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23.6 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 \([0,1]\).