# Estimating joint distributions from marginal

Suppose I have the marginal probability density functions of two random variables A and B, P(A), and P(B). Suppose I modelled P(A) and P(B) using a mixture model from some dataset D and obtained a closed form pdf for each.

I am interested in finding their joint density function P(A and B) and associated properties such as maximas, minimas, etc.

Ideally the joint density is expressed as a closed form 2D mixture model as well, but this is not critical.

I could do something perhaps by brute force by use of Baye's theorem:

ie. I can approximate

P(A and B) = P(A) P(B | A) = P(B) | P(A | B)

But eventually I need to extend this to higher dimensions, eg. P( A and B and C and D... etc) and this is certainly no trivial task.

Stephen Tashi
In general, you cannot determine a joint probability distribution when given only the marginal probability distributions, so if your problem can be solved the solution depends on special circumstances or information that you haven't mentioned. To get the best advice, you should describe the situation completely.

I could do something perhaps by brute force by use of Baye's theorem:

ie. I can approximate

P(A and B) = P(A) P(B | A) = P(B) | P(A | B)

That isn't a mere approximation. It is a theorem.

Are you saying that you have data that could be used estimate the conditional probability distributions?

Well A and B are two variables that specify (completely) the state of the system. Suppose i've sampled a whole bunch of data points (a,b) s.t. I can generate their PDFs.

I can approximate P(B | A=a1) and P(A | B=b1) as well by taking a slice of my dataset, (eg. B= b1+-0.1) and count the occurrences of A. However, this can be bad because my entire dataset may be quite small, and using only a subset of it will result in a lot of noise and error.

Stephen Tashi