## an elegant sampler

**F**ollowing an X validated question on how to simulate a multinomial with fixed average, W. Huber produced a highly elegant and efficient resolution with the compact R code

tabulate(sample.int((k-1)*n, s-n) %% n + 1, n) + 1

where *k* is the number of classes, *n* the number of draws, and *s* equal to *n* times the fixed average. The R function *sample.int* is an alternative to *sample* that seems faster. Breaking the outcome of

sample.int((k-1)*n, s-n)

as nonzero positions in an *n x (k-1)* matrix and adding a adding a row of *n* 1’s leads to a simulation of integers between 1 and *k* by counting the 1’s in each of the *n* columns, which is the meaning of the above picture. Where the colour code is added after counting the number of 1’s. Since there are *s* 1’s in this matrix, the sum is automatically equal to *s*. Since the *s-n* positions are chosen uniformly over the *n x (k-1)* locations, the outcome is uniform. The rest of the R code is a brutally efficient way to translate the idea into a function. (By comparison, I brute-forced the question by suggesting a basic Metropolis algorithm.)

January 16, 2020 at 7:51 am

How one makes that beautiful graphic: https://i.stack.imgur.com/QBAxP.png

January 15, 2020 at 8:16 am

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