Posterior Mean and Mode

Two common Bayesian point estimators are the posterior mean and the posterior mode.

The posterior mode is the most likely value of the parameter given the data. It is the highest point in the posterior pdf. (The value, y, that maximizes P(y | data)). With uninformative priors, the posterior mode is frequently the same as the MLE.

The posterior mean is the expected value of the parameter given the data. E(y | data)

The posterior mode is useful when the parameter itself is your interest. For example, when measuring the fraction of voters who support candidate x, the posterior mode is the most likely fraction after you did your survey.

The posterior mean is useful when you want to predict the future based on the parameter. For example, the probability that an individual voter will support candidate x is the posterior mean.