A new method to implement Bayesian inference on stochastic differential equation models

Abstract

Stochastic differential equations (SDEs) are widely used to model numerous real-life phenomena. However, transition densities of most of the SDE models used in practice are not known, making both likelihood based and Bayesian inference difficult. Methods for Bayesian inference have mainly relied on MCMC based methods which are computationally expensive. There is a need to develop a computationally efficient method which will provide accurate inference. This thesis introduces a new approach to approximate Bayesian inference for SDE models. This approach is not MCMC based and aims to provide a more efficient option for Bayesian inference on SDE models. This research problem was motivated by a civil engineering problem of modeling the force exerted by vehicles on the road surface as they traverse it. Proposed here two new methods to implement this approach. These methods have been named as the Gaussian Modified Bridge Approximation (GaMBA) and its extension GaMBA- Importance sampling (GaMBA-I). This thesis provides an easy to use algorithm for both these methods, discusses their consistency properties, describes examples where these methods provide efficient inference and also illustrates situations where these methods would not yield efficient and accurate inference. To illustrate how GaMBA-I could be used to model complex real life processes, this research attempts to model the dynamic force exerted by the vehicles on the road surface using SDE models. An SDE model based on one of the existing differential equation models was used to fit a simulated force data using GaMBA-I. This was considered as a ’proof of concept’ work to investigate if the SDE modeling of this problem is feasible.