# Parameter Fitting Loss Function

(Redirected from Loss Function)

## References

### 2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/loss_function Retrieved:2024-3-8.
• In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) 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. 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 Cramér in the 1920s. 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.

• NOTES:
1. **Fundamental Concept**: Loss functions, also known as cost or error functions, map the outcomes of models or decisions to real numbers, representing the "cost" associated with the outcomes. They are pivotal in mathematical optimization and decision theory, guiding towards minimizing the discrepancy between predicted and actual results.
2. **Optimization Objective**: The primary goal in an optimization problem is to minimize the loss function, which quantifies the errors or deviations from the desired outcomes. Conversely, in certain domains, the objective function may seek to maximize a reward, profit, utility, or fitness function.
3. **Application Across Fields**: Loss functions are utilized in various domains such as statistics for parameter estimation, economics for calculating economic cost or regret, actuarial science, optimal control, and financial risk management to quantify monetary losses.
4. **Statistical Significance**: In statistics, loss functions are crucial for estimating parameters, where the event often involves the difference between estimated and true values. The choice of loss function can significantly influence the statistical models and their predictive performance.
5. **Diverse Types and Examples**: Common loss functions include quadratic loss, 0-1 loss, and two-parameter loss functions. Each serves different purposes and applications, such as least squares methods for regression using quadratic loss or penalizing incorrect classifications in decision theory with 0-1 loss.
6. **Constructing Objective Functions**: The construction of loss and objective functions can be based on problem formulation or the elicitation of decision-makers' preferences. Methods for constructing these functions have evolved, allowing for more tailored approaches to specific decision-making contexts.
7. **Expected Loss**: The concept of expected loss plays a significant role in both frequentist and Bayesian statistical theories. It involves making decisions based on the expected value of the loss function, which varies depending on the statistical paradigm.
8. **Decision-Making Criteria**: Loss functions are integral to establishing decision rules based on optimality criteria such as minimax (minimizing the worst-case loss) and criteria that focus on minimizing the expected average loss. These rules guide the selection of optimal actions under uncertainty, reflecting various strategic considerations and preferences.
9. **Selection of Loss Functions**: The selection of a loss function is critical and influenced by the specific context of the application. It must align with the nature of the problem, the distribution of the data, and the desired outcomes of the model or decision-making process. Different loss functions are optimized under distinct circumstances, such as the mean for squared-error loss or the median for absolute-difference loss, emphasizing the importance of matching the loss function to the problem's specifics and objectives.