Statistics Definitions > Fisher Information

## What is Fisher Information?

Fisher information tells us **how much information about an unknown parameter we can get from a sample**. In other words, it tells us how well we can measure a parameter, given a certain amount of data. More formally, it measures the expected amount of information given by a random variable (X) for a parameter(Θ) of interest. The concept is related to the law of entropy, as both are ways to measure disorder in a system (Friedan, 1998).

Uses include:

- Describing the asymptotic behavior of maximum likelihood estimates.
- Calculating the variance of an estimator.
- Finding priors in Bayesian inference.

## Finding the Fisher Information

Finding the expected amount of information requires calculus. Specifically, a good understanding of differential equations is required if you want to derive information for a system.

Three different ways can calculate the amount of information contained in a random variable X:

- This can be rewritten (if you change the order of integration and differentiation) as:

- Or, put another way:

The bottom equation is usually the most practical. However, **you may not have to use calculus**, because expected information has been calculated for a wide number of distributions already. For example:

- Ly et.al (and many others) state that the expected amount of information in a
**Bernoulli distribution**is:

I(Θ) = 1 / Θ (1 – Θ). - For
**mixture distributions**, trying to find information can “become quite difficult” (Wallis, 2005). If you have a mixture model, Wallis’s book*Statistical and Inductive Inference by Minimum Message Length*gives an excellent rundown on the problems you might expect.

If you’re trying to find expected information, try an Internet or scholarly database search first: the solution for many common distributions (and many uncommon ones) is probably out there.

## Example

Find the fisher information for X ~ N(μ, σ^{2}). The parameter, μ, is unknown.

**Solution**:

For −∞ < x < ∞:

First and second derivatives are:

So the Fisher Information is:

## Other Uses

Fisher information is used for slightly different purposes in Bayesian statistics and Minimum Description Length (MDL):

**Bayesian Statistics**: finds a default prior for a parameter.**Minimum description length (MDL)**: measures complexity for different models.

**References**:

Frieden and Gatenby.(2010). Exploratory Data Analysis Using FI. Springer Science and Business Media.

Friedan (1998). Physics from Fisher Information: A Unification. Cambridge University Press.

Lehman, E. L., & Casella, G. (1998). Theory of Point Estimation (2nd edition). New York, NY: Springer.

Ly, A. et. al. A Tutorial on Fisher I. Retrieved September 8, 2016 from: http://www.ejwagenmakers.com/submitted/LyEtAlTutorial.pdf.

Wallis, C. (2005). Statistical and Inductive Inference by Minimum Message Length. Springer Science and Business Media.

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