Sts dev projection

[Nimrais]

This PR extends the base PR with projection based poisson likelihood model!

[Nimrais]

@albertpod it's ready for review.

[Nimrais]

Old prediction with auxilary node:

New prediction with projections:

[albertpod]

Going through the example.

1. Getting the warning in the first cell:
   WARNING: using Distributions.Product in module ClosedFormExpectations conflicts with an existing identifier.
2. Failed when running inference:

PosDefException: matrix is not positive definite; Factorization failed.

Stacktrace:
   [1] checkpositivedefinite
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/factorization.jl:68 [inlined]
   [2] #cholesky!#163
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:269 [inlined]
   [3] cholesky!
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:267 [inlined]
   [4] #cholesky!#164
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:301 [inlined]
   [5] cholesky! (repeats 2 times)
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:295 [inlined]
   [6] _cholesky
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:411 [inlined]
   [7] cholesky(A::Matrix{Float64}, ::NoPivot; check::Bool)
     @ LinearAlgebra ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:401
   [8] cholesky (repeats 2 times)
     @ ~/.julia/juliaup/julia-1.11.2+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:401 [inlined]
   [9] PDMat
     @ ~/.julia/packages/PDMats/8CRip/src/pdmat.jl:40 [inlined]
  [10] Wishart
     @ ~/.julia/packages/Distributions/psM3H/src/matrix/wishart.jl:65 [inlined]
  [11] convert(::Type{Wishart}, dist::ExponentialFamily.WishartFast{Float64, Matrix{Float64}})
     @ ExponentialFamily ~/.julia/packages/ExponentialFamily/prORE/src/distributions/wishart.jl:94
...
     @ ~/.julia/packages/RxInfer/UX7t9/src/inference/inference.jl:545 [inlined]
 [414] with_session(f::RxInfer.var"#312#314"{GraphPPL.ModelGenerator{typeof(rxsts_twist_poisson), @Kwargs{H::Vector{Float64}, X::Vector{Vector{Float64}}, R::Matrix{Int64}, priors::Dict{Symbol, ContinuousDistribution}}, GraphPPL.PluginsCollection{Tuple{}}, RxInfer.ReactiveMPGraphPPLBackend{Static.False}, String}, @NamedTuple{y::Vector{Union{Missing, Int64}}}, Nothing, Nothing, RxInfer.InitSpecification, GraphPPL.Constraints{Vector{GraphPPL.FactorizationConstraint}, Vector{GraphPPL.MarginalFormConstraint}, Vector{GraphPPL.MessageFormConstraint}, Dict{Function, GraphPPL.GeneralSubModelConstraints}, Dict{GraphPPL.FactorID, GraphPPL.SpecificSubModelConstraints}, String}, GraphPPL.MetaSpecification, @NamedTuple{limit_stack_depth::Int64}, KeepLast, Nothing, Nothing, Nothing, Int64, Bool, Tuple{RxInfer.ObjectiveDiagnosticCheckNaNs, RxInfer.ObjectiveDiagnosticCheckInfs}, Bool, Bool, Bool, Nothing, Nothing, DefaultPostprocess, Nothing, Bool, Bool, Bool}, session::RxInfer.Session, label::Symbol)
     @ RxInfer ~/.julia/packages/RxInfer/UX7t9/src/session.jl:253
 [415] #infer#311
     @ ~/.julia/packages/RxInfer/UX7t9/src/inference/inference.jl:515 [inlined]

[Nimrais]

Getting the warning in the first cell:
WARNING: using Distributions.Product in module ClosedFormExpectations conflicts with an existing identifier.

It's not really important julia just not happy that ClosedFormExpectations.jl has a struct Product and Distributions.jl also has it. It just mean that if you will use it in this notebook you would need to specify the package like ClosedFormExpectations.Product or Distributions.Product. But such situation would never happen in this notebook :).

Failed when running inference:

That is annyoing FastCholesky not happy with some small numbers, I will play with the precision. I see it also fails in the CI.

[Nimrais]

I am not really experienced with FastCholesky. Inference in my example is running and I do not see any strange results. But because of some numerical issues, a matrix that is supposed to be symmetric is slightly asymmetric. So I decided to just do the following at runtime:

### EXAMPLE_HIDDEN_BLOCK_START ###
delete!(ENV, "JULIA_FASTCHOLESKY_THROW_ERROR_NON_SYMMETRIC");
nothing
### EXAMPLE_HIDDEN_BLOCK_END ###

I do not know why this variable is set to throw an error for all examples if FastCholesky.jl itself does not throw an error by default. Should it not be set for the specific examples where we want to throw an error?

[Nimrais]

@bvdmitri maybe you are able to help.