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Optimal Particle-based Approximation of Discrete Distributions (OPAD)

Hadi Mohasel Afshar, Gilad Francis, Sally Cripps

TL;DR

This work proves that for discrete target distributions, the KL divergence of any particle-based approximation is minimized when particle weights are proportional to the target probabilities, defining the Optimal Particle-based Approximation of Discrete Distributions (OPAD). The authors establish a main theorem showing the minimum $D_{KL}$ equals $-\log(\pi^*(\mathcal{X}^P))$ and provide a Jensen-based proof, with OPAD+ further leveraging rejected proposals. The approach requires no extra computation and can be applied to existing MCMC outputs by reweighting, yielding substantial reductions in approximation error. Empirical evaluations on Ising models, Bayesian Variable Selection, and Bayesian Structure Learning demonstrate that OPAD/OPAD+ consistently outperform standard MCMC in KL divergence by orders of magnitude, highlighting the practical impact for high-dimensional discrete inference.

Abstract

Particle-based methods include a variety of techniques, such as Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC), for approximating a probabilistic target distribution with a set of weighted particles. In this paper, we prove that for any set of particles, there is a unique weighting mechanism that minimizes the Kullback-Leibler (KL) divergence of the (particle-based) approximation from the target distribution, when that distribution is discrete -- any other weighting mechanism (e.g. MCMC weighting that is based on particles' repetitions in the Markov chain) is sub-optimal with respect to this divergence measure. Our proof does not require any restrictions either on the target distribution, or the process by which the particles are generated, other than the discreteness of the target. We show that the optimal weights can be determined based on values that any existing particle-based method already computes; As such, with minimal modifications and no extra computational costs, the performance of any particle-based method can be improved. Our empirical evaluations are carried out on important applications of discrete distributions including Bayesian Variable Selection and Bayesian Structure Learning. The results illustrate that our proposed reweighting of the particles improves any particle-based approximation to the target distribution consistently and often substantially.

Optimal Particle-based Approximation of Discrete Distributions (OPAD)

TL;DR

This work proves that for discrete target distributions, the KL divergence of any particle-based approximation is minimized when particle weights are proportional to the target probabilities, defining the Optimal Particle-based Approximation of Discrete Distributions (OPAD). The authors establish a main theorem showing the minimum equals and provide a Jensen-based proof, with OPAD+ further leveraging rejected proposals. The approach requires no extra computation and can be applied to existing MCMC outputs by reweighting, yielding substantial reductions in approximation error. Empirical evaluations on Ising models, Bayesian Variable Selection, and Bayesian Structure Learning demonstrate that OPAD/OPAD+ consistently outperform standard MCMC in KL divergence by orders of magnitude, highlighting the practical impact for high-dimensional discrete inference.

Abstract

Particle-based methods include a variety of techniques, such as Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC), for approximating a probabilistic target distribution with a set of weighted particles. In this paper, we prove that for any set of particles, there is a unique weighting mechanism that minimizes the Kullback-Leibler (KL) divergence of the (particle-based) approximation from the target distribution, when that distribution is discrete -- any other weighting mechanism (e.g. MCMC weighting that is based on particles' repetitions in the Markov chain) is sub-optimal with respect to this divergence measure. Our proof does not require any restrictions either on the target distribution, or the process by which the particles are generated, other than the discreteness of the target. We show that the optimal weights can be determined based on values that any existing particle-based method already computes; As such, with minimal modifications and no extra computational costs, the performance of any particle-based method can be improved. Our empirical evaluations are carried out on important applications of discrete distributions including Bayesian Variable Selection and Bayesian Structure Learning. The results illustrate that our proposed reweighting of the particles improves any particle-based approximation to the target distribution consistently and often substantially.

Paper Structure

This paper contains 17 sections, 3 theorems, 32 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimal Particle-based Approximation of Discrete Distributions (OPAD)
  3. MCMC and OPAD
  4. OPAD+
  5. Experiments
  6. 1D Ising Model
  7. Reference MCMC for Ising Model
  8. Bayesian Variable Selection
  9. Reference MCMC for Bayesian Variable Selection
  10. Synthetic dataset
  11. Real-world dataset
  12. Bayesian Structure Learning
  13. Reference MCMC for Bayesian Structure Learning
  14. Synthetic dataset
  15. Evaluations
  16. ...and 2 more sections

Key Result

Proposition 1

For any discrete distribution, $P$, on a set, ${\mathcal{X}}^P$, any integrable function $g$ and any strictly convex function, $f: \mathbb{R} \to \mathbb{R}$, if $\exists \;{\boldsymbol{\mathbf{x}}}_1, {\boldsymbol{\mathbf{x}}}_2 \in {\mathcal{X}}^{P}$ such that ${\boldsymbol{\mathbf{x}}}_1 \neq {\b Otherwise, that is, if there exists a constant $c$ such that $\forall {\boldsymbol{\mathbf{x}}} \in

Figures (4)

  • Figure 1: KL divergence from the 1D Ising model (Equation \ref{['eq.ising1d']}) versus sampling iterations: Plotted for 20 chains of the reference MCMC (see Section \ref{['sect.ising1d.mcmc']}) (blue curves), as well as their OPAD and OPAD+ counterparts (red and green curves) (a) for 10K sampling iterations and (b) for 1 million iterations.
  • Figure 2: KL divergence from the Bayesian Variable Selection target distribution (Equation \ref{['eq:bvs.target']}) versus sampling iterations using a reference MCMC explained in Section \ref{['sect.bvs.mcmc']}: On the left, the divergence of 20 MCMC chains is plotted (blue curves), along with their corresponding OPAD and OPAD+ counterparts (red and black curves), whereas on the right, the mean and 95% confidence interval of these algorithms are plotted. In (a) & (b), the target posterior is w.r.t. 200 synthetic data points (see Section \ref{['sect.bvs.synthetic.data']}) while in (c) & (d), the target posterior is w.r.t. the real-world Mice nutrition dataset (Section \ref{['sect.bvs.real.data']}).
  • Figure 3: KL divergence from the Bayesian Structure Learning target distribution (Equation \ref{['eq:bsl.posterior']}) versus sampling iterations where the reference MCMC is Structure MCMC. The target posterior is synthesized per MCMC chain using 200 data points generated from a randomly constructed Erdős--Rényi ground truth DAG, denoted as $\text{ER}(n, d)$, where $n$ represents the number of nodes and $d$ indicates the expected vertex degree. The mean and 95% confidence intervals of 20 chains are plotted.
  • Figure 4: KL divergence from the Bayesian Structure Learning target distribution (Equation \ref{['eq:bsl.posterior']}) versus sampling iterations where the reference MCMC is Partition MCMC. The target posterior is synthesized per MCMC chain using 200 data points generated from a randomly constructed Erdős--Rényi ground truth DAG, denoted as $\text{ER}(n, d)$, where $n$ represents the number of nodes and $d$ indicates the expected vertex degree. The mean and 95% confidence intervals of 20 chains are plotted.

Theorems & Definitions (3)

  • Proposition 1: A variant of Jensen's inequality
  • Theorem 1: Main Theorem
  • Corollary 1