Suppose I have a variable Y that I want to predict with a model using predictor variables X1, X2 and X3. I have a large set of Y-data and from this I know with some certainty and accuracy the empirical distribution of Y.
Assume I have no data where X1...X3 is paired with Y. However, I know from domain knowledge that Y is (say) directly related to X1 and X2, and indirectly related to X3.
Now, given a new data set of say 1000 observations where each observation has values for X1, X2 and X3, I want to predict Y for each observation based on my domain knowledge such that the resulting distribution of predicted Y-values matches the known empirical distribution of Y.
Thus to summarize: I have no data on which to train a model, but I do have domain knowledge. My main interest is to get - for a set of predictor variables - a predicted Y-distribution that matches the previously observed Y-distribution.
What tools and methods can I use to approach this problem? Many thanks!