Consistent and transparent handling of 0 predictions in count noise.

[dylanhmorris]

I think make this configurable or constrain PoissonNoise to have strictly positive (versus non-negative) predictions and require the user to choose it.

Originally posted by @dylanhmorris in #644 (comment)

[damonbayer]

Why do you need to enforce any special behavior? The numpyro poisson distribution accepts a 0 rate parameter, which results in a degenerate distribution, which seems desireable.

[dylanhmorris]

log prob of a 0-mean Poisson evaluated at 0 is nan rather than 0.

Other cases: log prob of a non-zero-mean poisson evaluated at zero is defined, as expected. log prob of a zero-mean poisson evaluated at a non-negative integer is -inf. Log prob of a zero-mean poisson evaluated at a negative integer is nan.

import numpyro
import jax.numpy as jnp
my_poisson = numpyro.distributions.Poisson(jnp.array([1.5, 0.32, 0, 0, 0]))
my_poisson.log_prob(jnp.array([2, 0, 0, 1, -1]))
# Array([-1.3822172 , -0.32000047,         nan,        -inf,         nan],      dtype=float32)

[damonbayer]

To relate it back to my previous comment, you are saying that Poisson(0) doesn't actually work like a degenerate 0 distribution, even though you can sample from it like it is. Therefore, we shouldn't allow it.

>>> import jax.numpy as jnp
>>> import numpyro
>>> 
>>> test_values = jnp.array([-1, 0, 1])
>>> numpyro.distributions.Poisson(0).log_prob(test_values)
Array([ nan,  nan, -inf], dtype=float32)