# Multinomial Distribution Sampler Not Working

I am trying to sample from a multinomial distribution many times but with a different multinomial distribution each time. The straightforward

RandomVariate[MultinomialDistribution[1,{p1, p2, ..., pN}]]


is too slow for my application. I wrote my own sampler where I use Accumulate on the distribution and then use RandomReal[] to choose one possible outcome.

S = 300;
dis = Accumulate@Table[1/S, {k, 1, S}];
GetRand[dist_] := FirstPosition[dist, SelectFirst[dist, # >= RandomReal[] &]][[1]];
test = Table[GetRand[dis], {k, 1, 100000}];
Histogram[test, S, "PDF"]


Because dis is uniform, my expectation is that the histogram should be mostly uniform. Instead I get a histogram which is clearly not uniform. I've found that I get the correct result if I separately generate a table of all the random reals and then use them to sample from the multinomial distribution.

Why does this code not give a uniform histogram? Also, why does it appear to give a distribution from the Gaussian Orthogonal Ensemble?

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• Why do you think "dis is uniform"? Commented Jul 13, 2023 at 7:13

If you're really wanting to sample from a multinomial, then go with @Asim 's answer.

But the example you give isn't really a multinomial as the result from GetRand (after using the correction below) is just the position of the success category with a sample size of one.

I think the issue is that RandomReal[] is called internally for each element in dist rather than being a single call to RandomReal[]. Therefore, the element selected never gets much past 60 as there are multiple chances to get a low number. (I would bet that's by design rather than considered a bug.) The fix might be the following:

S = 300;
dis = Accumulate@Table[1/S, {k, 1, S}];
GetRand[dist_] := Module[{r},
r = RandomReal[];
FirstPosition[dist, SelectFirst[dist, # >= r &]][[1]]];
test = Table[GetRand[dis], {k, 1, 100000}];
Histogram[test, S, "PDF"]


If you only need the position of what is selected from a sample of size 1, then RandomChoice should be considered:

SeedRandom[12345];
AbsoluteTiming[test = Table[GetRand[dis], {k, 1, 100000}];]
(* {8.72437, Null} *)
SeedRandom[12345];
AbsoluteTiming[RandomChoice[Range[300], 100000];]
(* {0.0012076, Null} *)


You may want to look at the solution that I posted sometime ago. You will need a C compiler on your machine. Also note that the function does not check to ensure that the inputs for consistency with the requirements of a distribution.

Improving Speed of Multinomial Draws