## Sufficiency [BC]

**H**ere 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 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.The Bayesian ChoiceThere 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-1degree 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-1degree of freedom chi-square of .

**T**he 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 !*

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