Calculate $P[A,B,C]$ from $P[A,B]$ and $P[B,C]$

Question

I have 3 (not independent) events $A, B, C$ and I know everything about how any two of them correlate. For example, I know:

$$ P[A], P[B], P[C], P[A,B], P[A,C], P[B,C], P[A|B], P[A|C], P[B|C], P[B|A], ...$$

Is there any way to use this information to calculate a correlation for the three of them, i.e.

$$ P[A,B,C] \text{ or } P[A,B|C] \text{ or } P[A|B,C] $$

A numerical algorithm would also be fine if there isn't an exact formula. If it is not possible, is there a way to get a confidence interval for these values?

Answer 1

I think you can constrain $P[A,B,C]$ to an interval such as

$$\max(0,\, P[A,B]+P[A,C]-P[A], \,P[A,B]+P[B,C]-P[B], \,P[A,C]+P[B,C]-P[C]) \\ \le P[A,B,C] \le \\ \min(P[A,B],\,P[A,C],\,P[B,C],\,1+P[A,B]+P[A,C]+P[B,C]-P[A]-P[B]-P[C]).$$

As an example, if $P[A]=P[B]=P[C]=\frac12$ and $P[A,B]=P[A,C]=P[B,C]=\frac14$ then $0 \le P[A,B,C] \le \frac14$.