I’ve heard there may be some perception that to use MeasureTheory, you need to entirely drop Distributions.jl. That’s not true!
Calling Distributions from MeasureTheory
MeasureTheory actually has Distributions as a dependency, and exports it as Dists. So this works just fine:
julia> using MeasureTheory
julia> m = Normal(2,3)
Normal(μ = 2, σ = 3)
julia> d = Dists.Normal(2,3)
Distributions.Normal{Float64}(μ=2.0, σ=3.0)
We generally prefer logdensity over logpdf, because probability densities are a very special case. And we generally leave off the normalization constant. Well, “leave off” isn’t quite right, it’s really that we change the base measure to make the computation more efficient.
Currently, logdensity of a Distribution just calls logpdf:
julia> logdensity(d, 0)
-2.2397730440950046
julia> Dists.logpdf(d, 0)
-2.2397730440950046
and vice versa:
julia> logdensity(m, 0)
-1.3208345108903319
julia> Dists.logpdf(m, 0)
-1.3208345108903319
This last one is really not correct, we should instead do
julia> logdensity(m, Lebesgue(ℝ), 0)
-2.2397730440950046
I mean, it’s correct as a log-density, but not as a logpdf a user might expect to be compatible with Distributions.jl.
Because the density of m with respect to Lebesgue(ℝ) integrates to one. But there’s not always a base measure that makes this true. And even if we do know this base measure, bringing the constant back in will always slow down the computations. Anyway, please let me know if you have ideas for a better way for the interface between these to be set up.
Calling MeasureTheory from Distributions
Of course we can also go the other way:
julia> using Distributions
julia> import MeasureTheory
julia> m = MeasureTheory.Normal(2,3)
Normal(μ = 2, σ = 3)
julia> d = Normal(2,3)
Normal{Float64}(μ=2.0, σ=3.0)
julia> MeasureTheory.logdensity(m, 0)
-1.3208345108903319
julia> logpdf(m, 0)
-1.3208345108903319
julia> MeasureTheory.logdensity(d, 0)
-2.2397730440950046
julia> logpdf(d, 0)
-2.2397730440950046
julia> MeasureTheory.logdensity(m, MeasureTheory.Lebesgue(MeasureTheory.ℝ), 0)
-2.2397730440950046