So, I'm currently working through "Data Analysis: A Bayesian Tutorial 2nd Ed." by D.S. Sivia and I am stuck an equation, which I am not able to reproduce (p.40 - 42 for those who might have a copy).
A little background: we want to estimate amplitudes $A$, $B$ of a gaussian peak and its background of data points $N_{k}$. Additionally, the width $w$ and position $x_{0}$ are also variable parameters, resulting in the following posterior expression:
$$ prob(A, B, w, x_{0} | \{N_{k}\}, I) $$
The author then goes on to decompose the above expression with the help of Bayes Theorem:
$$ prob(A, B, w, x_{0} | \{N_{k}\}, I) \propto prob(\{N_{k}\} | A, B, w, x_{0}, I) \times prob(A, B, w, x_{0} | I). $$
Now is the point where I lose the author who claims that the 2nd part of this equation can be further decomposed as:
$$ prob(A, B, w, x_{0} | I) = prob(A, B | I) \times prob(w, x_{0} | I) $$
With the rules which were developped this far into the book, I actually cannot reproduce this result. I tried it with (what the author calls) the "product rule" stating
$$ prob(X, Y | I) = prob(X | Y, I) \times prob(Y, I). $$
However, this would result in $$ prob(A, B, w, x_{0} | I) = prob(A, B | w, x_{0}, I) \times prob(w, x_{0} | I) $$ and it would assume that setting $X = A, B$ and $Y = w, x_{0}$ is a valid thing to do.
If someone sees the point I am missing, it would be really great of you to help me out.