Simplest Way to Generate an Assymetric Gaussian?

numpyro
1

I feel like this should be easy and I’m just missing something, but what is the most straightforward way to define a skewed Gaussian distribution, i.e.

x \sim \frac{1}{\sqrt{2\cdot\pi\cdot(\frac{\sigma_1+\sigma_2}{2})^2}} \begin{cases} \exp(\frac{-1}{2}(\frac{x}{\sigma_1}))^2, x\lt 0 \\ \exp(\frac{-1}{2}(\frac{x}{\sigma_2}))^2, x \ge 0 \end{cases}

There doesn’t appear to be a standard distribution for this. The approaches I can think of are:

  1. Use some transformation to adjust x where x<0
  2. Define a new distribution
  3. Adjust the values after sample, i.e.
x = sample('x', dist.Normal(0,sig))
if x<0: x*=sigratio

Any direction as to the best way to approach this?

2

it depends what you’re doing but if you just need this distribution in the model and don’t need to sample from the distribution you only need to define log_prob in the custom distribution. and indeed the easiest route is to just use a factor statement with a custom log_prob. to compute the latter you can use distributions.Normal(...).log_prob(...) to make it readable. but #3 should also work fine if you use jnp.where. #1 would probably be complicated

p.s. your normalization constant looks wrong

3

I’m trying to constrain the values of \sigma_1 and \sigma_2 using MCMC or SVI, or at least find the “best” value. Specifically, I want to combine this with another continuous distribution via MixtureGeneral, so I need a method that offers the same support.
Obviously I could use factor to do this manually, but I try to avoid this hack as a last resort as it can play strangely with SVI.

Ideally, I want something like:

def model(Y,E):
    sig1 = sample('sig1', [SOME PRIOR])
    sig2 = sample('sig1', [SOME PRIOR])
    mean = sample('sig1', [SOME PRIOR])

    [SOME WORKING]
    with plate('data', len(Y)):
        sample('y', [SOME DISTRIBUTION])

sampler = MCMC(NUTS(model))
sampler.run(PRNGKey(1))

There is obviously also the issue of the gaussian errors being randomly distributed as well, which makes things a little messier.

4

If you want a fully fledged distribution your best bet is probably to create a custom transform roughly along the lines of AffineTransform