# How can a uniform prior make the posterior mean different from the MLE?

I read the following in Machine Learning: A Probabilistic Perspective:

How can a uniform prior move the posterior mean? Isn't a uniform distribution supposed to not bias the result? Are there any other examples where this happens?

Note: In my original question I had misread the paragraph. I fixed the question accordingly.

From a pragmatic perspective, the argument is a bit silly, as the two values differ by an order of $\frac {2}{N_1+N_0 }$ anyway. So the only time they are different is when you have a small sample size - in which case niether falls outside the "region of uncertainty" associated with the other.
• user023472 - posterior $\propto$ likelihood $\times$ prior. If prior is uniform, posterior $\propto$ likelihood. In that case, they necessarily share a mode. (For them not to, the previous argument has to break down somewhere...) – Glen_b Mar 16 '14 at 23:05
• This is due to the lack of invariance of the uniform prior. For example, The posterior mode for the odds $\gamma=\frac {\theta}{1-\theta}$ is not equal to the mle of the odds, under a uniform prior for $\theta$ – probabilityislogic Mar 17 '14 at 0:15