We use Markov categories to develop generalizations of the theory of Markov chains and hidden Markov models in an abstract setting. This comprises characterizations of hidden Markov models in terms of local and global conditional independences as well as existing algorithms for Bayesian filtering and smoothing applicable in all Markov categories with conditionals. We show that these algorithms specialize to existing ones such as the Kalman filter, forward-backward algorithm, and the Rauch-Tung-Striebel smoother when instantiated in appropriate Markov categories. Under slightly stronger assumptions, we also prove that the sequence of outputs of the Bayes filter is itself a Markov chain with a concrete formula for its transition maps. There are two main features of this categorical framework. The first is its generality, as it can be used in any Markov category with conditionals. In particular, it provides a systematic unified account of hidden Markov models and algorithms for filtering and smoothing in discrete probability, Gaussian probability, measure-theoretic probability, possibilistic nondeterminism and others at the same time. The second feature is the intuitive visual representation of information flow in these algorithms in terms of string diagrams.

翻译：我们利用马尔可夫范畴在抽象框架下发展了马尔可夫链与隐马尔可夫模型理论的推广。这包括基于局部与全局条件独立性对隐马尔可夫模型的刻画，以及适用于所有具备条件性的马尔可夫范畴的贝叶斯滤波与平滑现有算法。我们证明这些算法在特定马尔可夫范畴中实例化时将退化为卡尔曼滤波、前向-后向算法及Rauch-Tung-Striebel平滑器。在稍强假设下，我们还证明了贝叶斯滤波器输出序列本身构成一个马尔可夫链，并给出了其转移映射的具体公式。该范畴化框架具有两大特征：其一是普适性——可应用于任何具备条件性的马尔可夫范畴，尤其能同时为离散概率、高斯概率、测度论概率、可能性非决定论等情形下的隐马尔可夫模型及滤波平滑算法提供系统性的统一描述；其二是利用弦图直观展示这些算法中信息流动的可视化表示。