Robust inference using density-powered Stein operators

We introduce a density-power weighted variant of the Stein operator, called the γ-Stein operator, for robust inference with unnormalized probability models. The operator is motivated by the first variation of the γ-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 γ-score matching estimating equations and discuss their non-integrable, generalized-method-of-moments character. We further study two extensions: a γ-kernelized Stein discrepancy, interpreted as a robust diagnostic or contaminated-null goodness-of-fit procedure, and γ-Stein variational gradient descent for robust posterior approximation. Numerical examples on directional, mixture, and quartic-potential models illustrate the robustness--efficiency trade-off: positive γ can stabilize inference under targeted contamination, whereas γ=0 remains preferable under clean well-specified models.