoretically correct resampling schemes that cannot be implemented and the practical tree-based resampling schemes that were developed instead. After presenting the performance of our schemes in various applications, we show that two of the schemes are asymptotically consistent with the theoretically correct but unimplementable methods introduced earlier. Finally, we conclude the thesis with a discussion.