Loss and risk

In Bayesian decision theory a set $A$ of possible actions $a$ is considered, together with a function $l(x,y,a)$ describing the loss $l$ suffered in situation $x$ if $y$ appears and action $a$ is selected [16,128,185,201]. The loss averaged over test data $x$, $y$, and possible states of Nature ${h}$ is known as expected risk,

$$r(a,f) = \int \!dx\,dy\, p(x)\,p(y\vert x,f)\,l(x,y,a).$$ (41)

$$= < l(x,y,a) >_{X,Y\vert f}$$ (42)

$$= < r(a,{h}) >_{{H}\vert f}$$ (43)

where $< \cdots >_{X,Y\vert f}$ denotes the expectation under the joint predictive density $p(x,y\vert f)$ = $p(x) p(y\vert x,f)$ and

$$r(a,{h}) = \int \!dx\,dy\, p(x)\,p(y\vert x,{h})\,l(x,y,a).$$ (44)

The aim is to find an optimal action $a^*$

$$a^* ={\rm argmin}_{a\in A} r(a,f) .$$ (45)