Log-Normalization Constant Estimation using the Ensemble Kalman-Bucy Filter with Application to High-Dimensional Models

Abstract

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman-Bucy filter based estimates based upon several nonlinear Kalman-Bucy diffusions. Based upon new conditional bias results for the mean of the afore-mentioned methods, we analyze the empirical log-scale normalization constants in terms of their L n −errors and conditional bias. Depending on the type of nonlinear Kalman-Bucy diffusion, we show that these are of order (t 1/2 /N 1/2 )+t/N or 1/N 1/2 (L n −errors) and of order [t+t 1/2 ]/N or 1/N (conditional bias), where t is the time horizon and N is the ensemble size. Finally, we use these results for online static parameter estimation for above filtering models and implement the methodology for both linear and nonlinear models.

Multilevel Estimation of Normalization Constants Using the Ensemble Kalman-Bucy Filter

“…The purpose of this work was to apply MLMC strategies for normalizing constant (NC) estimation. In particular our aim was to extend the work of [12], which used the EnKBF, to its multilevel counterpart which is the MLEnKBF [10]. As stated, our motivation is primarily in the linear setting, for which the optimal filter is the Kalman-Bucy filter, where we could provide propagation of chaos-type results.…”

“…• We introduce and approximate a multilevel estimator for the computation of normalizing constants associated with the EnKBF. This formulation is based on the multilevel EnKBF introduced by Chada et al [10], which looks to extend the single-level estimator which was proposed by Crisan et al [12].…”

“…the signal process when the observation is fixed and independent of the signal. From the work of Crisan et al [12], the authors show that the normalizing constant is given by…”

“…This filter has been analyzed extensively of recent through various pieces of work which include, but not limited to, understanding stability, deriving uniform propagation of chaos bounds [5,6,17] and acquiring a multilevel estimator [10,19,20]. However in the context of estimating normalizing constants, recent work has been done by Crisan et al [12] where the authors provide a way to do so using the EnKBF. However despite this, there is a computational burden associated with such costs, for example with the EnKBF or other Monte-Carlo algorithms, to attain a mean squared error (MSE) of a particular order.…”

“…In particular we will propose the use of the MLEnKBF for the application of normalizing constant estimation. Our motivation for this, as mentioned, is that firstly applying ML techniques can reduce the cost, associated to attaining an MSE of a particular oder, compared to the normalizing constant estimator in [12] which uses the EnKBF. Due to this we will make use of the methodology discussed in [10].…”