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Conditional distributions for the nested Dirichlet process via sequential imputation

Evan Donald, Jason Swanson

TL;DR

The paper develops a rigorous sequential-imputation framework to perform Bayesian nonparametric inference for the nested Dirichlet process (NDP) under row-exchangeable arrays, addressing the infeasibility of exact posteriors. By constructing a simulation measure and using importance sampling with carefully designed weights, it provides a principled way to approximate the posterior distribution of row-distributions and, hence, the future behavior of agents and new agents. The approach generalizes Liu’s ideas to arbitrary state spaces, proves convergence with and without densities, and specializes to the NDP, yielding practical algorithms for posterior prediction. The authors illustrate the method across finite and infinite-state settings, including applications to coins, thumbtacks, Amazon reviews, and video-game leaderboards, highlighting robust inference and meaningful uncertainty quantification for comparing agents. Overall, the work offers a scalable, nonparametric alternative for posterior inference in complex hierarchical DP models with nested dependence.

Abstract

We consider an array of random variables, taking values in a complete and separable metric space, that exhibits a kind of symmetry which we call row exchangeability. Given such an array, a natural model for Bayesian nonparametric inference is the nested Dirichlet process (NDP). Exactly determining posterior distributions for the NDP is infeasible, since the computations involved grow exponentially with the sample size. In this paper, we present a new approach to determining these posterior distributions that involves the use of sequential

Conditional distributions for the nested Dirichlet process via sequential imputation

TL;DR

The paper develops a rigorous sequential-imputation framework to perform Bayesian nonparametric inference for the nested Dirichlet process (NDP) under row-exchangeable arrays, addressing the infeasibility of exact posteriors. By constructing a simulation measure and using importance sampling with carefully designed weights, it provides a principled way to approximate the posterior distribution of row-distributions and, hence, the future behavior of agents and new agents. The approach generalizes Liu’s ideas to arbitrary state spaces, proves convergence with and without densities, and specializes to the NDP, yielding practical algorithms for posterior prediction. The authors illustrate the method across finite and infinite-state settings, including applications to coins, thumbtacks, Amazon reviews, and video-game leaderboards, highlighting robust inference and meaningful uncertainty quantification for comparing agents. Overall, the work offers a scalable, nonparametric alternative for posterior inference in complex hierarchical DP models with nested dependence.

Abstract

We consider an array of random variables, taking values in a complete and separable metric space, that exhibits a kind of symmetry which we call row exchangeability. Given such an array, a natural model for Bayesian nonparametric inference is the nested Dirichlet process (NDP). Exactly determining posterior distributions for the NDP is infeasible, since the computations involved grow exponentially with the sample size. In this paper, we present a new approach to determining these posterior distributions that involves the use of sequential
Paper Structure (36 sections, 10 theorems, 147 equations, 5 figures, 7 tables)

This paper contains 36 sections, 10 theorems, 147 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Motivation and main objective
  3. The role of sequential imputation
  4. Other related models
  5. The dependent Dirichlet process
  6. Mixture models
  7. The infinite relational model
  8. Outline of paper
  9. Notation and background
  10. General notation
  11. Dirichlet processes
  12. Mixtures of Dirichlet processes
  13. Sequential imputation
  14. Importance sampling
  15. Effective sample size
  16. ...and 21 more sections

Key Result

Proposition 2.1

Let $\alpha$ be a nonzero, finite measure on $S$. Then for every $A \in \mathcal{S}^n$ and every Borel set $B \subseteq M_1$.

Figures (5)

  • Figure 1: Approximate distribution and density functions for $\theta_{m, \ell}$
  • Figure 2: Approximate density of $\mathcal{L}(\theta_{321, 1} \mid Y = y)$
  • Figure 3: Approximate densities for $\mathcal{L}(A(\theta_{m}) \mid Y = y)$
  • Figure 4: Asparagus Soda ($m = 9$) vs. Potato Log ($m = 2$)
  • Figure 5: Asparagus Soda ($m = 9$) vs. Pumpkins ($m = 7$)

Theorems & Definitions (22)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Remark 3.2
  • proof : Proof of Theorem \ref{['T:imp-samp']}
  • Theorem 3.4
  • proof
  • Theorem 3.6
  • ...and 12 more