How to Approximate Inference with Subtractive Mixture Models

Softmax Mixture Models (SMMs) are discrete $K$-component mixture models for the probabilities of selecting one of $p$ candidate feature vectors $X_1,\ldots,X_p\in\mathbb{R}^L$ in heterogeneous populations and are widely used in econometrics and scientific applications. Related softmax mixture mechanisms also appear in modern LLM architectures. We provide a theoretical and methodological study of SMMs, focusing on the Expectation-Maximization (EM) algorithm and the Method of Moments (MoM). We show that EM recovers the mixture atoms at the parametric rate, up to logarithmic factors, after $\mathcal{O}(\log N)$ iterations, provided atom separation is at least of order $\log K$. This improves on separation conditions in existing analyses of EM for high-dimensional Gaussian mixtures. We also develop MoM procedures for parameter and subspace estimation. Although MoM parameter estimates converge more slowly than EM and can deteriorate with $K$, they provide provable warm starts for EM and are useful for small $K$. For general $K$, we estimate the atom subspace via MoM and recommend running EM from multiple random initializations within this subspace. Finally, as $p\to\infty$, we show that SMMs approximate mixtures of exponential tilts of the feature distribution, yielding asymptotic identifiability.

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $M$-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $f_i$; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $\varphi$-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $\ell_2$-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.