Adversarial Estimation of Assortment Probabilities under Independence Structure

TL;DR

This paper introduces a regularized adversarial estimator for efficiently estimating assortment probabilities by exploiting independence structures, improving statistical and computational performance in high-dimensional settings.

Contribution

It develops a novel adversarial estimation framework that leverages independence structures, extending to covariate settings and outperforming traditional methods.

Findings

Achieves optimal convergence rates under regularity conditions

Demonstrates substantial finite-sample improvements in causal inference

Numerical studies validate theoretical advantages

Abstract

We consider the problem of estimating assortment probabilities, which is common in operations management applications, including product bundling, advertising, etc. Existing approaches typically model each assortment as a category and apply multinomial models to estimate the choice probabilities; while computationally convenient, these methods do not exploit independence structures in the joint distribution and may therefore be statistically inefficient when the total number of items is large. Using the representation from Bahadur (1959), we relate the sparsity of the generalized correlation coefficients to the independence structure of the binary components. We formulate the problem as estimating a high-dimensional vector of generalized correlation coefficients, together with low or moderate-dimensional nuisance parameters corresponding to the marginal probabilities. We develop a regularized adversarial estimator that attains the optimal rate under standard regularity conditions while remaining computationally feasible. The framework naturally extends to settings with covariates. We apply the proposed estimators to causal inference with multiple binary treatments and show substantial finite-sample improvements over non-adaptive methods. Numerical studies corroborate the theoretical results.