Fandom

Psychology Wiki

Risk function

34,203pages on
this wiki
Add New Page
Talk0 Share

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · 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:

 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

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

ExamplesEdit

  • For a scalar parameter \theta, a decision function whose output \hat\theta is an estimate of \theta, 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.\,

ReferencesEdit

  • 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.
This page uses Creative Commons Licensed content from Wikipedia (view authors).


This page uses Creative Commons Licensed content from Wikipedia (view authors).

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random Wiki