Minor notational point

[sethaxen]

In the paper, we define the $N$-simplex as being an $N+1$-length vector. In our model implementations, we use $N$ to refer to the number of elements in the vector on the simplex. We should harmonize these notations.

[mjhajharia]

made a comment and removed it (just me mixing up notation again, which makes the need to make it clear even more important). i keep thinking in terms of $\Delta^N$ and forgetting we're actually using $\Delta^{N-1}$ but defining $\Delta^{N}$ in the first section

@Raoul-Kima if you got a github email, ignore that!

[bob-carpenter]

We want the standard definition of unit simplex,

$\Delta^N = \{ \theta \in \mathbb{R}^{N+1} : \theta_n \geq 0, \textrm{sum}(\theta) = 1 \}.$

Then we just have to keep the dimensions consistent elsewhere. I'm about to take a notational pass through the doc.

[bob-carpenter]

Did everyone decide to just skip over doing the simplex version and go straight to full generality? The intro no longer mentions simplexes and Section 3 currently begins,

In this section, we consider transforms from unconstrained space to constrained vectors, including
unit-simplex, sum-to-zero, and unit-length constraints

Given the change in focus, I didn't want to get in an edit war of trying to change it back, so I just left it.

P.S. I didn't know where to comment with so many issues, so just chose this one at random.

P.P.S. I like the newly framed transforms intro section. That's about as clear as I think we'll be able to be.

[mjhajharia]

Did everyone decide to just skip over doing the simplex version and go straight to full generality? The intro no longer mentions simplexes and Section 3 currently begins,

In this section, we consider transforms from unconstrained space to constrained vectors, including
unit-simplex, sum-to-zero, and unit-length constraints

Given the change in focus, I didn't want to get in an edit war of trying to change it back, so I just left it.

P.S. I didn't know where to comment with so many issues, so just chose this one at random.

P.P.S. I like the newly framed transforms intro section. That's about as clear as I think we'll be able to be.

i thought we agreed on simplex. and not general. i was making rough patches of things about aitchison geometry in general to add, @spinkney adding the ilr as well makes it seem like a good prospect to me - so we get a chance to introduce the transforms idea for hmc in general as well as the compositional data analysis utility point of view is there

yeah i think it looks good enough!

PS - sorry for not responding a lot on the issues, missed 3 weeks of classes so had some catching up to do, will add my changes in a day or two, the cluster had some rotation shift going on, so will check check and then put the other transforms to sample as well.

[mjhajharia]

We want the standard definition of unit simplex,

ΔN={θ∈RN+1:θn≥0,sum(θ)=1}.

Then we just have to keep the dimensions consistent elsewhere. I'm about to take a notational pass through the doc.

yeah i continued this over the email thread - unit and standard simplex have similar definitions and some places use them interchangeably which isnt correct and then there's aitchison's definition too

[bob-carpenter]

The "unit" in "unit simplex" refers to the sum being unity (i.e., 1). In the Wikipedia entry for simplex, "standard N-simplex" and "unit N-simplex" are synonyms that both refer to $\Delta^{N}$. I suggest we just stick to the "unit simplex" terminology so as not to cause confusion.