I've been working on this problem a few years ago. It is far from trivial.

Basically, you don't have enough information to answer. Actually, assume you have three events $A$, $B$, $C$. What you known is:

- the probability of each individual event : $P(A), P(B), P(C)$ (1)
- the probability of each pariwise conjunctions: $P(A\cap B), P(A\cap C), P(B\cap C)$ (2)

If you consider the sigma (=boolean) algebra generated by $A,B,C$, you know everything except the missing information $P(A\cap B\cap C)$. With this missing information, you could calculate absolutely everything involving $A,B$ and $C$ like for example $P(A|B\cap C)$

Since you have some missing information, you need to estimate it. The solution I've found (that works very well on many problems) is to use the principle of maximum entropy on the sigma algebra generated by $A,B,C$:

- Call $P$ the probability measure on this sigma algebra
- Assume (1) and (2) as constraints
- Find the value of $P(A\cap B\cap C)$ that maximizes the entropy of the probability measure

The problem can be extended to $n>3$ events just the same, except that you have a high dimensional vector of missing information.

Again, the solution to this problem is far from trivial. But luckily maximum entropy has been intensively studied, and we know how it leads to exponential families solutions. The problem is very similar to logistic regression. You can read literature about fitting maximum entropy. Actually, the most efficient solution is to use standard convex optimization algorithms once you have transformed the problem into an unconstrained optimization problem. A good resource : https://web.cs.elte.hu/blobs/diplomamunkak/msc_mat/2013/biszak_elod.pdf

## Best Answer

Another approach would be:

Note the similarity to:

And there are many equivalent forms.

Taking U = (B, C, D) gives: P(A| B, C, D) = P(A, U)/P(U)

I'm sure they're equivalent, but do you want the joint probability of B, C & D given A?