Using the Observations to Estimate the Prior Distribution

Summary A parameter μ is to be estimated from an unbiased, unit-variance measurement x. The prior distribution is symmetrical about zero with unknown scale parameter σ. It is claimed that the observation x gives information about σ which is contained in the likelihood distribution p(dx|σ). An estimator μ̂ = γx is found, based on the posterior distribution of μ conditional on the most probable value of σ. For a prior distribution of arbitrary shape, γ is zero when x2 ≤ 1 and approaches 1 – x–2 when x2 ≥ 1. A minimum mean-square estimator is found based on an estimate of the prior variance. This gives the same result, independently of the shape of the prior distribution, as the previous estimator for a normal prior.