Bayesian Poisson regression with global-local shrinkage priors

Poisson regression is a fundamental tool for modeling count data, appearing ubiquitously in applications ranging from epidemiology (disease counts) and ecology (species abundance) to economics (patent counts) and social sciences (event frequencies). The classical generalized linear model (GLM) framework treats count outcomes as Poisson-distributed with a log-linear relationship to covariates. However, modern datasets often involve high-dimensional predictor spaces where the number of covariates p is large relative to, or even exceeds, the sample size n, presenting significant challenges for traditional estimation methods.

In high-dimensional settings, regularization becomes essential to prevent overfitting and improve prediction. Frequentist approaches like LASSO (ℓ1 penalization) and elastic net have proven effective for variable selection and sparse estimation. However, these methods face limitations including: (i) difficulty in uncertainty quantification, (ii) sensitivity to tuning parameter selection, and (iii) challenges in handling grouped or correlated predictors. Bayesian approaches offer a natural alternative, replacing penalization with prior distributions that encode sparsity assumptions while providing full posterior uncertainty quantification.

Global-local shrinkage priors represent a powerful class of hierarchical Bayesian priors that have revolutionized sparse estimation. These priors—including the horseshoe (Carvalho et al., 2010), the Dirichlet-Laplace (Bhattacharya et al., 2015), the generalized double Pareto (Armagan et al., 2013), and the R2-D2 prior (Zhang et al., 2022)—combine a global shrinkage parameter (controlling overall sparsity) with local parameters (allowing individual coefficients to escape shrinkage when warranted). This hierarchical structure provides near-optimal performance: strong shrinkage toward zero for negligible effects while minimal shrinkage for truly important signals.

The horseshoe prior, in particular, has gained prominence due to its theoretical properties: it concentrates posterior mass near zero for small coefficients while having heavy tails that resist over-shrinking large effects. Unlike LASSO's Laplace prior, which induces bias in large coefficients, the horseshoe's Cauchy-like tails preserve signal strength. Extending these powerful priors to Poisson regression with potentially overdispersed or zero-inflated counts presents both theoretical and computational challenges, including posterior intractability requiring advanced MCMC or variational inference methods.