The Uniform Prior Bayes Estimate

For the Gaussian sequence model Xi = μi + σZi, i = 1, 2, · · · , n, where Zi are iid standard normal, and σ is considered known, we give three equivalent exact formulas for the posterior mean of the mean vector μ = (μ1, · · · , μn) when μ has a uniform prior distribution in an n-dimensional solid ball of a general radius c. The exact formulas have interesting and even curious connections to noncentral chisquare large deviation probabilities and modified Bessel functions. These exact formulas are then used to derive approximations to the posterior mean in two different asymptotic paradigms, one for fixed ||x|| and n → ∞, and the other for fixed but arbitrary n and ||x|| → ∞. We then address the problem of calibrating the radius c of the ball to prior knowledge about μ. Specifically, we show concrete methods to calibrate c to prior information about either the L∞ norm or the L1 norm of μ. This leads us to derive two weak convergence results, which may be of independent interest. There could be some use of our results in sparse high dimensional Gaussian estimation, or in general shrinkage estimation.