If I need to generate N numbers from a given probability distribution, I am aware of how to do this.
But my problem is out of the N numbers, say, k numbers are already fixed. (k is very small than n).
I am wondering what is the best way of generating the other elements so that the overall distribution is close to the given distribution?
EDIT:
I will give a clear example of my problem:
I have to select 100 numbers from a set of 1000 numbers. The selection is done independently and with replacement. {p1, p2,...., p1000} is the vector of probabilities representing the sampling probability (all values are non zero) for each of the 1000 elements.
Now, say, I already have k elements decided without following the sampling procedure. i.e., only 100-k is left to be selected.
How should I design my new sampling probabilities {q1, q2,...,q1000} so that in effect we get the original distribution {p1,p2,...p1000}, or closest to the original distribution If KL divergence is my measure.