Risk function
Encyclopedia
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
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
and estimation theory
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...
, the risk function R of a decision rule
Decision rule
In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy in game theory....
, δ, is the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of a loss function
Loss function
In statistics and decision theory a loss function is a function that maps an event onto a real number intuitively representing some "cost" associated with the event. Typically it is used for parameter estimation, and the event in question is some function of the difference between estimated and...
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 measureProbability measureIn mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
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
, a decision function whose output 
is an estimate of 
, and a quadratic loss function,
- the risk function becomes the mean squared errorMean squared errorIn statistics, the mean squared error of an estimator is one of many ways to quantify the difference between values implied by a kernel density estimator and the true values of the quantity being estimated. MSE is a risk function, corresponding to the expected value of the squared error loss or...
of the estimate,
- In density estimationDensity estimationIn probability and statistics,density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function...
, the unknown parameter is probability densityProbability densityProbability density may refer to:* Probability density function in probability theory* The product of the probability amplitude with its complex conjugate in quantum mechanics...
itself. The loss function is typically chosen to be a norm in an appropriate function spaceFunction spaceIn mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
. For example, for
norm,
- the risk function becomes the mean integrated squared error
