New Preprint! | Areeb SM's website

We¹ have finally finished up a preprint (Fritz et al., 2024) for our work on hidden Markov models² in the context of Markov categories (woohoo!).

We’ve been at it for a while, I’d in fact presented this work at the Applied Category Theory 2023 conference³ back in the August of 2023. Of course, we’ve added a fair bit since then. For instance, the material on smoothing was all written after the conference. Still, it’s good to finally have something out :D

Tobias Fritz, Andreas Klingler, Drew McNeely, Yuwen Wang and I. ↩
Sometimes called state space models. ↩
The recording is on YouTube. ↩

References

Unpublished

Preprint

2024

Hidden Markov Models and the Bayes Filter in Categorical Probability

Tobias Fritz, Andreas Klingler, Drew McNeely, Areeb Shah-Mohammed, and Yuwen Wang

Jan 2024

Abs arXiv

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.

References

Unpublished

Preprint

2024

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