---
tag: 18MMYZ7922
kind: concept
name: Jeffreys prior
msc: [62F15]
specializes: [00NBBSXRQS]
sources:
  - url: https://en.wikipedia.org/wiki/Jeffreys_prior
same_as:
  wikidata: Q4068240
  wikipedia: Jeffreys prior
---

## Summary

The Jeffreys prior is a [[prior distribution|00NBBSXRQS]] whose density is proportional to the square root of the determinant of the [[Fisher information|G1Z3WSTE8Q]].
Its defining property is invariance: it gives the same prior no matter how the parameter is written.
It is a standard choice of a non-informative prior in [[Bayesian inference|0JWRJX2RMP]], and it is often improper.

## Explanations

### L1 human

A flat prior on a parameter is not flat on a transformed parameter, such as its square or its logarithm.
So "flat" depends on how the parameter is written down.
The Jeffreys prior avoids this by measuring distances with the Fisher information, which measures how well data can tell nearby parameter values apart.
For a coin with chance $p$ of heads, it gives the density proportional to $1/\sqrt{p(1 - p)}$, which puts a little more weight near $0$ and $1$.