Loss function

Dina statistik, téori kaputusan sarta ékonomi, loss function nyaéta fungsi nu metakeun hiji kajadian (sacara teknis hiji unsur dina rohangan sampel) kana wilangan riil keur ngagambarkeun biaya ekonomi atawa hanjelu pakait jeung hiji kajadian.

Loss functions sacara tipikal digambarkeun dina watesan "keuangan" dumasar kana ukuran biaya nu mungkin, contona kalahiran jeung "kematian" dina widang public health.

Loss functions are complementary to utility functions which represent benefit and satisfaction. Typically, for utility U, loss is equal to k-U, where k is some arbitrary constant.

Expected loss

Loss function satisfies the definition of a random variable so we can establish a cumulative distribution function and an nilai ekspektasi. However, more commonly, the loss function is expressed as a function of some other random variable. For example, the time that a light bulb operates before failure is a random variable and we can specify the loss, arising from having to cope in the dark and/or replace the bulb, as a function of failure time. For a continuous random variable, X with probability density function f(x) and loss function, λ(x), the expected loss is:

Minimum expected loss is widely used as a criterion for choosing between prospects. It is closely related to the criterion of maximum expected utility.

Loss functions in Bayesian statistics

One of the consequences of Bayesian inference is that in addition to experimental data, the loss function  does not in itself wholly determine a decision.  What is important is the relationship between the loss function and the prior probability.  So it is possible to have two different loss functions which léad to the same decision when the prior probability distributions associated with éach compensate for the details of éach loss function.

Combining the three elements of the prior probability, the data, and the loss function then allows decisions to be based on maximising the subjective expected utility, a concept introduced by Leonard J. Savage.  He also argued that using non-Bayesian methods such as minimax, the loss function should be based on the idéa of regret, i.e. the loss associated with a decision should be the difference between the consequences of the best decision that could have been taken had the underlying circumstances been known and the decision that was in fact taken before they were known.

Statistika