Sequence InvariantMonoidals

[vinayakpathak]

I don't think we need F[_] to be an Applicative to be able to sequence over it. InvariantMonoidal is all we need.

[joroKr21]

Hmm, that is interesting. Thanks for the PR 👍
This is not binary compatible however. Instead of changing the current implicit definitions you can just add new ones (at lower priority).

[vinayakpathak]

Yes, ok, I wasn't sure how to take care of the binary incompatibility. I'll try what you suggest and update

[vinayakpathak]

Ok, I've updated it. Seems like it's binary compatible now. I also added a few tests to test just the InvariantMonoidal part.

I'm not 100% sure that the way I've split things into different priorities is right. Please let me know if I've messed something up.

[joroKr21]

The inheritance relation between MkHNilSequencerLowPrio and MkHConsSequencerLowPrio also has to be inverted.

Thanks for the contribution! This is really complex stuff, but the use case with Semigroup looks really nice (although we already have derivation for that particular type class).

[joroKr21]

I made the changes because I'd like to merge and release 2.2.0 this weekend

[vinayakpathak]

Oh, thanks for merging this!

My original motivation for this was to be able to write some appropriate typeclass instances for probability distributions. So you can define a Rand[T] class to represent a probability distribution and then you can define a monad instance for Rand. Combined with kittens, this gives you a great way to define joint distributions. For example, if you have a Rand[A] and a Rand[B] you can use the type-level sequence to get a Rand[A :: B :: HNil] for free, which is awesome!

However, with the previous code, this would only work if you had an Applicative instance for Rand. When doing probabilistic modelling, you face situations where all you have is a Density[A] (i.e., an unnormalized probability distribution) that simply gives you a def apply(a: A): Double. And now you want to be able to define joint densities instead. It is fairly straight-forward to define it mathematically (density over A and B is simply the product of density over A and density over B), but Density is not an applicative. This is what prompted me to make this change.

Ideally I would write my tests in terms of Density but that's not a data type in cats, so I decided to go with Semigroup instead.