A unified approach for estimation of unnormalized models via Stein operators

Mitsuki Nagai; Keisuke Yano

stat.MLOct 21, 2026

A unified approach for estimation of unnormalized models via Stein operators

Authors: Mitsuki Nagai, Keisuke Yano

Organizations: Graduate Institute for Advanced Studies, SOKENDAI · The Institute of Statistical Mathematics

Abstract

In this paper, we develop a unified framework connecting proper scoring rules and Stein's method for the estimation of unnormalized models. We construct a class of proper scoring rules that encompasses score matching, ratio matching, pseudolikelihood, and noise contrastive estimation. We show that these proper scoring rules naturally induce Stein operators. This connection yields a new construction principle for Stein operators, and enables us to study estimators based on these scoring rules through the geometry associated with the induced Stein operators. Using this geometric perspective, we provide a unified characterization of the asymptotic efficiency of the resulting estimators.

Parameter Estimation

Parameter estimation studies how to infer unknown quantities in statistical, physical, and machine-learning models from observed data.

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Generalised Score Matching on Convex Domains

Nishanth Shetty, Saisuchith Mahajan, Chandra Sekhar Seelamantula

Department of Electrical Engineering Indian Institute of Science, Bengaluru 560012

Score Matching Parameter Estimation Exponential Family

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Score matching avoids computing the normalising constant that maximum-likelihood estimation requires. On constrained domains, its generalised variants weight the Fisher divergence so that boundary terms vanish. We derive generalised score matching on open convex subsets of

\mathbb{R}^{d}

as the small-neighbourhood limit of minimum probability flow, in which the geometry of the neighbourhoods determines the weight. Every

C^{2}

positive definite weight arises in this way, including those of classical score matching on

\mathbb{R}^{d}

and of its variants for non-negative data on

\mathbb{R}_{+}^{d}

. For exponential families, we extend the standard convexity, consistency and asymptotic normality results to every such weight and show that the estimator converges to the true parameter under certain boundary conditions. For a truncated Gaussian on a polytope and a Dirichlet distribution on the simplex, proposed estimators attain the lowest median error of all methods compared, in at least 42 of 50 ground-truth configurations.

Abstract

Parameter Estimation

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