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22.6 Uniform posteriors
Suppose your model includes a parameter ψ that is defined on [0,1] and is given a flat prior uniform(ψ∣0,1). Now if the data don’t tell us anything about ψ, the posterior is also uniform(ψ∣0,1).
Although there is no maximum likelihood estimate for ψ, the posterior is uniform over a closed interval and hence proper. In the case of a uniform posterior on [0,1], the posterior mean for ψ is well-defined with value 1/2. Although there is no posterior mode, posterior predictive inference may nevertheless do the right thing by simply integrating (i.e., averaging) over the predictions for ψ at all points in [0,1].