The minimax optimal convergence rate of posterior density in the weighted orthogonal polynomials

TL;DR

This paper establishes the minimax optimal convergence rate for Bayesian density estimation using orthogonal polynomial expansions in weighted Sobolev spaces, addressing challenges on unbounded domains without positive lower bounds.

Contribution

It introduces a novel shifting method to handle densities without positive lower bounds and constructs a Gaussian sieve prior achieving the minimax rate.

Findings

The proposed method attains the minimax rate of n^{-p/(2p+1)}.

Numerical results confirm the estimator's accuracy and theoretical convergence rate.

The shifting technique effectively extends the theory to unbounded domains without positive lower bounds.

Abstract

We investigate Bayesian nonparametric density estimation via orthogonal polynomial expansions in weighted Sobolev spaces. A core challenge is establishing minimax optimal posterior convergence rates, especially for densities on unbounded domains without a strictly positive lower bound. For densities bounded away from zero, we give sufficient conditions under which the framework of \cite{shen2001} applies directly. For densities lacking a positive lower bound, the equivalence between Hellinger and weighted $L_2$-norm distance fails, invalidating the original theory. We propose a novel shifting method that lifts the true density $g_0$ to a sequence of proxy densities $g_{0,n}$. We prove a modified convergence theorem applicable to these shifted densities, preserving the optimal rate. We also construct a Gaussian sieve prior that achieves the minimax rate $\varepsilon_n=n^{-p/(2p+1)}$ for any integer $p\geq1$. Numerical results confirm that our estimator approximates the true density well and validates the theoretical convergence rate.