How to get the conditional probabilities from joint probability table? I have a table of 3 binary variables whose joint probability is given.




a
b
c
p(a,b,c)




0
0
0
0.192


0
0
1
0.144


0
1
0
0.048


0
1
1
0.216


1
0
0
0.192


1
0
1
0.064


1
1
0
0.048


1
1
1
0.096




I see that there is a joint probabilities formula for 3 variable
$P(a,b,c) = P(c|a,b)P(a,b) = P(c|a,b)P(b|a)P(a)$
But when I cross-validate it with the table value and the formula it's not exact. What's happening here I am not getting it. It's not the same formula I assume. But how do I get the conditional probabilities from this table? If I can get the form, I can build the DAG.
I can see some patterns like 0.192 and 0.048 is twice even though $a$ is changed. Then I thought maybe $a$ is independent of $b$ and $c$?
I also find this formula $p(θ|X,α)=\frac{p(X|θ)p(θ|α)}{p(X|α)}$
So, I tried with a=1, b=1, c=1,
$
P(c|a, b) * P(b|a) * P(a)
= P(c|a, b) * \frac{P(b \cap a)}{P(a)} * P(a)
= \frac{P(c\cap a \cap b)}{P(a\cap b)} * \frac{P(b \cap a)}{P(a)} * P(a)
= \frac{1}{8}
$
 A: First, It appears that you are mixing up the definition of conditional probability for events and for random variables.
The definition $P(a,b,c)=P(c|a,b)P(a,b)=P(c|a,b)P(b|a)P(a)$ is for events.
For random variables, the definition is :
$$P(A = a,B = b,C =c)=P(C = c|A = a,B = b)P(B = b,A = a)$$
where $a,b,c$ is the values that the random variables $A$, $B$, $C$ can assume (in this case, $0$ or $1$).
Now, if you want  the conditional probability of $C$, given $A$ and $B$, you must obtain the probability $P(B = b,A = a)$.
Here is what you should do:
First, obtain $P(B = b, A = a)$ for all possible values of $a$ and $b$.
For example, to obtain $P(B = 0, A = 1)$, just sum all probabilities where $B = 0$ and $A = 1$. Do the same for the other possible values of $A$ and $B$.


*Now, you can calculate $P(C = c|A = a,B = b) = \frac{P(A = a,B = b,C =c)}{P(B = b,A = a)}$
you can follow the same reasoning for the other conditional probabilities.
A good exercise for learning is to obtain all possible combinations of conditional and unconditional probabilities for all variables.
