In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) [1] 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 hierarchy.

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. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramr 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.


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Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea 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 made had the underlying circumstances been known and the decision that was in fact taken before they were known.

The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t, then a quadratic loss function is

for some constant C; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss (SEL).[1]

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.

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.

where L {\displaystyle L} is the two-parameter (or ranged) loss function, a {\displaystyle a} is the high value parameter, b {\displaystyle b} is the low value parameter, and x {\displaystyle x} is the unknown value.

We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution, Ptag_hash_117, of the observed data, X. This is also referred to as the risk function[11][12][13][14] of the decision rule tag_hash_119 and the parameter tag_hash_120. Here the decision rule depends on the outcome of X. The risk function is given by:

One then should choose the action a* which minimises the expected loss. Although this will result in choosing the same action as would be chosen using the frequentist risk, the emphasis of the Bayesian approach is that one is only interested in choosing the optimal action under the actual observed data, whereas choosing the actual frequentist optimal decision rule, which is a function of all possible observations, is a much more difficult problem.

Sound statistical practice requires selecting an estimator consistent with the actual acceptable variation experienced in the context of a particular applied problem. Thus, in the applied use of loss functions, selecting which statistical method to use to model an applied problem depends on knowing the losses that will be experienced from being wrong under the problem's particular circumstances.[15]

A common example involves estimating "location". Under typical statistical assumptions, the mean or average is the statistic for estimating location that minimizes the expected loss experienced under the squared-error loss function, while the median is the estimator that minimizes expected loss experienced under the absolute-difference loss function. Still different estimators would be optimal under other, less common circumstances.

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.

The choice of a loss function is not arbitrary. It is very restrictive and sometimes the loss function may be characterized by its desirable properties.[16] Among the choice principles are, for example, the requirement of completeness of the class of symmetric statistics in the case of i.i.d. observations, the principle of complete information, and some others.

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. For example, a person who arrives before a plane gate closure can still make the plane, but a person who arrives after can not, a discontinuity and asymmetry which makes arriving slightly late much more costly than arriving slightly early. In drug dosing, the cost of too little drug may be lack of efficacy, while the cost of too much may be tolerable toxicity, another example of asymmetry. Traffic, pipes, beams, ecologies, climates, etc. may tolerate increased load or stress with little noticeable change up to a point, then become backed up or break catastrophically. These situations, Deming and Taleb argue, are common in real-life problems, perhaps more common than classical smooth, continuous, symmetric, differentials cases.[17]

The U.S. Food and Drug Administration (FDA) finalized its rule permitting the sale of over-the-counter (OTC) hearing aids directly to consumers over 18 with mild to moderate hearing loss in August 2022. Available since last October, these assistive devices may provide another pathway to care for some adults who could benefit from support with their hearing.

Dr. Kelly King is an audiologist and program officer at the National Institute on Deafness and Other Communication Disorders (NIDCD) of the National Institutes of Health (NIH). She oversees grants aimed at understanding hearing and hearing loss, as well as various treatments for communication disorders including hearing aids and cochlear implants. Dr. King also leads institute initiatives aimed at making hearing healthcare more accessible and affordable.

Some traditional financing instruments could be used to deal with loss and damage. Social protection, contingency finance, catastrophe risk insurance and catastrophe bonds can provide a certain buffer and rapid pay-outs after disasters. However, a broadened donor base and innovative finance tools would be needed to respond to the magnitude of loss and damage.

The UN Secretary-General Antnio Guterres, for example, has called for the use of windfall taxes on fossil fuel companies and diverting the money to people struggling with rising food and energy prices and to countries suffering loss and damage caused by the climate crisis. Others have called for debt for loss and damage swaps, international taxes and a dedicated finance facility for loss and damage under the UN Framework Convention on Climate Change. At COP 26 and 27 philanthropies and country governments pledged funds for loss and damage. These efforts could be scaled up considering the mounting challenges of the most vulnerable communities.

Sound is measured in units called decibels. Sounds at or below 70 A-weighted decibels (dBA), even after long exposure, are unlikely to cause hearing loss. However, long or repeated exposure to sounds at or above 85 dBA can cause hearing loss. The louder the sound, the shorter the amount of time it takes for NIHL to happen.

When you are exposed to loud noise over a long period of time, you may slowly start to lose your hearing. Because the damage from noise exposure is usually gradual, you might not notice it, or you might ignore the signs of hearing loss until they become more pronounced. Over time, sounds may become distorted or muffled, and you might find it difficult to understand other people when they talk or have to turn up the volume on the television. The damage from NIHL, combined with aging, can lead to hearing loss severe enough that you need hearing aids to magnify the sounds around you to help you hear, communicate, and participate more fully in daily activities.

Sometimes exposure to impulse or continuous loud noise causes a temporary hearing loss that disappears 16 to 48 hours later. Recent research suggests, however, that although the loss of hearing seems to disappear, there may be residual long-term damage to your hearing. 006ab0faaa

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