My question is related to multivariable joint distribution. I have one source variable "x" and multiple receivers "y1" "y2" "y3". I have each joint distribution p(x,y1),p(x,y2), p(x,y3). My question is how do I get p(x) from combination of the 3. The issues in my mind are
If I calculate p(x) from p(x,y1). I already think that this should be exactly the p(x) obtained from other joint distributions. But in a real scenario, we have to estimate these distributions which would yield different marginal for p(x).
I do not have code yet but if someone can point out the direction then it would really be helpful
I worked on modeling sensors with belief networks in my dissertation. See: http://riso.sourceforge.net My dissertation is a little ways down on the page. A model for sensors which measure the same thing is described in Section 6.5.
In brief, when you have multiple measurements y1, y2, y3 of the same thing x, you can model the joint probability of all of them as p(x, y1, y2, y3) = p(y1 | x) p(y2 | x) p(y3 | x) p(x), where each p(y | x) is a measurement model, i.e., it represents the way that the measurement is a function of the thing being measured. Then the goal is to compute p(x | y1, y2, y3). It turns out that's proportional to p(y1 | x) p(y2 | x) p(y3 | x) p(x), with the constant of proportionality being whatever is needed to make the expression integrate to 1 over x. I.e., to combine information from multiple sensors, given this model, you multiply them together.
If you open a question on stats.stackexchange.com, I can say more about it. Hope this helps.