Compositionality in algorithms for smoothing

2023-03-24

AI Summary

This paper addresses the lack of compositional semantics in probabilistic algorithms—particularly the bilateral filter-based Gaussian (BFFG) algorithm. To resolve this, the authors establish, for the first time, a rigorous connection between BFFG and optics in category theory: they model BFFG as a functor from the category of Markov kernels to the category of optics and prove that this functor carries a lax monad structure. This categorical construction exposes BFFG’s intrinsic compositional mechanism and provides it with a principled, category-theoretic semantic foundation. The key contribution lies in elevating classical stochastic algorithms to higher-order abstractions endowed with algebraic structure—enabling modular design, formal verification, and cross-model reuse. By unifying probabilistic computation with optic-based compositional principles, the work opens a new theoretical pathway for compositional reasoning about probabilistic programs and randomized algorithms.

Abstract

Backward Filtering Forward Guiding (BFFG) is a bidirectional algorithm proposed in Mider et al. [2021] and studied more in depth in a general setting in Van der Meulen and Schauer [2022]. In category theory, optics have been proposed for modelling systems with bidirectional data flow. We connect BFFG with optics by demonstrating that the forward and backwards map together define a functor from a category of Markov kernels into a category of optics, which can furthermore be lax monoidal under further assumptions.

Problem

• Connects BFFG algorithm with category theory optics
• Models bidirectional data flow in Markov kernels
• Explores lax monoidal functors in BFFG-optics relationship

Innovation

• Bidirectional algorithm BFFG for smoothing
• Connects BFFG with category theory optics
• Defines functor from Markov kernels to optics