Computing Fisher information via forward-mode automatic differentiation

fisher.jl

using Distributions
import ForwardDiff: Dual, value, partials

@generated function get_values(a::NTuple{N}) where {N}
    return ForwardDiff.tupexpr(i -> :(value(a[$i])),N)
end
ForwardDiff.value(p::ForwardDiff.Partials) =
    ForwardDiff.Partials(get_values(p.values))

fisherinf(f, x) = ForwardDiff.hessian(y -> EI(f(y)), x)
# should probably check we're getting the right type...

# "expected information" operator
#  - value:    technically entropy, but with an arbitrary constant, so choose 0
#  - gradient: 0 (expected score)
#  - hessian:  Fisher information
function EI(d::Normal{Dual{A,Dual{B,T,N},M}}) where {A,B,M,N,T}
    vσ = value(value(d.σ))
    Iμμ = 1/vσ^2
    Iσσ = 2/vσ^2
        
    Dual{A}(zero(value(d.μ)), # Dual{B,T.N}
            value(partials(d.μ)) * Dual{B}(zero(T), Iμμ*partials(value(d.μ))) +
            value(partials(d.σ)) * Dual{B}(zero(T), Iσσ*partials(value(d.σ))))
end


fisherinf(x -> Normal(x[1], x[2]), [0.0, 1.2])