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- This article is about the mathematical definition of risk in statistical decision theory. For a more general discussion of concepts and definitions of risk, see the main article Risk.
- θ is a fixed but possibly unknown state of nature;
- X is a vector of observations stochastically drawn from a population;
- 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.
- For a scalar parameter θ, a decision function whose output is an estimate of θ, and a quadratic loss function
- the risk function becomes the mean squared error of the estimate,
- In density estimation, the unknown parameter is probability density itself. The loss function is typically chosen to be a norm in an appropriate function space. For example, for L2 norm,
- the risk function becomes the mean integrated squared error
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