Sufficiency [BC]

Here is an email I received about The Bayesian Choice a few days ago:

I am an undergraduate student in Japan. I am self-studying your classical book The Bayesian Choice. The book is wonderful with many instructive examples. Although it is a little bit hard for me right now, I think it will be very useful for my future research.

There is one point that I do not understand in Example 1.3.2 (p.14-15). I know a standard result that the sample mean and sample variance are independent, with the sample mean follows

while follows a chi-square of n-1 degree of freedom. In this example is it correct that one must factorize the likelihood function to which must be the product of these two normal and chi-square densities, and which is free of ?

In the book I do not see why $g(T(x) | \theta)$ is the product of normal and chi-square densities. The first part correctly corresponds to the density of . But the second part is not the density of n-1 degree of freedom chi-square of .

The example, as often, skips a lot of details, meaning that when one starts from the likelihood

this expression only depends on T(x). Furthermore, it involves the normal density on and part of the chi-square density on s². One can then plug in the missing power of s² to make appear. The extra terms are then canceled by a function we can call … However, there is a typo in this example in that in the chi-square density should be !