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：我们运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，以及适用于所有带条件性的马尔可夫范畴中贝叶斯滤波与平滑的现有算法。研究表明，当在适当的马尔可夫范畴中实例化时，这些算法可具体化为卡尔曼滤波器、前向后向算法以及Rauch-Tung-Striebel平滑器等现有方法。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。该范畴框架具有两个主要特征。其一为通用性——可在任意带条件性的马尔可夫范畴中应用，尤其能系统统一地描述离散概率、高斯概率、测度论概率、可能性非决定论等框架下的隐马尔可夫模型及滤波平滑算法。其二则是通过弦图对这些算法中的信息流进行直观的可视化表征。