大阪大学　数理・データ科学教育研究センター
Center for Mathematical Modeling and Data Science,Osaka University

Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo

Ajay Jasra (National University of Singapore)

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4. Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo

大阪大学 数理・データ科学セミナー データ科学セミナーシリーズ 第20回

Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo

Ajay Jasra (National University of Singapore)

In this talk I consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using the Euler Maruyama method. Given this assumption, we show how one can use Markov chain Monte Carlo (MCMC) and particularly particle MCMC to implement a new approximation of the multilevel (ML) Monte Carlo (MC) collapsing sum identity. Our approach comprises constructing an approximate coupling of the posterior density of the joint distribution over parameter and hidden variables at two different discretization levels and then correcting by an importance sampling method. The variance of the weights are independent of the length of the observed data set. The utility of such a method is that, for a prescribed level of mean square error, the cost of this MLMC method is provably less than i.i.d. sampling from the posterior associated to the most precise discretization. However the method here comprises using only known and efficient simulation methodologies.
The theoretical results are illustrated by inference of the parameters of two prototypical processes given noisy partial observations of the
process: the first is an Ornstein Uhlenbeck process and the second is a more general Langevin equation.