Convergence Rates for Latent Mixing Measures in Infinite Homoscedastic Location-Scale Mixture Models

Thời gian: 9h30 sáng Thứ 5, ngày  16/07/2026

Địa điểm: Phòng 507 nhà A6

Tóm tắt: We study posterior contraction rates for mixing measures in homoscedastic location-scale mixture models with infinitely many components. While posterior convergence at the level of densities is well-understood, ensuring convergence of the latent mixing measure is more challenging and has remained an open problem in settings where both location and scale parameters are unknown. We address this by deriving novel lower-bounds that connect the L1 distance between mixture densities to discrepancies, based on the Wasserstein distances and the operator norm, between the underlying mixing measures and scale matrices. Our approach combines the dual formulation of the W1 distance with functional-analytic approximation techniques. This leads to general inequalities, whose strength is determined (i) by the smoothness of the mixture kernel via the rate of decay of its characteristic function, and (ii) by a key lower-bound on the L1 metric involving the operator norm discrepancy between scale parameters. Moreover, a novel PDE inversion condition yields a sharper inequality for important ordinary-smooth cases. We specialize these bounds to popular mixtures based on multivariate Gaussian, Cauchy, and Laplace kernels. As a consequence, we obtain first-of-their-kind contraction rates in the context of Dirichlet process mixtures with an unknown scale parameter shared across components. As a byproduct of our inequalities, we can distinguish the convergence behavior of the location mixing measure from that of the scale parameter across a range of kernel choices, leading to nuanced insights into their respective rates.