Fischer information is a concept of [[information theory]]. In the context of [[statistics]], the Fisher information tells us how much information a sample of data contains about an unknown parameter $\theta$. The Fisher information is the [[variance]] of the [[score function]] (note the [[expectation]] of the score function is 0 under regularity conditions so the term drops out). $I_n(\theta) := E \Big [ \Big (\frac{\delta}{\delta \theta} \ln f(\vec{x};\theta) \Big )^2 \Big]$ where the subscript $n$ indicates the sample size to reinforce the fact that the Fisher information is calculated for a sample of data. The interpretation of the Fisher information is that it measures the curvature of the graph of the log likelihood. Near the [[maximum likelihood estimator|MLE]], high Fisher information indicates the graph is "sharp", whereas low Fisher information indicates the graph is "smooth". Fisher information can also be written as the negated second derivative of the log likelihood function. $I_n(\theta) := -E \Big [ \Big (\frac{\delta^2}{\delta^2 \theta} \ln f(\vec{x};\theta) \Big ) \Big]$ With [[independent and identically distributed|iid]] data, Fisher information can be simplified to $n$ times the expectation of the log likelihood of the marginal pdf of $x_i$. $I_n(\theta) := -n E \Big [ \Big (\frac{\delta^2}{\delta^2 \theta} \ln f(x_i;\theta) \Big ) \Big]$ Fisher information is used to calculate the [[Cramér-Rao Lower Bound]] (CRLB) and [[Jeffreys prior]].