# Reference Prior

The notion of reference prior is similar to that of a non-informative prior, but there are subtle differences. The uniform prior is arguably non-informative, but is it not a good reference, because it is not always invariant under reparamaterization.

Consider a reparameterization $$\gamma = \log \theta$$, converting the support of the parameter to the real line. The prior on $$\gamma$$ is given by:

$$p_\gamma (\gamma) = p(\theta)|J| = |J|$$

where $$|J| = \frac{d \theta}{d \gamma}$$ from the Change of Variables Theorem. Then

$$p_\gamma (\gamma) = e^\gamma, - \infty < \gamma < + \infty$$

which is clearly not uniform. A prior that is invariant to transformation, and easy to compute, is the Jeffreys Prior.

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