multiply_log(0, 0) has infinite gradient

[jsocolar]

Description

For more details, see https://discourse.mc-stan.org/t/non-linear-modelling-michaelis-menten/22736/8

Example

The gradient for poisson_lpmf(0 | lambda) is infinite at lambda = 0.

Suggested fix

@nhuurre writes on discourse:

We should just special-case a=0.0 like we do with a=1.0

Current Version:

v4.0.1

[SteveBronder]

Ah interesting, it looks like for reverse mode we don't check this, but we do when working with doubles so we def should be doing it for reverse mode!

https://github.com/stan-dev/math/blob/develop/stan/math/rev/fun/multiply_log.hpp

[mhollanders]

Any chance this could be addressed? I think accounting for this is making formulating complex HMMs on the log scale a bit more verbose than it needs to be.