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Decision theory

*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.*

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

where

- 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*; - 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[edit | edit source]

- For a scalar parameter , a decision function whose output is an estimate of , 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 norm,

- the risk function becomes the mean integrated squared error

## References[edit | edit source]

- Template:SpringerEOM
- Berger, James O. (1985).
*Statistical decision theory and Bayesian Analysis*, 2nd, New York: Springer-Verlag. - DeGroot, Morris [1970] (2004).
*Optimal Statistical Decisions*, Wiley Classics Library. - Robert, Christian (2007).
*The Bayesian Choice*, 2nd, New York: Springer.

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