Sampling From A Multinomial Distribution
First, reorder the parameters such that they are sorted in descending order (this is only to speed up computation and not strictly necessary). Now, for each trial, draw an auxiliary variable X from a uniform (0, 1) distribution. The resulting outcome is the component
This is a sample for the multinomial distribution with n = 1. A sum of independent repetitions of this experiment is a sample from a multinomial distribution with n equal to the number of such repetitions.
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“There is the illusion of time, which is very deep; who has disposed of it? Mor come to the conviction that what seems the succession of thought is only the distribution of wholes into causal series.”
—Ralph Waldo Emerson (1803–1882)