Compute $P(A|B,C)$ knowing only $P(A|B)$ and $P(A|C)$ This is perhaps better explained with an example.
Say we have 2 models:


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*Model 1 takes a name, and outputs a probability whether it is a female name or male name. ie. $P(gender | name)$

*Model 2 takes an audio recording of speech, and outputs a probability whether it belongs to a woman or a man. ie. $P(gender | speech)$
Also assume that it's not trivial to compute marginals $P(gender)$, $P(name)$, and $P(speech)$.
I would like to compute $P(gender | name, speech)$. I want to use both signals to give me a better guess. How do I compute it from the two conditional distributions above?
(Intuitively, it seems like it should be a weighted combination of $P(gender | name)$ and  $P(gender | speech)$. However, I can't figure out what this weight should be.)
 A: From a strictly probabilistic perspective, that weight will depend on how much overlap there is between name and speech pattern variables, so there is not one right answer.
$P(gender|name,speech)$ can be equal to $P(gender|name)$ if speech pattern did not tell us anything about gender that name did not already tell us (in other words, if we took a name, all individuals with that name had the same speech pattern).  (And similarly with $P(gender|name,speech)$ equating to $P(gender|speech)$).
If, however, every instance of a particular name was associated with a different speech pattern and with a different gender, then it is possible for $P(gender|name,speech)$ to equal to $P(gender|name) + P(gender|speech)$ or to $P(gender|name) - P(gender|speech)$.
Having said all that, you may want to try ensemble methods / model stacking to try to get better predictions from your separate models.  It can be as simple as taking a simple average of predictions from different models, but it can also involve implementing  second-level (and higher order) prediction models using predictions from the original models as input.
