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Speeding up the ordered allocation sampler

Maria F. Gil-Leyva, Fidel Selva, Pierpaolo De Blasi

Abstract

The ordered allocation sampler is a Gibbs sampler designed to explore the posterior distribution in nonparametric mixture models. It encompasses both infinite mixtures and finite mixtures with random number of components, and it has be shown to possess mixing properties that pair well with collapsed, or marginal, samplers that integrate out the mixing distribution. The main advantage is that it adapts to mixing priors that do not enjoy tractable predictive structures needed for the implementation of marginal sampling methods. Thus it is as widely applicable as other conditional samplers while enjoying better algorithmic performances. In this paper we provide a modification of the ordered allocation sampler that enhances its performances in a substantial way while easing its implementation. In addition, exploiting the similarity with marginal samplers, we are able to adapt to the new version of the sampler the split-merge moves of Jain and Neal. Simulation studies confirm these findings.

Speeding up the ordered allocation sampler

Abstract

The ordered allocation sampler is a Gibbs sampler designed to explore the posterior distribution in nonparametric mixture models. It encompasses both infinite mixtures and finite mixtures with random number of components, and it has be shown to possess mixing properties that pair well with collapsed, or marginal, samplers that integrate out the mixing distribution. The main advantage is that it adapts to mixing priors that do not enjoy tractable predictive structures needed for the implementation of marginal sampling methods. Thus it is as widely applicable as other conditional samplers while enjoying better algorithmic performances. In this paper we provide a modification of the ordered allocation sampler that enhances its performances in a substantial way while easing its implementation. In addition, exploiting the similarity with marginal samplers, we are able to adapt to the new version of the sampler the split-merge moves of Jain and Neal. Simulation studies confirm these findings.

Paper Structure

This paper contains 10 sections, 4 theorems, 59 equations, 1 figure, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Ordered allocation sampler
  3. Efficient ordered allocation sampler
  4. Empirical comparison of Gibbs samplers
  5. Split and merge moves
  6. Split-merge moves for non size-biased ordered weights
  7. Illustration
  8. Final remarks
  9. Derivation of the ordered allocation sampler
  10. Distributional symmetries of weights in size-biased random order

Key Result

Theorem 1

Let $\bm{\theta} = (\theta_i)_{i=1}^\infty$ be a species sampling sequence driven by the species sampling process $P = \sum_{j=1}^\infty p_j \delta_{x_j}$, with base measure $\nu$. Define the $j$th distinct value to appear in $\bm{\theta}$ through $\tilde{x}_j = \theta_{M_j}$, where $M_{j} = \min\{i

Figures (1)

  • Figure 1: Plot of $f$ (solid line) and $\hat{f}$ via the OAS with (dashed line) and without (dot-dashed line) split-merge moves. The graph on the left corresponds to the trimodal mixture using a DP prior and the one on the right pertains the bimodal mixture using a GP prior.

Theorems & Definitions (11)

  • Theorem 1
  • Definition 1
  • Remark 5.1
  • Remark A.1
  • Lemma B.1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 1 more