How do I best find a best fit line if my y data is probability functions rather than single points? Obviously, in every regression problem you can include uncertainty on the xs and ys, but I have a dataset where I have a single point for each of my xs but my y data is probabilistic (that is for each x, I have some function representing the likelihood of y being a range of value). I could sample n points from each of the probability functions and then do a least squares line but that seems slow and not quite right. Are there any methods of doing this efficiently/in a particularly good way (I am also interested in any particularly useful metrics or data visualization techniques that are applied to this kind of data--the obvious solutions, like using colors to represent the probability on an x-y plot, which I am using so far don't seem like the easiest to read solution).
As a related question, what if x were not just a single number but rather coordinates (x1,x2,x3). Given probability functions at some set of points, is there a method to find a best fit function over space in general?