On the influence of the prior distribution in im- age reconstruction

Two measures of the influence of the prior distribution p(θ) in Bayes estimation are proposed. Both involve comparing with alternative priors proportional to p(θ), for s ≥ 0. The first one, the influence curve for the prior distribution is simply the curve of parameter values which are obtained as estimates when the estimation is made using p(θ) instead of p(θ). It measures the overall influence of the prior. The second one, the influence rate for the prior, is the derivative of this curve at s = 1, and quantifies the sensitivity to small changes or inaccuracies in the prior distribution. We give a simple formula for the influence rate in marginal posterior mean estimation, and discuss how the influence measures may be computed and used in image processing with Markov random field priors. The results are applied to an image reconstruction problem from visual field testing and to a stylized image analysis problem.