# Adversarial Variational Bayes Methods for Tweedie Compound Poisson Mixed Models

## TL;DR

This paper introduces an adversarial variational inference method for Bayesian Tweedie mixed models, effectively handling intractable likelihoods and hierarchical structures, with improved bias and predictive performance demonstrated on real and simulated data.

## Contribution

It develops a novel adversarial variational inference approach for Tweedie mixed models, addressing intractability and hierarchical complexities with enhanced accuracy.

## Key findings

- Smaller estimation bias on random effects compared to MCMC.
- Achieves state-of-the-art predictive performance.
- Provides richer variance function estimation.

## Abstract

The Tweedie Compound Poisson-Gamma model is routinely used for modeling non-negative continuous data with a discrete probability mass at zero. Mixed models with random effects account for the covariance structure related to the grouping hierarchy in the data. An important application of Tweedie mixed models is pricing the insurance policies, e.g. car insurance. However, the intractable likelihood function, the unknown variance function, and the hierarchical structure of mixed effects have presented considerable challenges for drawing inferences on Tweedie. In this study, we tackle the Bayesian Tweedie mixed-effects models via variational inference approaches. In particular, we empower the posterior approximation by implicit models trained in an adversarial setting. To reduce the variance of gradients, we reparameterize random effects, and integrate out one local latent variable of Tweedie. We also employ a flexible hyper prior to ensure the richness of the approximation. Our method is evaluated on both simulated and real-world data. Results show that the proposed method has smaller estimation bias on the random effects compared to traditional inference methods including MCMC; it also achieves a state-of-the-art predictive performance, meanwhile offering a richer estimation of the variance function.