Adversarial Estimation of Assortment Probabilities under Independence Structure

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.

💡 Research Summary

The paper addresses the estimation of assortment probabilities—probabilities of observing particular combinations of binary items—a problem that arises in many operations‑management contexts such as product bundling, advertising, clinical trials, and causal inference with multiple binary treatments. Traditional approaches treat each possible assortment as a separate category and fit a multinomial model. While computationally convenient, this strategy ignores any independence structure among the items and becomes statistically inefficient when the number of items M is large, because the parameter space grows as 2^M − M − 1.

To exploit latent independence, the authors build on Bahadur’s (1959) representation of a multivariate binary distribution. For a binary vector Y∈{0,1}^M they write the joint probability in terms of marginal probabilities α_j = P(Y_j=1) and a set of generalized correlation coefficients r_ℓ, one for each subset ℓ⊆{1,…,M} with |ℓ|≥2. The vector r∈ℝ^p, where p=2^M−M−1, captures all dependence; if a subset of variables is independent (conditionally on covariates), the corresponding r_ℓ equals zero. Hence, in many realistic settings r is sparse. The authors treat α as low‑dimensional nuisance parameters and r as the high‑dimensional target of interest.

They first examine the maximum‑likelihood estimator (MLE) for (α,r). The log‑likelihood is non‑concave in α because the standardized components z_j(y,α) appear inside a product, and the dimensionality of r grows exponentially, leading to over‑fitting and computational infeasibility. A natural two‑step “plug‑in” approach fixes α by its sample means and then estimates r via an ℓ₁‑penalized convex program. Although this yields a tractable estimator, the error in α propagates to r, and the resulting convergence rate is sub‑optimal (order M·s/N, where s is the number of non‑zero r components).

To achieve optimal rates, the authors propose an adversarial estimation framework. They define a worst‑case inner minimization over α belonging to a hyper‑rectangle 𝒜⊂(0,1)^M and an outer maximization over r with an ℓ₁ penalty:

  max_{r} min_{α∈𝒜}