We develop a Gaussian comparison theory for posterior inference in non-Gaussian high-dimensional models. The framework allows for model misspecification and does not require the posterior distribution itself to be approximately Gaussian. Working directly with likelihood processes on separable function spaces, we establish explicit nonasymptotic free energy bounds controlled by first two moment discrepancies and local third moments. A perturbation-and-convexity argument transfers these comparisons to sandwich bounds for posterior mean squared error and posterior variance, including one-sided conclusions at nondifferentiable phase transitions. In high-dimensional product models, these conditions reduce to local moment controls and a canonical feature-radius scaling. Under weak regularity requirements, the resulting universality theory applies to a broad class of likelihood models, ranging from exponential families to more general regular local log-likelihoods. The framework is illustrated on sparse Bernoulli hypergraph inference.
We consider a general model for high-dimensional empirical risk minimization whereby the data xi are d-dimensional Gaussian vectors, the model is parametrized by Θ∈Rd×k and the loss depends on the data via the projection ΘTxi. This setting covers as special cases classical statistics methods (e.g., multinomial regressi...
Kiana Asgari, Andrea Montanari, Basil Saeed· Annals of Statistics· 0 citations
This work addresses a longstanding gap in the statistical foundations of marginal maximum likelihood estimation for high-dimensional latent variable models. Marginal maximum likelihood estimation is widely used to fit latent variable models across the social sciences, ecology, and machine learning. Despite its broad us...
Local Gaussian models of constant-step learning predict output variability and expected losses, but weak convergence alone does not justify these moment predictions. We establish moment-accurate Gaussian mixtures by matching stationary energy with local Ornstein--Uhlenbeck limits, ruling out quadratic tail mass invisib...
We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos a...
We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on...
Emanuele Dolera, Stefano Favaro, M. Giordano· 0 citations
We develop a rank-based framework for high-dimensional two-sample testing that detects marginal distributional differences beyond means and variances. Three marginal likelihood-ratio statistics generate SUM tests for widespread differences, MAX tests for concentrated departures, and Cauchy combinations for unknown sign...
Xiao-Xu Zhang, Long Feng· 0 citations
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