Maximum Likelihood Estimator and Posterior Mean What is it that determines whether the MLE of a parameter will be larger than its posterior mean? Is it always larger?
 A: Several things can make a posterior mean larger than the maximum of the likelihood.


*

*imagine a symmetric unimodal likelihood for a moment. If the prior tends to take higher values than the middle of the likelihood, then the mean of the posterior can be pulled up, past where the likelihood is maximized. So clearly the prior can have an impact

*now imagine a flat prior, but a likelihood that has a tail to the left (i.e. if you scale the likelihood to a density, it's left-skew, formally in the sense of first-Pearson-skewness, i.e. mode-skewness). Then the mode of the posterior will be the mode of the likelihood, but the mean of the posterior will be to its left.

*so both the shape of the likelihood and the relationship between prior and likelihood are both relevant; they act in combination and either individually or together they can make that happen.
It isn't always the case -- just flip the above situations about to make the relationship between mode of likelihood and mean of posterior go the opposite direction. 
A: By definition, the maximum likelihood estimator is obtained as the result of a maximisation program$$\arg\,\max_{\theta\in\Theta}\ell(\theta|x)$$and as such


*

*does not depend on the volume of the parameter space $\Theta$

*does not account for a prior measure on that parameter space $\Theta$

*does not depend on the parameterisation of the model: the MLE of the transform of $\theta$ by the one-to-one function $\Psi$ is the transform by $\Psi$ of the MLE of $\theta$

*only characterises the largest value of $\ell(\theta|x)$, rather than its general shape

*does not exist for unbounded likelihoods


Conversely, the posterior mean$$\mathbb{E}[\theta|x]$$


*

*depends on the prior measure $\pi$ on the parameter space $\Theta$

*depends on the parameterisation of the model: the posterior mean of the transform of $\theta$ by the one-to-one function $\Psi$ is not the transform by $\Psi$ of the posterior mean $\mathbb{E}[\theta|x]$. For instance for some transforms that produce fat tails posteriors, the posterior mean does not exist

*depends on the entire shape of $\ell(\theta|x)$, as for instance in the case of multimodal likelihoods where the posterior mean may stand in a low likelihood region in-between those modes


While both estimates converge to the "true" value of $\theta$ as the sample size $n$ grows to infinity, there is no generic reason for them to be in a particular relation.
