Asymptotic Theory of Bayes Factor in Stochastic Differential Equations: Part I

Abstract

Research on asymptotic model selection in the context of stochastic differential equations (SDEs) is almost non-existent in the literature. In particular, when a collection of SDEs is considered, the problem of asymptotic model selection has not been hitherto investigated. Indeed, even though the diffusion coefficients may be considered known, questions on appropriate choice of the drift functions constitute a non-trivial model selection problem. In this article, we develop the asymptotic theory for comparisons between collections of SDEs with respect to the choice of drift functions using Bayes factors when the number of equations (individuals) in the collection of SDEs tend to infinity while the time domains remain bounded for each equation. Our asymptotic theory covers situations when the observed processes associated with the SDEs are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). In particular, we allow incorporation of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function; we also permit different initial values and domains of observations for the SDEs. Our model selection problem thus encompasses selection of a set of appropriate time-dependent covariates from a set of available time-dependent covariates, besides selection of the part of the drift function free of covariates. For both iid and non-iid set-ups we establish almost sure exponential convergence of the Bayes factor. Furthermore, we demonstrate with simulation studies that even in non-asymptotic scenarios Bayes factor successfully captures the right set of covariates.