Exact Kalman Filter for Binary Time Series

Non-Gaussian state-space models arise routinely in several applications. Within this framework, the binary time series setting provides a source of constant interest due to its relevance in a variety of studies. However, unlike Gaussian state-space models---where filtering and predictive distributions are available in closed form---binary state-space models require either 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 probit or logistic likelihood induced by the observation equation for the binary data. In this article we prove that the filtering and predictive distributions of dynamic probit models with Gaussian state variables belong to the class of unified skew-normals (SUN) and the associated parameters can be updated recursively via analytical expressions. Also the functionals of these filtering and predictive distributions depend on known functions, but their calculation requires Monte Carlo integration. Leveraging the SUN results, we address this point by proposing methods to draw independent and identically distributed samples from filtering and predictive distributions, thereby improving state-of-the-art approximate or sequential Monte Carlo inference in small-to-moderate dimensional dynamic models. A scalable and optimal particle filter which exploits SUN properties is also developed and additional exact expressions for the smoothing distribution are provided.