Sequential Bayesian Filtering aims to estimate the current state distribution of a Hidden Markov Model, given the past observations. The problem is well-known to be intractable for most application domains, except in notable cases such as the tabular setting or for linear dynamical systems with gaussian noise. In this work, we propose a new class of filters based on Gaussian PSD Models, which offer several advantages in terms of density approximation and computational efficiency. We show that filtering can be efficiently performed in closed form when transitions and observations are Gaussian PSD Models. When the transition and observations are approximated by Gaussian PSD Models, we show that our proposed estimator enjoys strong theoretical guarantees, with estimation error that depends on the quality of the approximation and is adaptive to the regularity of the transition probabilities. In particular, we identify regimes in which our proposed filter attains a TV $\epsilon$-error with memory and computational complexity of $O(\epsilon^{-1})$ and $O(\epsilon^{-3/2})$ respectively, including the offline learning step, in contrast to the $O(\epsilon^{-2})$ complexity of sampling methods such as particle filtering.

翻译：序贯贝叶斯滤波旨在根据历史观测估计隐马尔可夫模型的当前状态分布。这一问题在多数应用领域中已知难以处理，仅在表格场景或具有高斯噪声的线性动力系统等特例中可解。本文提出一类基于高斯PSD模型的新型滤波器，其在密度近似与计算效率方面具有多重优势。我们证明，当转移与观测过程均为高斯PSD模型时，滤波可高效地以闭式完成。若转移与观测过程由高斯PSD模型近似，本文提出的估计器具备强理论保障：估计误差取决于近似质量，且对转移概率的正则性具有自适应性。特别地，我们识别出若干机制，其中所提滤波器在包含离线学习步骤的情况下，以记忆复杂度$O(\epsilon^{-1})$与计算复杂度$O(\epsilon^{-3/2})$达到TV $\epsilon$误差，而粒子滤波等采样方法的复杂度为$O(\epsilon^{-2})$。