Is the Jeffreys prior a physically correct prior?
To have a physically correct prior, you need it to have the right dimensions, i.e. the only powers of σ physically possible in a non-parametric prior are: π(μ, σ) ∼ 1 / σ2 and π(μ, σ2) ∼ 1 / σ3 . 1 σ3 is the Jeffreys prior.
How to calculate Jeffreys prior for a normal distribution?
According to my calculations, the following holds for Jeffreys prior: p(μ, σ2) = √det(I) = √det(1 / σ2 0 0 1 / (2σ4)) = √ 1 2σ6 ∝ 1 σ3. Here, I is Fisher’s information matrix. p(μ, σ2) ∝ 1 / σ2 see Section 2.2 in Kass and Wassermann (1996). as Jeffreys prior for the case of a normal distribution with unkown mean and variance.
When to use Jeffreys prior for binomial probability?
If I use a Jeffreys prior for a binomial probability parameter θ then this implies using a θ ∼ beta(1 / 2, 1 / 2) distribution. If I transform to a new frame of reference ϕ = θ2 then clearly ϕ is not also distributed as a beta(1 / 2, 1 / 2) distribution.
What is the density function of the Jeffreys prior?
In Bayesian probability, the Jeffreys prior, named after Sir Harold Jeffreys, is a non-informative (objective) prior distribution for a parameter space; its density function is proportional to the square root of the determinant of the Fisher information matrix:
Why are Jeffreys priors considered in a Bayesian model?
The Jeffreys prior coincides with the Bernardo reference prior for one-dimensional parameter space (and “regular” models). Roughly speaking, this is the prior for which the Kullback-Leibler divergence between the prior and the posterior is maximal. This quantity represents the amount of information brought by the data.
What happens to a Jeffreys prior under a transformation?
What happens with Jeffreys’ prior under a transformation is that the Jacobian from the transformation gets sucked into the original Fisher information, which ends up giving you the Fisher information under the new parameterization. No magic (in the mechanics at least), just a little calculus and linear algebra.