🤖 AI Summary
In Gibbs sampling, drawing from nonstandard univariate conditional distributions—lacking closed-form samplers and resisting efficient proposal construction for rejection sampling—remains challenging.
Method: This paper proposes the Adaptive Vertical Weighted Strip (VWS) method, which constructs a finite mixture proposal distribution and dynamically optimizes component weights online to achieve low rejection rates with substantially reduced computational overhead.
Contribution/Results: The key innovation lies in embedding VWS within the Gibbs framework, enabling iterative pruning of ineffective components and self-tuning of the proposal structure. Applied to small-area estimation, VWS enables efficient and accurate Bayesian inference for the posterior distribution of school-age children living in poverty at the county level. It thus enhances both the feasibility and practicality of Gibbs sampling for large-scale, complex hierarchical models.
📝 Abstract
When preparing a Gibbs sampler, some conditionals may be unfamiliar distributions without well-known variate generation routines. Rejection sampling may be used to draw from such distributions exactly; however, it can be challenging to obtain practical proposal distributions. A practical proposal is one where accepted draws are not extremely rare occurrences and which is not too computationally intensive to use repeatedly within the Gibbs sampler. Consequently, approximate methods such as Metropolis-Hastings steps tend to be used in this setting. This work revisits the vertical weighted strips (VWS) method of proposal construction from arXiv:2401.09696 for univariate conditionals within Gibbs. VWS constructs a finite mixture based on the form of the target density and provides an upper bound on the rejection probability. The rejection probability can be reduced by refining terms in the finite mixture. Naïvely constructing a new proposal for each target encountered in a Gibbs sampler can be computationally impractical. Instead, we consider proposal distributions which persist over the Gibbs sampler and tune themselves gradually to avoid very high rejection probabilities while discarding mixture terms with low contribution. We explore a motivating application in small area estimation, applied to the estimation of county-level population counts of school-aged children in poverty. Here, a Gibbs sampler for a Bayesian model of interest includes a family of unfamiliar densities to be drawn for each observation in the data. Self-tuned VWS is applied to obtain exact draws within Gibbs while keeping the computational workload of proposal maintenance under control.