feat: add Levy distribution

[Qazalbash]

This PR adds the implementation of Levy distribution. There is some numerical error in checking PDF integrate to be equal to cdf. I revisited the implementation several times and found no ambiguity. Therefore, I am pushing to seek some guidance.

[YeungOnion]

Hmm, while our check for consistency being an integrator is likely susceptible to the usual nuance of general forward integrators.

The error message, below, indicates that two steps into the integration it's already off by 20%. Since it checks at each step this is pretty significant.

failures:

---- distribution::levy::tests::test_continuous stdout ----
Integral of pdf doesn't equal cdf!
Integration from 1 by 0.003182932192553111 to 1.0063658643851063 = 0.006119699097797403
cdf = 0.005069737329462489

I'm not certain this is the case, but the implementations appear correct to me as well.

Options I expect could be

• inappropriate step size for that part of the function, which I don't think is the case considering it's not terribly steep near the mean
• the erfc implementation isn't sufficiently accurate, but we use that elsewhere, so we would have expected to see it elsewhere
• my first thought, the integrator is insufficient. The current one integrates a linear approximation to the function, and perhaps we need better (but again the function isn't very steep here, so I'm unsure this is the case).

We may need to update the check for continuous distributions to be more robust integration or rely on other constraints. I can open this part as a different issue.

[Qazalbash]

Hi, I am also contributing a little bit to NumPyro, a Python-based PPL. Their test suites are big and quite good. Looking at them may help.

[YeungOnion]

That's a very cool project, along with JAX. I think I may have seen JAX before and forgot about it.

I took a pass through by skimming, searching, and with an LLM. Couldn't find one in that test suite.

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I've also been thinking, perhaps this is a little unnecessary: testing values for the function will isolate precision effects to and precision checking (done by approx).

Perhaps if we had a default impl of PDF for a CDF then this may have been a stronger requirement. But instead, an alternative sanity check might be going the other way and verifying a finite difference in the derivative over some "well-chosen" interval.

@Qazalbash would you be willing to write an internal function analogous to the integrator for this and use that test instead? You could use it standalone in the test and for later development we could see if it works replacing the integrator within internal::check_continuous_distribution

[Qazalbash]

Perhaps if we had a default impl of PDF for a CDF then this may have been a stronger requirement. But instead, an alternative sanity check might be going the other way and verifying a finite difference in the derivative over some "well-chosen" interval.

I am in favor of checking whether the gradient of a CDF equals a PDF. Is there any good autograd we can use?

Qazalbash would you be willing to write an internal function analogous to the integrator for this and use that test instead? You could use it standalone in the test and for later development we could see if it works replacing the integrator within internal::check_continuous_distribution

I will experiment with some, Just to let you know it will not be soon.