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TL;DR

A measure of the information a random variable carries about a parameter: \( I(\theta) = E\left[\left(\frac{\partial \log f(X|\theta)}{\partial \theta}\right)^2\right] \). It determines the precision of parameter estimation.

By Valenke Exam Prep Team·Last updated 2026-06-03

Fisher Information

A measure of the information a random variable carries about a parameter: \( I(\theta) = E\left[\left(\frac{\partial \log f(X|\theta)}{\partial \theta}\right)^2\right] \). It determines the precision of parameter estimation.

Why it matters for interviews

Fisher information sets the Cramer-Rao lower bound on estimation variance. In item response theory (used in adaptive testing) and in options pricing (via Ito calculus), Fisher information quantifies the value of observations.

Definition and Mathematical Foundation

A measure of the information a random variable carries about a parameter: \( I(\theta) = E\left[\left(\frac{\partial \log f(X|\theta)}{\partial \theta}\right)^2\right] \). It determines the precision of parameter estimation.

Application in Quantitative Finance

Fisher information sets the Cramer-Rao lower bound on estimation variance. In item response theory (used in adaptive testing) and in options pricing (via Ito calculus), Fisher information quantifies the value of observations.

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Frequently Asked Questions

What is the Cramer-Rao lower bound?▼
For any unbiased estimator: \( \text{Var}(\hat{\theta}) \geq 1/I(\theta) \). No unbiased estimator can have variance below this bound. The MLE achieves it asymptotically.
How does Fisher information relate to the Hessian?▼
Under regularity conditions, \( I(\theta) = -E[\frac{\partial^2 \log f}{\partial \theta^2}] \). The Fisher information is the negative expected curvature of the log-likelihood. Sharp peaks (high curvature) mean more information.
What is the Fisher information matrix?▼
For vector parameters, the matrix with entries \( I_{ij} = E[\frac{\partial \log f}{\partial \theta_i} \frac{\partial \log f}{\partial \theta_j}] \). Its inverse gives the asymptotic covariance matrix of the MLE.