Robust inference using density-powered Stein operators

Abstract

We introduce a density-power weighted variant of the Stein operator, called the $\gamma$-Stein operator, for robust inference with unnormalized probability models. The operator is motivated by the first variation of the $\gamma$-divergence under infinitesimal escort transport and weights the usual Stein field by a positive power of the model density. This weighting down-weights observations in low model-density regions, providing a principled robustness mechanism while retaining the normalizing-constant-free structure of score matching. We develop the resulting $\gamma$-score matching estimating equations and discuss their non-integrable, generalized-method-of-moments character. We further study two extensions: a $\gamma$-kernelized Stein discrepancy, interpreted as a robust diagnostic or contaminated-null goodness-of-fit procedure, and $\gamma$-Stein variational gradient descent for robust posterior approximation. Numerical examples on directional, mixture, and quartic-potential models illustrate the robustness--efficiency trade-off: positive $\gamma$ can stabilize inference under targeted contamination, whereas $\gamma=0$ remains preferable under clean well-specified models.