I have nonlinear data of function y(x), which is let's say parabolic. At some points of x there are several y's (look at the picture).
Is it possible to train a probabilistic model to return several distributions (when needed) i.e. several means and variances. For example: when I feed a (x=a) to the model -> it returns 2 red distributions (2 means and 2 variances), and when I feed b (x=b) to the model -> it returns 1 blue distribution.
Thanks.
That data can be modeled as a statistical process, where the distributions and parameters change as a function of x. This is in contrast to modeling it as a typical statistical distribution which assumes the same distribution and parameters throughout a range.
The "a" range could be modeled as a bimodal distribution and the "b" range could be modeled as a unimodal distribution.
The conditional distribution of $Y$ when $X=a$ is bimodal. The mean is in the middle, and reporting it as such is correct (the mean of $1,2,3,91,92,93$ is indeed $47$).
This will be reflected somewhat in the variance being extremely wide. If you want to model a full distribution, you could consider quantile regression at many quantiles. I found this article on neural networks to predict multiple quantiles.
Such a quantile regression approach would, correctly, show that the conditional median is in the middle of the parabola, but then you’ll have a big jump to the, say, 45th and 55th percentiles before the quantiles bunch up around each mode.
