MCMC Methods in Wavelet Shrinkage: Non-Equally Spaced Regression, Density and Spectral Density Estimation

Abstract

We consider posterior inference in wavelet based models for non-parametric regression with unequally spaced data, density estimation and spectral density estimation. The common theme in all three applications is the lack of posterior independence for the wavelet coefficients d _jk . In contrast, most commonly considered applications of wavelet decompositions in Statistics are based on a setup which implies a posteriori independent coefficients, essentially reducing the inference problem to a series of univariate problems. This is generally true for regression with equally spaced data, image reconstruction, density estimation based on smoothing the empirical distribution, time series applications and deconvolution problems.

We propose a hierarchical mixture model as prior probability model on the wavelet coefficients. The model includes a level-dependent positive prior probability mass at zero, i.e., for vanishing coefficients. This implements wavelet coefficient thresholding as a formal Bayes rule. For non-zero coefficients we introduce shrinkage by assuming normal priors. Allowing different prior variance at each level of detail we obtain level-dependent shrinkage for non-zero coefficients.

We implement inference in all three proposed models by a Markov chain Monte Carlo scheme which requires only minor modifications for the different applications. Allowing zero coefficients requires simulation over variable dimension parameter space (Green, 1995). We use a pseudo-prior mechanism (Carlin and Chib, 1995) to achieve this.