Design a Bayesian hierarchical model for analyzing user behavior data across different regions.

Official answer

For a defined binary engagement event, one candidate model is y_ir ~ Bernoulli(p_ir) with logit(p_ir) = α_r + x_irᵀβ, where r is region and x contains appropriately scaled predictors. Use α_r ~ Normal(μ, τ²) to partially pool region intercepts.

An illustrative prior specification is μ ~ Normal(0, 1.5²), each β_j ~ Normal(0, 1²), and τ ~ half-Normal(0, 1²). These are starting assumptions, not universal defaults: check prior predictive implications against the event rate and feature units, and conduct sensitivity analysis. At μ=0 and zero covariates the central probability is 0.5, which may be implausible for a rare event.

Fit with a suitable Bayesian engine, check convergence, effective sample sizes and divergences, and use posterior predictive checks. Compare prediction for held-out regions and later observations as appropriate.

Posterior region probabilities and contrasts express uncertainty under the model. Sparse regions tend to borrow more information, but pooling is inappropriate if the shared structure badly misrepresents them. Repeated user outcomes need added dependence structure. Regional associations are not automatically causal effects.

Reference: Stan hierarchical regression.