Implement gammalogccdf for ForwardDiff

I am modeling in Turing.jl some process in which I use Gamma prior to parametrize Truncated Poisson and I use NUTs sampler.

However, my model has nested dependence and I cannot fully vectorize the loop, so Zygote would be slow. The default ForwardDiff throws at me an error

ERROR: MethodError: no method matching _gammalogccdf(::ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}, ::ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}, ::ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8})
Stacktrace:
  [1] gammalogccdf(k::Int64, θ::Int64, x::ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8})
    @ StatsFuns ~/.julia/packages/StatsFuns/atA5S/src/distrs/gamma.jl:71
  [2] poislogcdf(λ::ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}, x::Int64)
    @ StatsFuns ~/.julia/packages/StatsFuns/atA5S/src/distrs/pois.jl:30
  [3] logcdf(d::Poisson{ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}}, x::Int64)
    @ Distributions ~/.julia/packages/Distributions/Ufrz2/src/univariates.jl:638
  [4] _logcdf_noninclusive(d::Poisson{ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}}, x::Int64)
    @ Distributions ~/.julia/packages/Distributions/Ufrz2/src/censored.jl:441
  [5] truncated(d::Poisson{ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}}, l::Int64, u::Int64)
    @ Distributions ~/.julia/packages/Distributions/Ufrz2/src/truncate.jl:67
  [6] Truncated(d::Poisson{ForwardDiff.Dual{ForwardDiff.Tag{Turing.TuringTag, Float64}, Float64, 8}}, l::Int64, u::Int64)
    @ Distributions ./deprecated.jl:105
  [7] macro expansion

To my understanding this error suggests that the model is encountering issues with automatic differentiation, specifically when evaluating the complementary cumulative distribution function (CCDF) of the Gamma distribution. Seems like the error might be related to the Truncated Poisson distribution, as the truncation operation might involve the CCDF of the underlying distribution.

Can that be solved in ForwardDiff AD ?

Start by reproducing the shown MethodError and inspect the gammalogccdf entry point in StatsFuns/src/distrs/gamma.jl, along with the ForwardDiff.Dual arguments in the stack trace. Trace how the truncated Poisson path reaches this call; done means that path no longer raises the reported method error for ForwardDiff inputs.