Loss function
In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s.
[3] In optimal control, the loss is the penalty for failing to achieve a desired value.
In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables.
Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions.
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.
[4] The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences.
[5][6] 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.
[7][8] Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities[9] and the European subsidies for equalizing unemployment rates among 271 German regions.
Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms.
It is obtained by taking the expected value with respect to the probability distribution, Pθ, of the observed data, X.
The risk function is given by: Here, 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.
Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ.
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.
Some commonly used criteria are: Sound statistical practice requires selecting an estimator consistent with the actual acceptable variation experienced in the context of a particular applied problem.
In economics, when an agent is risk neutral, the objective function is simply expressed as the expected value of a monetary quantity, such as profit, income, or end-of-period wealth.
Other measures of cost are possible, for example mortality or morbidity in the field of public health or safety engineering.
W. Edwards Deming and Nassim Nicholas Taleb argue that empirical reality, not nice mathematical properties, should be the sole basis for selecting loss functions, and real losses often are not mathematically nice and are not differentiable, continuous, symmetric, etc.
may tolerate increased load or stress with little noticeable change up to a point, then become backed up or break catastrophically.
