Estimating P(C|A,B) from P(C|A) and P(C|B): Bayes Rule? Bayes Net? Classifier?

Question

Doing some ecommerce analytics, I want to understand click propensity broken out by different features present in users' profiles.

In this scenario, it's easy to test click propensity $p(C)$ broken out by any single feature, but it becomes difficult to test multiple features at once. This is to say, for features $A$ and $B$, I could have good sample sizes to estimate $p(C|A)$ and $p(C|B)$, but not necessarily for the joint probability $p(C|A,B)$.

So, I'd like to estimate $p(C|A,B)$ from more easily measured quantities. I can easily test $p(C)$, $p(C|A)$, and $p(C|B)$. I can also easily query how often different features occur or co-occur among users, this is to say I can easily query $p(A)$, $p(B)$, and $p(A,B)$.

How can I relate $p(C|A,B)$ to these quantities, and what assumptions do I make about the interactions and dependence between events?

So far, I can use Bayesian updating to make some progress:

$p(C|A,B) = \dfrac{p(A|B,C)p(C)}{p(A,B)}$

$ = \dfrac{p(A|B,C) p(B|C) p(C)}{p(A,B)}$

Then by applying Bayes to the $p(B|C)$ term in the numerator:

$ = \dfrac{p(A|B,C) p(C|B) p(B)}{p(A,B)}$

Or the more canonical form:

$ = \dfrac{p(A|B,C) p(C|B)}{p(A|B)}$

This has a lot of the quantities I can easily find, but the $p(A|B,C)$ term is still a pain. There doesn't seem to be a way to manipulate it that doesn't have all three events occurring in the same $p(...)$.

Second part of the question:

Can a Bayes Net be fruitfully used to model this probability? The features among users are not explicitly causally related.

Or, is this just a strange way of thinking about a classifier with response $C$ and predictors ${A,B}$? In this case is there a classifier that might perform well given the difficulty of testing the response against multiple features at once? What assumptions would it be making?

Answer 1

I don't see how. But, since you say

The features among users are not explicitly causally related

perhaps you can try the Naïve Bayes assumption.

The probability $$P(A,B,C) = P(C)P(A|C)P(B|C,A)$$ can be simplified by naively assuming that every feature is conditionally independent on every other, so that gets us $$P(A,B,C) = P(C)P(A|C)P(B|C)$$

By Bayes' theorem you can compute $P(A|C)$ and $P(B|C)$ with your information.

So now we have:

$$P(C|A,B) = \frac{P(A,B,C)}{P(A,B)} \approx \frac{P(C)P(A|C)P(B|C)}{P(A,B)}$$

which can be calculated.

Notice however that we made an assumption that might be invalid for your problem.