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Loss function

Contents
    0-1 loss function
Expected loss
        Bayes Risk
        Examples in statistics
    Economic choice under uncertainty
Decision rules
Further reading

In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchyt, then a quadratic loss function is

losses are used when the data has many large outliers.

Effect of using different loss functions, when the data has outliers

0-1 loss function

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In statistics and decision theory, a frequently used loss function is the 0-1 loss function

In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and the European subsidies for equalizing unemployment rates among 271 German regions.

Expected loss

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⁠X. This is also referred to as the risk function of the decision rule X. The risk function is given by:

X is a vector of observations stochastically drawn from a population, dPθ is a probability measure over the event space of X.

Bayes Risk

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In a Bayesian approach, the expectation is calculated using the prior distribution ρ(π∗,a)=∫Θ∫𝐗L(θ,a(𝐱))dP(𝐱|θ)dπ∗(θ)=∫𝐗∫ΘL(θ,a(𝐱))dπ∗(θ|𝐱)dM(𝐱)

where π∗(θ|𝐱) is the posterior distribution, and the order of integration has been changed. One then should choose the action a also minimizes the overall Bayes Risk. This optimal decision, θ.

Examples in statistics

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  • For a scalar parameter θ, and a quadratic loss function (squared error loss) R(f,f̂)=E⁡(‖f−f̂‖2).

Economic choice under uncertainty

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In economics, decision-making under uncertainty is often modelled using the von Neumann–Morgenstern utility function of the uncertain variable of interest, such as end-of-period wealth. Since the value of this variable is uncertain, so is the value of the utility function; it is the expected value of utility that is maximized.

Decision rules

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A decision rule makes a choice using an optimality criterion. Some commonly used criteria are:

Further reading

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