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Exact finite mixture representations for species sampling processes

Ramsés H. Mena, Christos Merkatas, Theodoros Nicoleris, Carlos E. Rodríguez

TL;DR

This work establishes that every proper species sampling process can be represented exactly as a finite mixture through a latent truncation index K and atom reweighting, preserving the original prior after marginalizing K. This permits distributionally exact simulation of SSP priors and enables standard finite-mixture posterior inference without ad hoc truncation, via a per-observation augmentation (k_i,z_i) and straightforward Gibbs updates. The authors provide rigorous total-variation bounds for truncation error, analyze special cases (including Dirichlet, Pitman–Yor, and geometric SSPs), and demonstrate practical performance on simulated and galaxy data, highlighting how the choice of the decreasing sequence \{\xi_j\} affects mixing and computation. The approach clarifies the distinctions among K, c_n, and m, offering a unifying, tractable framework that broadens the applicability of SSP-based Bayesian nonparametrics, including exact prior simulation and efficient posterior computation. Overall, the paper contributes a robust methodology for exact finite representations and practical inference in SSP-based mixture models, with implications for scalable Bayesian nonparametrics and beyond.

Abstract

Discrete random probability measures are central to Bayesian inference, particularly as priors for mixture modeling and clustering. A broad and unifying class is that of proper species sampling processes (SSPs), encompassing many Bayesian nonparametric priors. We show that any proper SSP admits an exact conditional finite-mixture representation by augmenting the model with a latent truncation index and a simple reweighting of the atoms, which yields a conditional random finite-atom measure whose marginalized distribution matches the original SSP. This yields at least two consequences: (i) distributionally exact simulation for arbitrary SSPs, without user-chosen truncation levels; and (ii) posterior inference in SSP mixture models via standard finite-mixture machinery, leading to tractable MCMC algorithms without ad hoc truncations. We explore these consequences by deriving explicit total-variation bounds for the conditional approximation error when this truncation is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs.

Exact finite mixture representations for species sampling processes

TL;DR

This work establishes that every proper species sampling process can be represented exactly as a finite mixture through a latent truncation index K and atom reweighting, preserving the original prior after marginalizing K. This permits distributionally exact simulation of SSP priors and enables standard finite-mixture posterior inference without ad hoc truncation, via a per-observation augmentation (k_i,z_i) and straightforward Gibbs updates. The authors provide rigorous total-variation bounds for truncation error, analyze special cases (including Dirichlet, Pitman–Yor, and geometric SSPs), and demonstrate practical performance on simulated and galaxy data, highlighting how the choice of the decreasing sequence \{\xi_j\} affects mixing and computation. The approach clarifies the distinctions among K, c_n, and m, offering a unifying, tractable framework that broadens the applicability of SSP-based Bayesian nonparametrics, including exact prior simulation and efficient posterior computation. Overall, the paper contributes a robust methodology for exact finite representations and practical inference in SSP-based mixture models, with implications for scalable Bayesian nonparametrics and beyond.

Abstract

Discrete random probability measures are central to Bayesian inference, particularly as priors for mixture modeling and clustering. A broad and unifying class is that of proper species sampling processes (SSPs), encompassing many Bayesian nonparametric priors. We show that any proper SSP admits an exact conditional finite-mixture representation by augmenting the model with a latent truncation index and a simple reweighting of the atoms, which yields a conditional random finite-atom measure whose marginalized distribution matches the original SSP. This yields at least two consequences: (i) distributionally exact simulation for arbitrary SSPs, without user-chosen truncation levels; and (ii) posterior inference in SSP mixture models via standard finite-mixture machinery, leading to tractable MCMC algorithms without ad hoc truncations. We explore these consequences by deriving explicit total-variation bounds for the conditional approximation error when this truncation is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs.
Paper Structure (11 sections, 5 theorems, 59 equations, 6 figures, 2 tables)

This paper contains 11 sections, 5 theorems, 59 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. From infinite to finite
  3. Exact simulation of SSP
  4. Total variation comparisons
  5. SSP mixture modeling and posterior computation
  6. Truncation, clustering, and finite mixture models
  7. Interpreting c_n in finite and nonparametric models
  8. Illustrations
  9. Simulated data example
  10. Galaxy data
  11. Conclusions

Key Result

Theorem 1

Let $G$ be a proper SSP on $(\mathbb{X},\mathcal{B}_{\mathbb{X}})$ admitting the a.s. representation where $\boldsymbol{w}=(w_j)_{j\ge1}$ satisfies $w_j\ge0$ and $\sum_{j=1}^\infty w_j=1$ a.s., and $\boldsymbol{\theta}=(\theta_j)_{j\ge1}$ are independent and identically distributed draws from $G_0$, a diffuse measure on $(\mathbb{X},\mathcal{B}_{\mathbb{X}})$. Let $\boldsymbol{\xi}:=(\xi_j)_{j\ge

Figures (6)

  • Figure 1: DP simulation comparison under three scenarios. The figure displays the pointwise mean of $G([0,x])$ across repeated simulations and pointwise $95\%$ bands.
  • Figure 2: Empirical validation of the TV bounds: $d_{\mathrm{TV}}(G,G^\star_k)$ and $d_{\mathrm{TV}}(G,G_\varepsilon)$. Plots are based on 5,000 simulations with $\alpha = 6$, $\varepsilon=0.01$ and $\eta = 0.01$.
  • Figure 3: Histogram of the simulated data with Monte Carlo density estimators and $95\%$ credible intervals for different choices of $\xi_j$ and $\eta$. Panel A: DPFinite vs. DPSlice. Panel B: GSBFinite vs. GSBSlice.
  • Figure 4: Ergodic means of the occupied–cluster count $c_n$ over iterations. Panel A: DPFinite vs. DPSlice. Panel B: GSBFinite vs. GSBSlice, for different choices of $\xi_j$ and $\eta$.
  • Figure 5: Galaxy data: histogram with Monte Carlo density estimators and $95\%$ credible intervals for different choices of $\xi_j$ and $\eta$. Panel A: DPFinite vs. DPSlice. Panel B: GSBFinite vs. GSBSlice.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Proposition 1
  • Remark 1
  • proof
  • Proposition 2
  • proof