Fisher Efficient Inference of Intractable Models

TL;DR

This paper introduces a Discriminative Likelihood Estimator (DLE) that achieves Fisher efficiency for intractable models by leveraging density ratio estimation and Stein operators, providing a practical alternative to maximum likelihood estimation.

Contribution

The paper derives a new DLE method that attains the efficiency bound for intractable models and offers a dual formulation for easier optimization.

Findings

DLE is consistent and asymptotically efficient.

Numerical studies confirm theoretical properties.

DLE successfully estimates complex intractable models.

Abstract

Maximum Likelihood Estimators (MLE) has many good properties. For example, the asymptotic variance of MLE solution attains equality of the asymptotic Cram{\'e}r-Rao lower bound (efficiency bound), which is the minimum possible variance for an unbiased estimator. However, obtaining such MLE solution requires calculating the likelihood function which may not be tractable due to the normalization term of the density model. In this paper, we derive a Discriminative Likelihood Estimator (DLE) from the Kullback-Leibler divergence minimization criterion implemented via density ratio estimation and a Stein operator. We study the problem of model inference using DLE. We prove its consistency and show that the asymptotic variance of its solution can attain the equality of the efficiency bound under mild regularity conditions. We also propose a dual formulation of DLE which can be easily optimized. Numerical studies validate our asymptotic theorems and we give an example where DLE successfully estimates an intractable model constructed using a pre-trained deep neural network.