Add natural gamma manifold

[Nimrais]

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[codecov]

Codecov Report

Attention: Patch coverage is 96.55172% with 3 lines in your changes missing coverage. Please review.

Project coverage is 96.66%. Comparing base (d59a096) to head (9e80b86).
Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
src/normal_gamma/natural.jl	96.15%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #26      +/-   ##
==========================================
- Coverage   96.69%   96.66%   -0.04%     
==========================================
  Files          25       28       +3     
  Lines         333      420      +87     
==========================================
+ Hits          322      406      +84     
- Misses         11       14       +3     

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[ismailsenoz]

A more generic implementation of Christoffel symbol.

function christoffel(::Type{D}, ::NaturalParametersSpace, η) where D
    # Pre-compute values used multiple times
    ef = ExponentialFamilyDistribution(D, η, nothing)
    G = fisherinformation(ef)
    G⁻¹ = cholinv(G)
    dG = partial_metric(D, NaturalParametersSpace(), η)
    
    T = eltype(G)  # Get the element type from G
    
    # Pre-allocate output array
    rep_size = sum(representation_size(get_natural_manifold(D, size(η))))
    Γ = zeros(T, rep_size, rep_size, rep_size)
        
    # Pre-compute the sum of dG terms
    dG_sum = similar(dG)
    
    @tensor begin
        dG_sum[ℓ,j,i] = dG[ℓ,j,i] + dG[ℓ,i,j] - dG[i,j,ℓ]
    end
    
    # Use tensor operations with pre-computed sum
    @tensor begin
        Γ[k,i,j] = T(0.5) * G⁻¹[k,ℓ] * dG_sum[ℓ,j,i]
    end
    
    return Γ
end