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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 $$[0,1]$$.