How do they define $r(x;\xi)$, $p(x|y;\xi)$,$Pr(A|y;\xi)$

Question

I am reading Methods of Information Geometry by Shun-ichi Amari and Hiroshi Nagaoka.
In section 2.2 The Fisher metric, I do not understand the notions $r(x;\xi)$, $p(x|y;\xi)$, $\mathrm{Pr}(A|y;\xi)$, how are they defined?
If $r(x;\xi)= p(x;\xi)/q(F(x);\xi)$ then $p(x|y;\xi)=r(x;\xi)\delta_{F(x)}(y)$, how is it getting ?

Here is my try :

Answer 1

I will drop the $\xi$ to simplify the notation.

You are correct that $p(x|y) = \frac{p(x, y)}{q(y)}$, however from there you need to remember that $Y$ fully depends on $X$ since it is a function of $X$, so that if $X=x$, then $Y=F(x)$.

As such, the joint probability $p(x, y)$ is always zero unless $y=F(x)$, in which case the pair $(x, y)$ "matches". If they match, you can then write $p(x, y) = p(x, F(x)) = p(x)$. Combining those two observations yields $p(x, y)=p(x)\cdot\delta_{F(x)}(y)\quad\forall x, y$.

Using this in the conditional probability, you get $$p(x|y) = \frac{p(x, y)}{q(y)} = \frac{p(x)\cdot\delta_{F(x)}(y)}{q(y)} = \frac{p(x)\cdot\delta_{F(x)}(y)}{q(F(x))}.$$