Scaled Inverse Chi-squared Distribution - Bayesian Estimation of The Variance of A Normal Distribution

Bayesian Estimation of The Variance of A Normal Distribution

The scaled inverse chi-squared distribution has a second important application, in the Bayesian estimation of the variance of a Normal distribution.

According to Bayes theorem, the posterior probability distribution for quantities of interest is proportional to the product of a prior distribution for the quantities and a likelihood function:

where D represents the data and I represents any initial information about σ2 that we may already have.

The simplest scenario arises if the mean μ is already known; or, alternatively, if it is the conditional distribution of σ2 that is sought, for a particular assumed value of μ.

Then the likelihood term L(σ2|D) = p(D|σ2) has the familiar form

Combining this with the rescaling-invariant prior p(σ2|I) = 1/σ2, which can be argued (e.g. following Jeffreys) to be the least informative possible prior for σ2 in this problem, gives a combined posterior probability

This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = n and τ2 = s2 = (1/n) Σ (x_i-μ)2

Gelman et al remark that the re-appearance of this distribution, previously seen in a sampling context, may seem remarkable; but given the choice of prior the "result is not surprising".

In particular, the choice of a rescaling-invariant prior for σ2 has the result that the probability for the ratio of σ2 / s2 has the same form (independent of the conditioning variable) when conditioned on s2 as when conditioned on σ2:

In the sampling-theory case, conditioned on σ2, the probability distribution for (1/s2) is a scaled inverse chi-squared distribution; and so the probability distribution for σ2 conditioned on s2, given a scale-agnostic prior, is also a scaled inverse chi-squared distribution.