Add ChainRules definitions for pdf of PoissonBinomial

[devmotion]

I am in the process of updating DistributionsAD to ChainRules 1. I thought it would be a good opportunity to reduce the amount of type piracy and move rules for functions in Distributions here. Currently, there's only one such rule in DistributionsAD. It defines the pullback for the pdf of the PoissonBinomial distribution, so the PR adds and tests only this rule.

ChainRulesCore 1 is a very lightweight package (see @oxinabox's comment for more details) and Distributions already depends on ChainRulesCore indirectly via SpecialFunctions and StatsFuns (support for ChainRules was added in version 0.9.10 which was released today). DistributionsAD does not support ChainRulesCore 1 yet, so the definitions in Distributions won't lead to any conflicts with DistributionsAD if we remove the rrules in DistributionsAD before we tag a CR1-compatible release.

The implementation is based on the same dynamic programming "trick" as the implementation of poissonbinomial_pdf. I didn't find any reference for it, I only derived it a while ago and added it to DistributionsAD. However, I am pretty confident that the ChainRules tests utilities would have revealed any errors.

/ cc: @bdeonovic who added the primal definition in #1198

[codecov-commenter]

Codecov Report

Merging #1390 (bed3009) into master (59df675) will increase coverage by 0.09%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master    #1390      +/-   ##
==========================================
+ Coverage   82.44%   82.54%   +0.09%     
==========================================
  Files         116      116              
  Lines        6911     6950      +39     
==========================================
+ Hits         5698     5737      +39     
  Misses       1213     1213              

Impacted Files	Coverage Δ
src/univariate/discrete/poissonbinomial.jl	93.57% <100.00%> (+2.48%)	⬆️

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[matbesancon]

Looks good, I had tried to start something in https://github.com/matbesancon/DistributionsDiff.jl/ but never got around sitting to continue this

[matbesancon]

do you want to implement frule in this PR or a following one?

[devmotion]

I'll add the frules in this PR 👍 I did not include them in DistributionsAD since the only AD backend in DistributionsAD with CR support is Zygote, so it didn't seem necessary. However, I guess we should add frules in Distributions as well.

[devmotion]

I added them 👍

[storopoli]

Great! Is this is the first AD in Distributions.jl?

[devmotion]

AFAIK yes. gradlogpdf has existed for a long time but there were no AD backends used in the package. There are many tests with ForwardDiff though.

[bdeonovic]

Great! Is this is the first AD in Distributions.jl?

Yes; but only because AD generally works off the shelf without having to implement anything extra.

[oschulz]

Awesome!