How to find P(A and B and C) given all the marginal probabilities and two conditional probabilities w/r/t to same random variable A?

Question

Given marginal probabilities $P(A)$, $P(B)$, and $P(C)$ and conditional probabilities $P(A|B)$, $P(B|A)$, $P(A|C)$, and $P(C|A)$ is it possible to find $P(A\cap B \cap C)$ ?

Attempt: $P(A \cap B \cap C) = P(A \cap B)\cap P(A \cap C)= P(B|A)\cdot P(A)\cap P(C|A)\cdot P(A)= P(A) [P(B|A) \cap P(C|A)] = P(A) \cdot [P(B|A | C|A)] \cdot P(C|A) = P(C) \cdot [P(B|A | C|A)] $

Can I distribute the conditioning variable? Are there useful properties I should know about? Any help is appreciated. My intuition tells me it should be possible to calculate $P(A\cap B \cap C)$. Any help appreciated.

Answer 1

This is actually asking if we can find the joint with marginals and $P(A|C),P(A|B)$, i.e. knowing $P(C|A)$ and $P(B|A)$ doesn't provide additional information since they can be found via Bayes rule. First of all, we know nothing about the dependence relation between $B$ and $C$, and also we don't know what happens when we condition on two of the events, which should give you a heads up already. We can't do it.

Let's find an example. Assume event $A$ has nothing to do with the other events both jointly and mutually, i.e. $P(A|B)=P(A),P(A|C)=P(A),P(A|B,C)=P(A)$. (e.g. $A$ can represent the event of raining in Amazons today, where $B,C$ represents the probability of heads of the same (maybe biased) coin in different tosses). Then, $P(A,B,C)=P(A)P(B,C)$, and we can't calculate $P(B,C)$ with only $P(B)$ and $P(C)$.

Also, your solution attempt is not meaningful because of @MauritusM 's comment.

Answer 2

Let's say for simplicity that each variable $A$, $B$ and $C$ has only two states such that they jointly have $8$ states. As described in the answers to What is the number of parameters needed for a joint probability distribution? you will need $8-1=7$ independent parameters to describe this distribution.

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You only have $5$ parameters $P(A)$, $P(B)$, $P(C)$, $P(B \vert A)$ and $P(C \vert A)$. Note that $$P(C \vert A) = P(A \vert C) P(C)/P(A)$$ and $$P(B \vert A) = P(A \vert B) P(B)/P(A)$$ are not independent from the other $5$ parameters.

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Counter-example

Let the probabilities $P(A=a \cap B=b \cap C=c)$ be expressed by the numbers $n_{abc} $ with subscripts $a$, $b$, $c$ taking the values $0$ or $1$ depending on whether $A$, $B$, $C$ are true or not. Then we have $8$ numbers, but only $6$ equations to define them.

$$\begin{array}{rcl} n_{000}+n_{001}+n_{010}+n_{011}+n_{100}+n_{101}+n_{110}+n_{111} &=&1 \\ n_{100}+n_{101}+n_{110}+n_{111} &=& P(A=1)\\ n_{010}+n_{011}+n_{110}+n_{111} &=& P(B=1)\\ n_{001}+n_{011}+n_{101}+n_{111} &=& P(C=1)\\ (n_{110}+n_{111})/(n_{100}+n_{101}+n_{110}+n_{111}) &=& P(B=1 \vert A=1) \\ (n_{101}+n_{111})/(n_{100}+n_{101}+n_{110}+n_{111}) &=& P(C=1 \vert A=1) \\ \end{array}$$

So this will have multiple solutions that can be expressed by two additional numbers. Let $p_a = 1-q_a = P(A=1)$ (and similar for $B$ and $C$)

$$\begin{array}{rcl} n_{000}=q_a q_b q_c + r + s + t + u \\ n_{001}=q_a q_b p_c + r - s - t - u \\ n_{010}=q_a p_b q_c - r + s - t - u \\ n_{011}=q_a p_b p_c - r - s + t + u \\ n_{100}=p_a q_b q_c - r - s + t - u \\ n_{101}=p_a q_b p_c - r + s - t + u \\ n_{110}=p_a p_b q_c + r - s - t + u \\ n_{111}=p_a p_b p_c + r + s + t - u \\ \end{array}$$

In these equations you can see the expression of the numbers $n_{abc}$ as independent (the product of terms $q$ and $p$) plus extra terms that make the numbers deviate from independence. Those terms are $r$, $s$, $t$ and $u$ which will leave probabilities like $P(A=1)$ invariant (you can see it has the higher dimensional version of this case: https://stats.stackexchange.com/a/363777/164061 ). When we put these $8$ solutions into the before mentioned 6 equations you get:

$$\begin{array}{rcl} 1 &=&1 \\ p_a &=& P(A=1)\\ p_b &=& P(B=1)\\ p_c &=& P(C=1)\\ (p_b p_a + 2 r)/(p_a) &=& P(B=1 \vert A=1) \\ (p_c p_a + 2 s )/(p_a) &=& P(C=1 \vert A=1) \\ \end{array}$$

So you can find counter examples by using different values for the parameters $t$ and $u$ which will render different solutions that still satisfy your conditions $P(A)$, $P(B)$, $P(C)$, $P(B \vert A)$ and $P(C \vert A)$.

For example the two cases below:

$$n_{000}=n_{001}=n_{010}=n_{011}=n_{100}=n_{101}=n_{110}=n_{111} =0.125$$ or $$\begin{array}{rcl} n_{000}=n_{011}=n_{101}=n_{110}&=&0.1 \\ n_{001}=n_{010}=n_{100}=n_{111} &=&0.15 \end{array} $$

are an example of two different distributions with the same $P(A) = P(B) = P(C) = P(B \vert A) = P(C \vert A) = 0.5$ and in this particular case also $P(C \vert B) = 0.5$.

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Graphical

The image below sketches a cube which is how I intuitively imagine these variables $r$, $s $, $t $, as a crossed pattern that changes the distribution while leaving the marginal distributions (including some conditional) invariant. And the variable $u $ will leave all marginal distributions invariant (when you take the sum of $n $ allong any rib then you get both a $+u $ and $-u $ term).