Posterior consistency of P\'olya trees for deconvolution under the linear model

Abstract

Several recent works have addressed the problem of deconvolution under a linear model, where the goal is to estimate a completely unknown $G_0$ from a vector of noisy observations $\boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\epsilon}$, assuming the coefficients $\beta_j$ are i.i.d. unobserved realizations from $G_0$. Assuming $G_0$ has a density $g_0$, we study theoretically a Bayesian nonparametric method proposed in Weinstein et al. (2025) that postulates a P\'olya tree prior $\Pi$ on $g_0$ and bases a deconvolution estimate on the posterior distribution $\Pi(\cdot|\boldsymbol{Y})$. Our main result asserts that under the true model (fixed and unknown $g_0$), and under a suitable condition on the minimum eigenvalue of $X^\top X$, the posterior $\Pi(\cdot|\boldsymbol{Y})$ concentrates around $g_0$ in sup-norm. The analysis presented builds on and extends results from Castillo (2017), where posterior consistency of P\'olya trees was proved for density estimation, the simpler problem of estimating $g_0$ when observing the coefficients $\beta_j$ directly.