densities

[cscherrer]

Say we have a normal distribution,

julia> μ = Normal()
Normal()

julia> logdensity(μ, 1.0)
-0.5

and a Cauchy distribution

julia> ν = Cauchy()
Cauchy()

julia> logdensity(ν, 1.0)
-0.6931471805599453

Note that the base measures here are different, because each accounts for the normalizing factor:

julia> basemeasure(μ)
0.398942 * Lebesgue(ℝ)

julia> basemeasure(ν)
0.31831 * Lebesgue(ℝ)

But we can still get density functions between them using a three-argument form I just added in dev:

julia> logdensity(μ, ν, 1.0)
0.41893853320467267

And just to check...

julia> Dists.logpdf(Dists.Normal(), 1.0) - Dists.logpdf(Dists.Cauchy(),1.0)
0.41893853320467267

We might want to have the Radon-Nikodym derivative as a separate operator, so we also can do this:

julia> 𝒹(μ, ν)(1.0)
0.41893853320467267

[cscherrer]

Here's how it works:

function logdensity(μ::AbstractMeasure, ν::AbstractMeasure, x)
    bμ = basemeasure(μ)
    bν = basemeasure(ν)
        
    result = logdensity(μ,x) + logdensity(bμ,x)
    result -= logdensity(ν,x) + logdensity(bν, x) 
    result += _logdensity(basemeasure(bμ), basemeasure(bν), x)
end

The idea is that the basemeasure of a basemeasure should be relatively simple - in this case both are Lebesgue(ℝ). So working in terms of these means we don't need to implement so many methods of _logdensity