feat: Implementing the PartitionedDistributions interface

[sethaxen]

From @lrnv at JuliaStats/Distributions.jl#418 (comment):

@sethaxen A few questions:

1. Have you seen what we did in Copulas.jl on the same subject ?

2. Do you have specific bindings for multivriate distribution simplemented in Distributions.jl, or generic paths for any multivariate dist, or both ?

3. Do you think we could implement some package extension to allow our interface to use your methods and/or vice-versa ?

RE (3), yes, I think implementing either/both interfaces in an extension would be very straightforward. To me it makes more sense to implement the PartitionedDistributions interface (in particular conditional/marginal) for Copula's distributions than the other way around (since I would guess any package using your interface does so because they assume a copula structure). The overloads could live anywhere, but I also think it makes more sense to be an extension here.

I'm working on a refactor to make PartitionedDistributions a little easier to extend, so if this plan sounds good, I can open a PR here with an extension when it's ready.

A few comments/notes:

• PartitionedDistributions.conditional takes as arguments a point x in the support as well as the indices keep of the elements that remain in the support after conditioning. This is because for higher-dimensional array-variate distributions (Matrix-variate, etc), I found it much easier to type-inferrably determine the type of the output distribution if we can infer it directly from the user-provided input x[keep]. These indices are the inverse of those accepted by Copulas.condition.
• I believe PartitionedDistributions.marginal and Copulas.subsetdims describe the same operation.
• PartitionedDistributions.pointwise_conditional_logpdfs!! has a fallback method that works for anything that implements conditional, but often a more efficient implementation is available. I wonder if there's anything in your API or internal functions that would support this.

[lrnv]

I'm working on a refactor to make PartitionedDistributions a little easier to extend, so if this plan sounds good, I can open a PR here with an extension when it's ready.

Yes please do, tell me if i can be of any help. Your conditional() function does exactly the same thing as our condition() function, with arguments expected reversed from each others as you noted already. Moreover, you deal with any dist::Distributions.Distribution{Distributions.ArrayLikeVariate{N}} while we only have dist::Union{Copula,SklarDist} which is more specific to our package. It would make sense to :

• Either unify the interface so that we can agree on it, but looks like you prefer yours and i prefer mine ;)
• Or, we could simply extend your interface to dispatch to mine when dist::Union{Copula,SklarDist} and mine to dispatch to yours when not dist::Union{Copula,SklarDist}. That would allow users to pick the interface they want and have it working on everything.

Same thing for marginal() and subsetdims(): will be easy to have them dispatch to each others when needed.

For partial logpdf, our fallback is the _partial_cdf() function there that simply leverages ForwardDiff and is not very smart. The efficient way of doing this operation on copula models is the fallback logpdf(condition(...),...) as it will dispatch to the right thing, so i'd say that no, we do not have fast-paths for this operation.

Last but not least, for the elliptical copulas (Gaussian and T), I am basically using the conditionning of Gaussian and T random vectors, and i bet your implementation is better than mine... so maybe there is something to do here too, but this can wait a second round.