Imprecision Attenuates Updating

This paper studies how imprecision in noisy signals attenuates Bayesian posterior means toward the prior mean, a property underlying many comparative-statics results in information economics. We introduce the precision order, characterized by the attenuation effect across symmetric location experiments: noise density $\tilde g$ is more precise than $g$ if and only if the posterior mean under $\tilde g$ is closer to the signal than under $g$ for all signal realizations and all symmetric, log-concave priors. We apply the precision order to derive comparative statics for prior precision, the value of information, and voting.