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# Q-conjugate Message Passing for Efficient Bayesian Inference

*Proceedings of The 12th International Conference on Probabilistic Graphical Models*, PMLR 246:295-311, 2024.

#### Abstract

Bayesian inference in nonconjugate models such as Bayesian Poisson regression often relies on computationally expensive Monte Carlo methods. This paper introduces {Q}-conjugacy, a generalization of classical conjugacy that enables efficient closed-form variational inference in certain nonconjugate models. {Q}-conjugacy is a condition in which a closed-form update scheme expresses the solution minimizing the Kullback-Leibler divergence between a variational distribution and the product of two potentially unnormalized distributions. Leveraging {Q}-conjugacy within a local message passing framework allows deriving analytic inference update equations for nonconjugate models. The effectiveness of this approach is demonstrated on Bayesian Poisson regression and a model involving a hidden gamma-distributed latent variable with Gaussian-corrupted logarithmic observations. Results show that {Q}-conjugate triplets, such as (Gamma, LogNormal, Gamma), provide better speed-accuracy trade-offs than Markov Chain Monte Carlo.