I'm working on an application for which it would be great to have the following functionality:

Say that you have a collection $C$ of $n$ events, for now let's set $n = 3$ and call the events $a, b,$ and $c$

Given $k$, say that for any subset $K \subset C$ s.t. $||K|| = k < n$ of the set of events, you can produce the complete set of joint and conditional probabilities for any combination of the $k$ events. So, if $n = 3$ and $k = 2$, you would have complete information about

$p(a | b)$, $p(b | a)$, $p(a | \neg b)$, $p(a \wedge b)$, $p(\neg (a \wedge b))$, etc., etc.

$p(a | c)$, $p(c | a)$, $p(a | \neg c)$, $p(a \wedge c)$, $p(\neg (a \wedge c))$, etc., etc.

$p(c | b)$, $p(b | c)$, $p(c | \neg b)$, $p(c \wedge b)$, $p(\neg (c \wedge b))$, etc., etc.

Is there a 'correct' way to estimate the corresponding values for the joint and conditional probabilities for a, b, AND c?

Things such as $p(a \wedge b \wedge c)$, $p(\neg (a \wedge b \wedge c))$, $p(a | b \wedge c)$, $p(a \wedge b | c)$, etc., etc.

I am especially interested in the general case.

# Edit

I should have been more expansive in my questions, as regardless of whether or not there is a 'correct' way of answering this question, if there are methods that allow one to say anything interesting about the bounds, or anything interesting at all, really, about the joint & conditional probabilities of a, b, and c, than those methods will be very useful to me.