# Risk function

In decision theory and estimation theory, the risk function R of a decision rule δ, is the expected value of a loss function L:

$R(\theta,\delta) = {\mathbb E}_\theta L\big(\theta,\delta(X) \big)= \int_\mathcal{X} L\big( \theta,\delta(X) \big) \, dP_\theta(X)$

where

• θ is a fixed but possibly unknown state of nature;
• X is a vector of observations stochastically drawn from a population;
• ${\mathbb E}_\theta$ is the expectation over all population values of X;
• dPθ is a probability measure over the event space of X, parametrized by θ; and
• the integral is evaluated over the entire support of X.

## Examples

• For a scalar parameter θ, a decision function whose output $\hat\theta$ is an estimate of θ, and a quadratic loss function
$L(\theta,\hat\theta)=(\theta-\hat\theta)^2,$
the risk function becomes the mean squared error of the estimate,
$R(\theta,\hat\theta)=E_\theta(\theta-\hat\theta)^2.$
$L(f,\hat f)=\|f-\hat f\|_2^2\,,$
the risk function becomes the mean integrated squared error
$R(f,\hat f)=E \|f-\hat f\|^2.\,$