Definition of Bayesian network

Question

In this PDF, page 5 says:

Given a set of functions $f(x_i,pa(x_i))$ non-negative and sum to 1, we define a joint probability as follow: $$p(x_1,\dots, x_n)=\prod_i f(x_i,pa(x_i))$$ [...] what is the relationship between the conditional probability $p(x_i|pa(x_i))$ and $f(x_i,pa(x_i))$, a function with has the properties of a conditional function but it is arbitrary? As will see, this function is in the fact one and same. That is, under the definition of joint probability, the function $f(x_i,pa(x_i))$ is $p(x_i|pa(x_i))$.

How can it go from $f(x_i,pa(x_i))$ to $p(x_i|pa(x_i))$ and state that it is the only possible function that can be assigned to $f(x_i,pa(x_i))$?

Answer 1

The author of the book explains in Section 2.1.3 why, given the particular constraints on the function $f(x_i, pa(x_i))$, one can view the function $f(x_i, pa(x_i))$ as equivalent to the conditional probability $P(x_i | pa(x_i))$. This equivalence only holds because of the particular property of the function and the structure of the graphical model.