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20.6 Uniform Posteriors

Suppose your model includes a parameter $\psi$ that is defined on $[0,1]$ and is given a flat prior $\textsf{uniform}(\psi \mid 0,1)$. Now if the data don’t tell us anything about $\psi$, the posterior is also $\textsf{uniform}(\psi \mid 0,1)$.

Although there is no maximum likelihood estimate for $\psi$, 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 $\psi$ 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 $\psi$ at all points in $[0,1]$.