A Closed-Form Filter for Binary Time Series.

Non-Gaussian state-space models arise in several applications. Within this framework, the binary time series setting is a source of constant interest due to its relevance in many studies. However, unlike Gaussian state-space models, where filtering, predictive and smoothing distributions are available in closed-form, binary state-space models require approximations or sequential Monte Carlo strategies for inference and prediction. This is due to the apparent absence of conjugacy between the Gaussian states and the likelihood induced by the observation equation for the binary data. In this article we prove that the filtering, predictive and smoothing distributions in dynamic probit models with Gaussian state variables are, in fact, available and belong to a class of unified skew-normals (SUN) whose parameters can be updated recursively in time via analytical expressions. Also the functionals of these distributions depend on known functions, but their calculation requires intractable numerical integration. Leveraging the SUN properties, we address this point via new Monte Carlo methods based on independent and identically distributed samples from the smoothing distribution, which can naturally be adapted to the filtering and predictive case, thereby improving state-of-the-art approximate or sequential Monte Carlo inference in small-to-moderate dimensional studies. A scalable and optimal particle filter which exploits the SUN properties is also developed to deal with online inference in high dimensions. Performance gains over competitors are outlined in a real-data financial application.