A.25 Distribution of the correlation coefficient

If the prior density of the correlation coefficient is ρ, then its posterior density, given n pairs of observations (Xi, Yi) with sample correlation coefficient r, is given by

on writing . It can also be shown that

When r=0 the density simplifies to

and so if the prior is of the form

it can be shown that

has a Student’s t distribution on degrees of freedom.

Going back to the general case, it can be shown that

Expanding the term in square brackets as a power series in u, we can express the last integral as a sum of beta functions. Taking only the first term, we have as an approximation

On writing

it can be shown that

and hence that for large n

approximately [whatever is]. A better approximation is