Imprecision Attenuates Updating

This paper studies how imprecision in noisy signals attenuates Bayesian updating toward the prior. This phenomenon is well-known under a normal prior and normal noise, where less precise signals yield posterior means closer to the prior mean. We show this effect extends to any symmetric, log-concave prior and any symmetric, quasi-concave location experiment, using a newly introduced precision order. Our main result is that for any such prior and any signal realization, the posterior mean under location experiment S is closer to the prior mean than is the posterior mean under S', if and only if S is less precise than S'. We discuss applications to cognitive imprecision, prior precision, and overconfidence.