In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information.

The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized and explored by the statistician Sir Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test.

In Bayesian statistics, the Fisher information plays a role in the derivation of non-informative prior distributions according to Jeffreys' rule. It also appears as the large-sample covariance of the posterior distribution, provided that the prior is sufficiently smooth (a result known as Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The same result is used when approximating the posterior with Laplace's approximation, where the Fisher information appears as the covariance of the fitted Gaussian.

Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have been shown to obey maximum Fisher information. The level of the maximum depends upon the nature of the system constraints.

The Fisher information is a way of measuring the amount of information that an observable random variable ${\displaystyle X}$ carries about an unknown parameter ${\displaystyle \theta }$ upon which the probability of ${\displaystyle X}$ depends. Let ${\displaystyle f(X;\theta )}$ be the probability density function (or probability mass function) for ${\displaystyle X}$ conditioned on the value of ${\displaystyle \theta }$. It describes the probability that we observe a given outcome of ${\displaystyle X}$, given a known value of ${\displaystyle \theta }$. If ${\displaystyle f}$ is sharply peaked with respect to changes in ${\displaystyle \theta }$, it is easy to indicate the "correct" value of ${\displaystyle \theta }$ from the data, or equivalently, that the data ${\displaystyle X}$ provides a lot of information about the parameter ${\displaystyle \theta }$. If ${\displaystyle f}$ is flat and spread-out, then it would take many samples of ${\displaystyle X}$ to estimate the actual "true" value of ${\displaystyle \theta }$ that would be obtained using the entire population being sampled. This suggests studying some kind of variance with respect to ${\displaystyle \theta }$.

Formally, the partial derivative with respect to ${\displaystyle \theta }$ of the natural logarithm of the likelihood function is called the score. Under certain regularity conditions, if ${\displaystyle \theta }$ is the true parameter (i.e. ${\displaystyle X}$ is actually distributed as ${\displaystyle f(X;\theta )}$), it can be shown that the expected value (the first moment) of the score, evaluated at the true parameter value ${\displaystyle \theta }$, is 0:

The Fisher information is defined to be the variance of the score:

Note that ${\displaystyle {\mathcal {I}}(\theta )\geq 0}$. A random variable carrying high Fisher information implies that the absolute value of the score is often high. The Fisher information is not a function of a particular observation, as the random variable X has been averaged out.