Is Mean for NegativeBinomial distribution correct?

[dsyme]

Playing around with distributions and it seems that the Mean of the NegativeBinomial distribution doesn't match the mean of samples.

open MathNet.Numerics
let b7 = Distributions.NegativeBinomial(2.0, 0.5) 

/// This gives about 4.0
[ for i in 0 .. 100 -> b7.Sample() ]  |> List.averageBy float

//This says 2.0
b7.Mean

[cdrnet]

That looks suspicious, thanks for reporting!

[cdrnet]

It turns out the mean is correct, but the samples return the number of trials (for reaching r successes) instead of the number of failures (until r successes are reached). Both are common definitions of the distribution, but they of course should be consistent within the implementation. We chose the second definition since it can be extended to all positive real numbers of r instead of just integers.

Quick workaround until it is fixed: subtract r (here: 2.0) from all the generated samples.

Unfortunately since r is real, we cannot actually subtract it from our samples since the discrete distribution interface assumes samples are integers. We may need to split this distribution if one accepting only integer r and thus also samples integers, and an extended one for real r with another interface.

[thamilto]

I'd like to point out that the quick work-around doesn't seem to work. In the example given by dsyme if I run this on my PC I get positive integer and 0 samples. I don't follow how this can be interpreted as "the number of trials (for reaching r successes)". Knocking off 2.0 from a 0 sample, regardless of types, doesn't seem to make sense.

Can someone elaborate?

[vermorel]

I might have pinpointed the problem in NegativeBinomial, the code is:

        static int SampleUnchecked(System.Random rnd, double r, double p)
        {
            var lambda = Gamma.SampleUnchecked(rnd, r, p);
            var c = Math.Exp(-lambda);
            var p1 = 1.0;
            var k = 0;
            do
            {
                k = k + 1;
                p1 = p1*rnd.NextDouble();
            }
            while (p1 >= c);
            return k - 1;
        }

The first code line should be replaced by

var lambda = Gamma.SampleUnchecked(rnd, r, (1 - p)/p);

Also, it would be nice to point-out in the code that the second part is actually drawing a Poisson sample (which could benefit from the optimized implementation in PoissonDistribution.cs which distinguish small lambdas from big lambdas).