Which book has the right conditional independence formula? I'm getting crazy.
I've just started to learn probability and, after it, Bayesian networks. I don't know so much about probability, that is why I'm getting crazy.
I'm using this book to study a subject of my master's degree. On pages 32 and 33 (PDF's pages 44 and 45), formula (1.50), it talks about conditional independence. And it says that:


*

*$I_G(A,B|C) => P(a,b|c) = P(a|c)·P(b|c)$
And in the next page, in the formula (1.51) it says that:
$I_G(A,B|C) => I_P(A,B|C)$
But in Learning Bayesian Networks by Neapolitan book, page 19 (PDF's page 30), it says that:


*$P(a|b,c)$ can be written as $I_p(A,B|C)$ when $P(a|b,c) = P(a|c)$
Clarification:
$I_p$ is the conditional independence given the probability distribution $p$ (joint probability function).
Which book is wrong? (or maybe both are correct because $P(a,b|c)$ and $P(a|b,c)$ are the same "thing" or I haven't understood anything).
 A: Ignoring notation issues, both definitions of C.I. are equivalent, although the first definition in your question is the standard way of defining it.
To see that they are equivalent observe that if $\mathbb{P}(a,b|c) = \mathbb{P}(a|c)\mathbb{P}(b|c)$, then
$$\begin{align*}
\mathbb{P}(a|bc) &= \frac{\mathbb{P}(abc)}{\mathbb{P(bc)}}\\
& = \frac{\mathbb{P}(ab|c)\mathbb{P}(c)}{\mathbb{P(bc)}}\\ 
& = \frac{\mathbb{P}(a|c)\mathbb{P}(b|c)\mathbb{P}(c)}{\mathbb{P(bc)}} & (\text{by hypothesis})\\
& = \frac{\mathbb{P}(a|c)\mathbb{P}(bc)}{\mathbb{P(bc)}}\\
& = \mathbb{P}(a|c)
\end{align*}$$
Likewise, you can multiply both sides of $\mathbb{P}(a|bc) = \mathbb{P}(a|c)$ by $\mathbb{P}(b|c)$ to show that the second definition implies the first.
Note that this is completely analogous to the non-conditional case where:
$$\mathbb{P}(ab)=\mathbb{P}(a)\mathbb{P}(b)\Leftrightarrow \mathbb{P}(a|b)=\mathbb{P}(a)$$
What the second definition of independence or, in your case, conditional independence, is saying is that, given $c$, the probability of $a$ also given $b$ is the same as the probability of $a$ if you didn't know $b$ (only $c$), i.e.,
Given $c$, knowing $b$ tells you nothing about the probability of $a$
