# Loss definition

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. For most optimization algorithms, it is desirable to have a loss function that is globally continuous and differentiable. 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. A loss is an excess of expenses over revenues, either for a single business transaction or in reference to the sum of all transactions for an accounting period. The presence of a loss for an accounting period is closely watched by investors and creditors, since it can signal a decline in the creditworthiness of a business.

1. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data.
2. For most optimization algorithms, it is desirable to have a loss function that is globally continuous and differentiable.
3. In financial risk management, the function is mapped to a monetary loss.
4. 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.

In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century.[2] In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example.

## Translations of loss

This is particularly the case when the loss is derived from just the operational activities of a business. These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘loss.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. It has received negative reception from critics and webcomic creators, especially for the shift in tone in the webcomic, and as an example of “fridging”—showing a killed or injured female character with the intention of provoking a male character. It has been adapted and parodied by numerous other creators and garnered a legacy as an internet meme.

The quadratic loss function is also used in linear-quadratic optimal control problems. 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 the context of stochastic control, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber, Log-Cosh and SMAE losses are used when the data has many large outliers.

## Constructing loss and objective functions

Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function. 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. For risk-averse or risk-loving agents, loss is measured as the negative of a utility function, and the objective function to be optimized is the expected value of utility. “Loss”, sometimes referred to as “Loss.jpg”,[1] is a strip published on June 2, 2008, by Tim Buckley for his gaming-related webcomic Ctrl+Alt+Del. Set during a storyline in which the main character Ethan and his fiancée Lilah are expecting their first child, the strip—presented as a four-panel comic with no dialogue—shows Ethan entering a hospital, where he sees Lilah weeping in a hospital bed after suffering a miscarriage.

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 financial risk management, the function is mapped to a monetary loss. Still different estimators would be optimal under other, less common circumstances.