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Bayesian Decision Theory

A posterior is a belief. A decision requires combining it with consequences.

The Bayes action
a∗=arg⁡min⁡a  Eθ∣D[L(a,θ)]a^* = \arg\min_{a}\; E_{\theta \mid D}\left[L(a,\theta)\right]

Minimise expected loss under the posterior. The loss function is where your actual preferences enter the maths.

Choose the action minimising expected loss, averaged over everything you still do not know.

The loss function picks the estimate

This is the part worth internalising, because it explains a question people usually answer by convention.

Squared loss L=(a−θ)2L = (a - \theta)^2 is minimised by the posterior mean.

Absolute loss L=∣a−θ∣L = |a - \theta| is minimised by the posterior median.

0-1 loss is minimised by the posterior mode.

So "should I report the mean or the median?" is not a matter of taste. It depends entirely on how you are penalised for being wrong, and the loss function answers it.

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