Convergence of numerical time-averaging and stationary measures via Poisson equations

Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Bayesian Model Averaging in the M-Open Framework

This chapter presents a model averaging approach in the M-open setting using sample re-use methods to approximate the predictive distribution of future observations. It first reviews the standard M-closed Bayesian Model Averaging approach and decision-theoretic methods for producing inferences and decisions. It then reviews model selection from the M-complete and M-open perspectives, before formulating a Bayesian solution to model averaging in the M-open perspective. It constructs optimal weights for MOMA:M-open Model Averaging using a decision-theoretic framework, where models are treated as part of the ‘action space’ rather than unknown states of nature. Using ‘incompatible’ retrospective and prospective models for data from a case-control study, the chapter demonstrates that MOMA gives better predictive accuracy than the proxy models. It concludes with open questions and future directions.