Cdf of multivariate normal in distributions.jl

I’m trying to use the cdf of a multivariate normal distribution using Distributions.jl.

d = Normal(0,1)
d2 = MvNormal([0.;0.], [1. 0.;0. 1.])
x = [1. ; 2.]
e1 = cdf(d,x[1])
e2 = cdf(d,x[2])

works (as long as the mean vector and the var-cov matrix for the multivariate normal are Floats ), I get an error whenever I write
e3 = cdf(d2,x)
Here is the error message:

ERROR: MethodError: no method matching cdf(::Distributions.MvNormal{Float64,PDMats.PDMat{Float64,Array{Float64,2}},Array{Float64,1}}, ::Array{Float64,1})
Closest candidates are:
  cdf(::Distributions.Distribution{Distributions.Univariate,S<:Distributions.ValueSupport}, ::AbstractArray{T,N}) at /Applications/

Does the cdf function support multivariate distributions? I couldn’t find anything on this in the documentation of Distributions.jl.

No, there is not a cdf for the multivariate normal. Generally, it is a slightly complicated computation. We have some code evaluating the bi- and trivariate case but it hasn’t been used for a long time. For some time, I’ve wanted a dedicated package for multivariate distributions that would be using StaticArrays for storage. The cdf code for fit well in such a package.


That’s a shame. We really should have something in Julia for this. I’ll work around it using PyCall for the time being.

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Are you mainly interested in 2D and 3D or higher dimensions? Do you know how higher dimensions are handled in Python?

My interest is currently for 2D. I have no idea how higher dimensions are handled in Python right now but hopefully I’ll learn a little bit about it in the next few days.

If you are interested in spending a little time on it, we could probably get the 2D Julia version working again. If we do it with StaticArrays it will also be extremely fast.

I am also interested in a julia version for the cdf for 2D normal. I might be able to spend a bit time on this, but where can I find the version that exists already? I am not familiar with StaticArrays though, but I can certainly give it a try.

It is here

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I think it would be very helpful to have a CDF defined for MvNormal, analog to MATLAB’s mvncdf. It would definitely help when dealing with truncated Gaussian priors, etc.

I’m working on it (shh…)

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I was surprised to find that there is a whole book about this:

I have the bivariate case in


Ah, I did not know we had that!

Yes, Prof. Genz literally “wrote the book on it”!

I apologize if this is a necro-bump, but came across this thread as the first Google Search entry.

Has the CDF for Multivariate normals been implemented in Julia now? I essentially looking for an analogous to just sample the CDF of a normal distribution, maybe something akin to R’s pnorm.

Edit: Found what I wanted with the combination of Univariate Distributions · Distributions.jl and Univariate Distributions · Distributions.jl


julia> dist = Distributions.Normal(0, 1)
Distributions.Normal{Float64}(μ=0.0, σ=1.0)

julia> Distributions.cdf(dist, 3/640) - Distributions.cdf(dist, -1/128)

MvNormalCDF.jl at GitHub - PharmCat/MvNormalCDF.jl: Quasi-Monte-Carlo numerical computation of multivariate normal probabilities

Thanks for the link!

Looks like here is the open issue: CDF of Multivariate Normal Distribution · Issue #260 · JuliaStats/Distributions.jl · GitHub