Bayesian priors and monotone shrinkage in the normal location model

We study the relationship between Bayesian posterior means and frequentist shrinkage estimators in the normal location model x|θ∼N(θ,1). Consider estimators of θ of the form t(x)=λ(x)x, where the shrinkage factor λ(x) is symmetric, bounded between zero and one, and nondecreasing in |x|. When the estimator is the posterior mean m(x) under a symmetric prior π(θ), the shrinkage factor satisfies λ(x)=m(x)/x. We derive a simple sufficient condition on the prior distribution ensuring that λ(x) is monotonic. The key step is an identity expressing the score of the marginal density as the posterior expectation of the score of the prior. Monotonicity holds whenever the ratio ψ(θ)/θ, where ψ(θ)=−π′(θ)/π(θ), is nonincreasing on (0,∞). The condition is satisfied by a broad class of heavy-tailed priors used in robust Bayesian analysis.