[Paper Review] $A$-Hypergeometric Distributions and Newton Polytopes

Nobuki Takayama, Satoshi Kuriki+1 more|arXiv (Cornell University)|Oct 8, 2015

Advanced Statistical Methods and Models10 references3 citations

TL;DR

This paper establishes a bijection between a quotient space of parameters and the moment space for $A$-hypergeometric distributions using Newton polytopes, enabling algorithmic inverse moment mapping via the holonomic gradient method (HGM). It proves asymptotic equivalence between the moment map and iterative proportional scaling (IPS), providing a new approximation for $A$-hypergeometric polynomials and enabling conditional maximum likelihood estimation in statistics.

ABSTRACT

We give a bijection between a quotient space of the parameters and the space of moments for any $A$-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method to solve a conditional maximum likelihood estimation problem in statistics. Our interplay between the theory of hypergeometric functions and statistics gives some new formulas of $A$-hypergeometric polynomials.

Motivation & Objective

• To establish a mathematical correspondence between the parameter space and the moment space for $A$-hypergeometric distributions.
• To develop an algorithmic method for computing the inverse of the moment map, crucial for conditional maximum likelihood estimation in statistics.
• To explore the asymptotic equivalence between the moment map and iterative proportional scaling (IPS), linking statistical estimation with algebraic geometry.
• To derive new formulas for $A$-hypergeometric polynomials using the interplay between hypergeometric functions and Newton polytopes.

Proposed method

• Define the $A$-hypergeometric distribution as a conditional distribution under a Poisson model with fixed sufficient statistics $Au = \beta$.
• Use the log-partition function $\psi(\xi) = \log Z(\beta; p(\xi))$ to express moments as $E[U_i] = \partial_{\xi_i} \psi(\xi)$, establishing the moment map from $\xi$-space to $\eta$-space.
• Introduce a quotient space of parameters via the torus action of $A$, identifying equivalent $\xi$-values that yield the same moments.
• Prove that the image of the moment map is the relative interior of the Newton polytope of $Z(\beta; p)$, establishing a bijection with the quotient space.
• Apply the holonomic gradient method (HGM) to numerically invert the moment map, enabling conditional MLE computation.
• Demonstrate asymptotic equivalence between the HGM iteration and iterative proportional scaling (IPS), using this to initialize the HGM and approximate the normalizing constant $Z(\beta; p)$.

Experimental results

Research questions

• RQ1 What is the image of the moment map $\xi \mapsto E[U]$ in the $\eta$-space, and how is it characterized geometrically?
• RQ2 How can the inverse of the moment map be computed efficiently for statistical inference in $A$-hypergeometric models?
• RQ3 What is the relationship between the holonomic gradient method and iterative proportional scaling (IPS) in the context of $A$-hypergeometric distributions?
• RQ4 Can the normalizing constant $Z(\beta; p)$ be asymptotically approximated using the IPS procedure, and what is the error behavior?

Key findings

• The image of the moment map $E[U]$ is the relative interior of the Newton polytope defined by $A\eta = \beta$, $\eta \geq 0$, establishing a geometric characterization of the moment space.
• A bijection is established between the quotient space of parameters (modulo the torus action of $A$) and the Newton polytope in the $\eta$-space, providing a new isomorphism in information geometry.
• The holonomic gradient method (HGM) provides an effective algorithmic approach to invert the moment map, enabling conditional maximum likelihood estimation for $A$-hypergeometric distributions.
• The moment map is asymptotically equivalent to iterative proportional scaling (IPS), with convergence to the same limit as $k \to \infty$ for scaled problems $k\beta$.
• Theorem 6 provides an asymptotic approximation for $Z(k\beta; p)$ using the IPS solution $m$, with error decreasing as $k$ increases, validated numerically for $k=9, 200, 300$.
• For $k=9$, the HGM approximation of $\log Z$ has an error of 1.8052, decreasing to 0.3734 at $k=300$, showing convergence to the exact value.