Change PoissonADSampler threshold to 10

[k88hudson]

Another one that came up when I was going through some of the older papers for sampling algorithms:

In the Poisson sampler, the threshold at which PoissonADSampler kicks in is currently μ >= 6, but the Ahrens-Dieter PD algorithm says it is designed for μ >= 10. I am not sure why 6 was chosen (it looks like it was originally from PoissonRandom.jl, see #1021 but I don't see anything beyond that). To verify that this is actually a problem: if you plot the hat function (equation 24) against the residual between the discretized normal and the poisson PMF, you will see that at 6 it definitely goes outside the bounds. With the following:

pmf_diff <- function(x, mu) {
  s <- sqrt(mu)
  dpois(floor(x), lambda = mu) - 
    pnorm(floor(x+1), mean = mu, sd = s) +
    pnorm(floor(x), mean = mu, sd = s)
}

hat <- function(t, mu) (0.1069 / mu) * exp(-abs(t - 1.8))

In terms of demonstrating the bias, right now the sampler tests only check down to 10.0, not 6.0:

Distributions.jl/test/samplers.jl

Line 63 in a7ed7b6

(PoissonADSampler, [10.0, 15.0, 20.0, 30.0])]

You can detect it at 6, but I found that you need to use n=10^8 (I included a regression test in the PR which fails on master). At μ=7 it requires n=10^9, I didn't go further than that.

Note: I think you would expect performance regressions for this change for values of μ from 6 to 9, since PoissonADSampler should be faster down to 6.

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 86.63%. Comparing base (a7ed7b6) to head (b022b0e).
⚠️ Report is 5 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2060      +/-   ##
==========================================
- Coverage   86.65%   86.63%   -0.02%     
==========================================
  Files         148      149       +1     
  Lines        8919     8921       +2     
==========================================
  Hits         7729     7729              
- Misses       1190     1192       +2     

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[devmotion]

Minor nitpick: Could you change this to a @testset? It would also be a bit more natural to use @test ... ≈ ... atol = ....

[devmotion]

Thank you!