Imagine that a point is space x can be characterized by either A,B,C,D,E,F,G in the c_A, c_B .. classifications of x as A,B,C,... My data gives:
P(x = A | c_A) = f_A(x)
P(x = B | c_B) = f_B(x) and so on.
What I want to ask is what x most probably is in this realization, given the probabilities that it can be all these A,B,C... (and then marginalize out the realizations). The latter I know how to do, the former I am a bit confused as to how to formulate mathematically.
One could choose the maximum out of P(x = A), P(x = B) etc., but I would like to do it in a Bayesian way and I think what is needed for this is a proper Bayesian error propagation, I just don't understand how to do it in this context.
