Proper way to combine conditional probability distributions of the same random variable conditioned on a discrete variable ? (based on assumptions)

Question

My question is the following. Let's say I have two probability distributions:

$f(x|b), g(x|c)$

$b$ and $c$ are discrete events while $x$ is a continuous variable, i.e., when the button b is pressed there is some distribution for the amount of rain fall the next day, $x$.

When the button $c$ is pressed there is a different distribution of rain fall the next day, $x$. Are there any strategies for estimating the distribution of rain fall if both buttons are pressed, i.e.,

$h(x|b,c)$ ?

And, what assumptions do those strategies rest on?

Answer 1

The solution is indeterminant. Even using p(b and c)= p(b) p(c) all we have is that the conditional density h(x|b and c) = h(x and b and c)/p(b and c)= h(x and b and c)/[p(b) p(c)]=h(x and c|b)/p(c). But this does nothing to relate the distribution h(x and c|b) to f(x|b) and g(x|c)

Answer 2

"10 'experts' forecast some pdf of risk (or an event related to risk) - how do you combine these to make a decision?

Assuming the experts come up with their pdfs using independent pieces of information, the unique correct way to combine the evidence is using the pointwise product of the density functions, just as we do when doing Bayesian estimation.