Non-Gaussian state-space models arise in several applications, and within this framework the binary time series setting provides a relevant example. However, unlike for 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 key functionals of these distributions are, in principle, available, but their calculation requires the evaluation of multivariate Gaussian cumulative distribution functions. Leveraging sun properties, we address this issue via novel Monte Carlo methods based on independent samples from the smoothing distribution, that can easily be adapted to the filtering and predictive case, thus improving state-of-the-art approximate and sequential Monte Carlo inference in small-to-moderate dimensional studies. Novel sequential Monte Carlo procedures that exploit the sun properties are also developed to deal with online inference in high dimensions. Performance gains over competitors are outlined in a financial application.

非高斯状态空间模型出现在多个应用中，在此框架内，二进制时间序列设置提供了一个相关示例。然而，与高斯状态空间模型不同——过滤、预测和平滑分布以封闭形式提供——二元状态空间模型需要近似或顺序蒙特卡罗策略来进行推理和预测。这是由于在高斯状态和二进制数据的观察方程引起的似然性之间明显不存在共轭。在本文中，我们证明了具有高斯状态变量的动态概率模型中的过滤、预测和平滑分布实际上是可用的，并且属于一类统一的偏斜法线（sun) 其参数可以通过解析表达式及时递归更新。这些分布的关键函数原则上也是可用的，但它们的计算需要评估多元高斯累积分布函数。利用太阳属性，我们通过基于来自平滑分布的独立样本的新颖蒙特卡罗方法解决了这个问题，该方法可以轻松适应过滤和预测情况，从而改进了最先进的近似和顺序蒙特卡罗推理小到中等维度的研究。利用太阳的新型顺序蒙特卡罗程序还开发了属性来处理高维的在线推理。在财务应用程序中概述了与竞争对手相比的性能提升。