DOC: Covariance matrix for ZeroSumNormal incorrect?

[covertg]

Issue with current documentation:

The docs state that ZSN(sigma) = N(0, sigma^2 (I - (1/n)J) where n is the nbr of zero-sum axes".

It had been my understanding that the marginal variance of any one dimension of a ZSN variable would be sigma^2 (1 - 1/K). And the covariance between any two (different) dimensions would be, I think, -sigma^2/K. (This is supposing K dimensions and just 1 zero-sum axis.)

I could be wrong, but to the best of my understanding I think that this indicates that 1/K should be the scaling term on J, not 1/n. (And then the notation will get messier if expressing multiple zero-sum axes.)

I'd be glad to submit a PR on the matter, but wanted to confirm the issue first, as I'm not sure I 100% understand the statistical machinery going on for pymc's ZSN implementation.

Idea or request for content:

No response

[ricardoV94]

@aseyboldt @lucianopaz

[lucianopaz]

The mathematical expression written there makes sense for a 1 dimensional zero sum normal. In that 1D case, n is the number of elements in the single dimension. The covariance matrix ends up being sigma^2 (I - (1/n)J), where I is the identity matrix of rank with length n, and J is a square 2D matrix with n rows and columns that's filled with ones.

I guess that the terms "axes", "dimension", and elements of an ND-array's axis or dimension are confusing. The text should not talk about the nbr of zero-sum axes. It should talk about the size of the zero-sum axes instead.

Now, what you say about the different dimensions is wrong in the sense that we take the ND-arrays' axis and dimensions to mean the same thing. In code, what you say is that K = array.ndim, but that's wrong and it should instead be K = array.shape[zero_sum_axes]. The mathematical representation of the zero sum normal is a bit different when we have more than a single zero summed dimension, and I don't think it's worth it to include that into the docstring because it will be more confusing than the standard 1D case.