Loss Function

In statistics, decision theory and economics, a loss function is a function that maps an event (technically an element of a sample space) onto a real number representing the economic cost or regret associated with the event.Loss functions in economics are typically expressed in monetary terms. For example:Other measures of cost are possible, for example mortality or morbidity in the field of public health or safety engineering.
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The loss function (the term loss was first used by Wald, 1939) is the function that is minimized in the process of fitting a model, and it represents a selected measure of the discrepancy between the observed data and data "predicted" by the fitted function. For example, in many traditional general linear model techniques, the loss function (commonly known as least squares) is the sum of squared deviations from the fitted line or plane. One of the properties (sometimes considered to be a disadvantage) of that common loss function is that it is very sensitive to outliers .
A common alternative to the least squares loss function (see above) is to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function; the term maximum likelihood was first used by Fisher, 1922a). These functions are typically used when fitting non-linear models. In most general terms, the likelihood function is defined as:

In theory, we can compute the probability (now called L, the likelihood) of the specific dependent variable values to occur in our sample, given the respective regression model.

An common alternative to the least squares loss function is to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function; the term maximum likelihood was first used by Fisher, 1922a). These functions are typically used when fitting non-linear models. In most general terms, the likelihood function is defined as:

Loss function
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