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Direct sampling from conditional distributions by sequential maximum likelihood estimations

Shuhei Mano

TL;DR

This work addresses direct sampling from the conditional distribution of discrete log-affine models by revealing that the sampling transition equals the UMVUE ratio to the total count, a computation dominated by $Z_A(b;x)$. It introduces an approximate direct sampling algorithm that substitutes the UMVUE with the MLE, enabling practical sampling without solving holonomic or Gröbner-basis problems. It proves that the UMVUE and the rational MLE coincide iff the model is decomposable (in which case the MLE is rational), but not in general, and demonstrates the approach on two-way and three-way contingency-table models. Numerical experiments show the approximate method can outperform Metropolis in certain decomposable cases and remains feasible when exact direct sampling is prohibitive, with asymptotically vanishing bias as sample size grows.

Abstract

We can directly sample from the conditional distribution of any log-affine model. The algorithm is a Markov chain on a bounded integer lattice, and its transition probability is the ratio of the UMVUE (uniformly minimum variance unbiased estimator) of the expected counts to the total number of counts. The computation of the UMVUE accounts for most of the computational cost, which makes the implementation challenging. Here, we investigated an approximate algorithm that replaces the UMVUE with the MLE (maximum likelihood estimator). Although it is generally not exact, it is efficient and easy to implement; no prior study is required, such as about the connection matrices of the holonomic ideal in the original algorithm.

Direct sampling from conditional distributions by sequential maximum likelihood estimations

TL;DR

This work addresses direct sampling from the conditional distribution of discrete log-affine models by revealing that the sampling transition equals the UMVUE ratio to the total count, a computation dominated by . It introduces an approximate direct sampling algorithm that substitutes the UMVUE with the MLE, enabling practical sampling without solving holonomic or Gröbner-basis problems. It proves that the UMVUE and the rational MLE coincide iff the model is decomposable (in which case the MLE is rational), but not in general, and demonstrates the approach on two-way and three-way contingency-table models. Numerical experiments show the approximate method can outperform Metropolis in certain decomposable cases and remains feasible when exact direct sampling is prohibitive, with asymptotically vanishing bias as sample size grows.

Abstract

We can directly sample from the conditional distribution of any log-affine model. The algorithm is a Markov chain on a bounded integer lattice, and its transition probability is the ratio of the UMVUE (uniformly minimum variance unbiased estimator) of the expected counts to the total number of counts. The computation of the UMVUE accounts for most of the computational cost, which makes the implementation challenging. Here, we investigated an approximate algorithm that replaces the UMVUE with the MLE (maximum likelihood estimator). Although it is generally not exact, it is efficient and easy to implement; no prior study is required, such as about the connection matrices of the holonomic ideal in the original algorithm.

Paper Structure

This paper contains 10 sections, 10 theorems, 65 equations, 1 figure, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithm
  3. Markov transitions and the UMVUE
  4. The MLE
  5. The UMVUE and the rational MLE
  6. The approximate algorithm without using connection matrices
  7. Numerical experiments
  8. Two-way contingency tables
  9. No-three-way interaction model of three-way table
  10. Discussion

Key Result

Lemma 2.3

The UMVUE $\tilde{\mu}$ of the vector of expected counts $\mu$ in the log-affine model $\mathcal{M}_A$ is given as for $b\in\mathbb{N}A$, where $Z_A(b;x)$ is the $A$-hypergeometric polynomial defined by Apol. Moreover, it is the unique unbiased estimator that is a function of $b$.

Figures (1)

  • Figure 1: The Markov lattice for the matrix in Example \ref{['exam:2x2:1']} with the maximum $b=(1,2,2,1)^\top$. A sample path is shown by solid edges.

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2: MT21+
  • Example 1.3: Two-way contingency tables of the independence model
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4: Lau94
  • Proposition 2.5
  • proof
  • ...and 16 more