Bayesian sigmoid shrinkage with improper variance priors and an application to wavelet denoising

The normal Bayesian linear model is extended by assigning a flat prior to the δδth power of the variance components of the regression coefficients 0<δ⩽12 in order to improve prediction accuracy. In the case of orthonormal regressors, easy-to-compute analytic expressions are derived for the posterior distribution of the shrinkage and regression coefficients. The expected shrinkage is a sigmoid function of the squared value of the least-squares estimate divided by its standard error. This gives a small amount of shrinkage for large values and, provided δδ is small, heavy shrinkage for small values. The limit behavior for both small and large values approaches that of the ideal coordinatewise shrinker in terms of the expected squared error of prediction, when δδ is close to 0. In a simulation study of wavelet denoising, the proposed Bayesian shrinkage model yielded a lower mean squared error than soft thresholding (lasso), and was competitive with two recent wavelet shrinkage methods based on mixture prior distributions.