Question about Gamma and Inverse Gamma logpdf, values outside support

[bratslavia]

I was doing some testing with MixtureModels and logsumexp functions, and I happened to notice a couple of things about the logpdf/loglikelihood functions for Gamma and Inverse Gamma that had me scratching my head.

Specifically, they seem to handle variables outside the support differently. E.g.,

using Distributions

logpdf(Gamma(), 0) # 0.0, but shouldn't this be -Inf?
logpdf(InverseGamma(), 0) # NaN, but again, shouldn't it be -Inf?

logpdf(Gamma(), -1) # -Inf (seems correct)
logpdf(InverseGamma(), -1) # Throws a domain error for "log", but shouldn't it just return -Inf?

Support for both of those distributions is supposed to be the positive reals, right? I would have expected that anything outside the support would just return -Inf for the logpdf.

[NOTE: I realize the Gamma logpdf may be correct -- I discovered that some implementations (e.g., Boost) allow a random variate of zero in the function. But in that case, perhaps the documentation might need to be updated to indicate that the range is [0, Inf), not (0, Inf)? Which is what the wikipedia link in the documentation says, and the way many (most?) implementations work, afaik.]

The same issues seem to affect the overall loglikelihood computation. E.g.,

loglikelihood(Gamma(), [1.0, 0.0]) # Gives -1.0, when I would expect -Inf, given the stated support
loglikelihood(InverseGamma(), [1.0, 0.0]) # Gives NaN, when I would expect -Inf

loglikelihood(Gamma(), [1.0, -1.0]) # Gives -Inf, which seems correct
loglikelihood(InverseGamma(), [1.0, -1.0]) # Domain error, in "log" function

[3f6a]

Another example:

julia> logpdf(Gamma(0.8, 18), 0.0)
Inf

This causes problems for me with inference of Hidden Markov Models.

[andreasnoack]

I hadn't noticed that Wikipedia defines the distribution for x > 0. That surprises me and it also differs from R and Wolfram. IMO, the current behavior for Gamma is fine. Why is it a problem for the linked package?

The situation is different for InverseGamma. Here something needs to be adjusted to avoid the NaN and the exception.

[3f6a]

Note that Wolfram Mathematica gives zero:

In[17]:= PDF[GammaDistribution[0.8, 18], 0]
Out[17]= 0.

[andreasnoack]

Looks like Wolfram is inconsistent here. I don't think it changes the point, though.

For the record then R does use the right limit

> dgamma(0.0, 0.8, 5.0)
[1] Inf