Question: Let's say we have a problem of predicting whether a storm is coming or not. We have a model `P(storm is coming | how many clouds are outside)`, and have another model `P(storm is coming | how scared the dogs are)`.

My question in general is how to combine these two models in a sensible way. It seems to make sense if I just multiply the two together - however, I don't know what this expression evaluates to:

`P(storm is coming | how many clouds are outside) * P(storm is coming | how scared the dogs are) = ?`

Or, with simpler notation:

`P(A|B)P(A|C) = ?` (where B and C are independent)

~~I have a feeling the answer is `P(A|B)P(A|C) = P(A|B,C)` but I can't prove it.~~ (As many pointed out, this is wrong)

Edit: Sorry for the confusion, the question here is "what is `P(A|B)P(A|C) = ?`"

Answer: 
In my opinion your problem is not about that expression but about modelling. You have different clues (clouds, scared dogs) which provide evidence for a forthcoming event (rain). In other words, if I understand you, your question is actually more about: how to combine different clues?.

This question is dealt with in the field of graphical models. Let me refer to the corresponding [Wikipedia site](https://en.wikipedia.org/wiki/Graphical_model), and the references therein. Specially the paper by David Heckerman ([A Tutorial on Learning With Bayesian Networks](https://www.microsoft.com/en-us/research/publication/a-tutorial-on-learning-with-bayesian-networks/?from=http%3A%2F%2Fresearch.microsoft.com%2Fen-us%2Fum%2Fpeople%2Fheckerman%2Ftutorial.pdf)), and the reference to Christopher Bishop's Machine Learning book.

The example you give is very similar to those presented to describe the explain away effect, that is nicely described in [this video](https://www.youtube.com/watch?v=pyxyYWNo8Qw).
As JonMark Perry has already mentioned, Bayes' theorem prohibits that your initial suspicion is true. The rules for conditional probabilities specifically allow for multiplication of probabilities only if conditioning under the same event (either B or C or both simultaneously). To show you a visualistion of the 2 probabilities take a look at [this image of a tree diagram.](http://maths.nayland.school.nz/Year_13_Maths/3.13_Probability/Images_3.3_Probability/Probab8.jpg). Within a branch of the tree, you may multiply the probabilities which is interpreted as both events having to happen simultaneously P(A and B) = P(A|B) \* P(B) or P(A and C) = P(A|C) \* P(C) . In order to combine to two different branches (the directions would be B and C) the probabilities have to be added. P(A|B) \* P(B)+ P(A|C) \* P(C) is then the probability that the storm comes up, independent of the clouds coming up or how scared the dogs are.