Compositionality in algorithms for smoothing

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 is furthermore lax monoidal in the case when the guiding kernels coincide with the generative dynamics