---
tag: RE02MP1H45
kind: definition
name: Loss function
msc: [62C05]
defines: 3BTG6VVCGZ
uses: [MSC0MR8VMH]
sources:
  - url: https://en.wikipedia.org/wiki/Loss_function
---

## Definition

Let $\Theta$ be a set of states of nature, for example the parameter set of a [[parametric model|MSC0MR8VMH]].
Let $\mathcal{A}$ be a set of actions.
A *loss function* is a function $L \colon \Theta \times \mathcal{A} \to [0, \infty)$.
The value $L(\theta, a)$ is the loss incurred by taking the action $a$ when the true state is $\theta$.

## Remarks

For estimating a real quantity $g(\theta)$, the action set is $\mathcal{A} = \R$ and standard choices are the following.
The *squared error loss* is $L(\theta, a) = (a - g(\theta))^2$.
The *absolute error loss* is $L(\theta, a) = \abs{a - g(\theta)}$.
In a test of $\theta \in \Theta_0$ against $\theta \notin \Theta_0$, with $\mathcal{A} = \set{0, 1}$, the *$0$-$1$ loss* is $1$ for a wrong decision and $0$ for a right one.
Some authors allow real-valued losses that are only bounded below; this changes little in the theory.

Example: $\Theta = \mathcal{A} = \R$ and $L(\theta, a) = (a - \theta)^2$.
It is a function of the pair $(\theta, a)$ with values in $[0, \infty)$, and it is $0$ exactly when $a = \theta$.

Non-example: $L(\theta, a) = a - \theta$ on $\R \times \R$.
It takes negative values, so it is not a loss function in the sense above.
It would also reward underestimating $\theta$ without limit.