Posterior median ($F(m^{-})\le 1/2\le F(m)$)

A posterior median is a median of the posterior distribution after observing data. Writing its cumulative distribution function as $F$, a value $m$ is a posterior median when $F(m^{-})\le 1/2\le F(m)$. This definition permits atoms and non-unique medians.

If the posterior has finite first absolute moment, its medians minimize posterior absolute-error loss $\mathbb{E}[|\theta -a|\mid x]$. For a continuous posterior, differentiating with respect to $a$ gives $2F(a)-1$; the general case uses the corresponding left and right derivatives. Thus a minimizer is exactly a point with at least half the posterior mass on either side.

For a posterior placing probability $0.7$ at zero and $0.3$ at ten, the median is zero and its expected absolute loss is three. The posterior mean is three, whose absolute loss is $0.7(3)+0.3(7)=4.2$. Choosing a Bayes estimator requires specifying the loss, not merely selecting a conventional summary of the posterior.