Does quadratic loss find the median of the prior distribution?

Question

Does quadratic loss find the median of the prior distribution? Someone told me linear loss finds the mean, all-nothing loss function finds the mode of the prior.

Answer 1

Minimizing the absolute error will lead to the median of the distribution. I assume "linear loss" is the same as "absolute error".

Minimizing the squared error will lead to the mean of the distribution. This is a standard result in introductory probability theory or statistics.

Minimizing the all-or-nothing loss will lead to the mode of the distribution (assuming it can take only a finite or countable number of values - if it is continuous, all-or-nothing loss has an expected value of zero for all single values).