GP Periodic Kernel uses uncommon formula

[AlexAndorra]

As discussed on the Discourse, PyMC3's Periodic kernel is implemented as:

But, in other sources such as David Duvenaud’s Kernel Cookbook, as well as the NB for mean and cov functions, the formula is given by:

According to @bwengals, this changes the interpretation of the lengthscale: a constant factor changes how you think of the lengthscale and what priors you’d set on it.
So, ideally, PyMC3's implementation should be changed to the second equation above -- note that this would not be backwards-compatible then, so it should be accompanied by a warning. Does anybody disagree?

A final note: it seems like we’re using another slightly different implementation in PyMC4:

@tirthasheshpatel do you confirm it is equivalent to the standard formula? I notice there is a new sigma ** 2 parameter and the distance between x and x' is squared 🤔

[tirthasheshpatel]

@tirthasheshpatel do you confirm it is equivalent to the standard formula? I notice there is a new sigma ** 2 parameter and the distance between x and x' is squared 🤔

Where did you see this equation in PyMC4? Is it present in the docs or notebooks? Actually PyMC4 doesn't use this. Instead, it uses the implementation from the kernel cookbook. And yes, the sigma ** 2 is the amplitude which was implemented in PyMC3 by externally multiplying the kernel with it.

[tirthasheshpatel]

Oh sorry! It is present in the notebooks. I will change that. Thanks for noticing!

[AlexAndorra]

Yes, it was in the NB with all the cov functions, in your PR.
FWIW, I find it better to integrate to amplitude parameter into the kernel formula, as we do in PyMC4 👍