For Bayesian model involving discrete variables, it is possible to sample a value of the discrete parameter at each iteration using methods such as Gibbs Sampling. With algorithms that do not support sampling discrete parameters, an alternative is to marginalise out the discrete parameters from the likelihood. This repo contains scripts which explore which approach is more computationally efficient.
Preprint of our analysis is available on arXiv: https://arxiv.org/abs/2204.06313