Mixing up of $\Xi$ and $\Psi$ in priors vignette

[mikemc]

The inverse Wishart is first defined with $\Xi$ as the scale matrix parameter,

fido/vignettes/picking_priors.Rmd

Line 96 in 1b33c57

$$\Sigma \sim W^{-1}(\Xi, \upsilon)$$

but the subsequent line and equation calls it $\Psi$. Then, starting in the next section, $\Psi$ starts to instead be used for the linear transformation that defines a compositional transformation. So perhaps the first two uses of $\Psi$ are typos for $\Xi$? I think it would help a lot to consistently use $\Xi$ and $\Psi$ to mean distinct matrices - e.g., $\Xi$ for the scale matrix, and $\Psi$ for the transformation matrix. I guess the benefit of $\Psi$ for the scale matrix is that is what is used in Wikipedia (which I'd guess is the source of the original typo).

[jsilve24]

Thanks for finding this! You are correct there were two times where I use \Psi but meant \Xi.

Fixed in: 1340768

Please let me know if you find any other errors or if I have not found all the typos you were referring to.

If all good, you can go ahead and close the issue.

[mikemc]

I will record some other possible mistakes in this vignette here so that they can be corrected together.

1. The Inverse Wishart is first introduced as having a parameterization with the scale matrix first, then the degrees of freedom parameter. But subsequent lines seem to be using a flipped order, e.g.

   fido/vignettes/picking_priors.Rmd

   Line 117 in 1340768

   It turns out that if we have $\Omega \sim W^{-1}(\gamma, S)$ for a $D\times D$ covariance matrix $\Omega$ then for $M\times D$ matrix $\Psi$ we have $\Psi \Omega \Psi^T \sim W^{-1}(\upsilon, \Psi S \Psi^T)$.

2. Also in

   fido/vignettes/picking_priors.Rmd

   Line 117 in 1340768

   It turns out that if we have $\Omega \sim W^{-1}(\gamma, S)$ for a $D\times D$ covariance matrix $\Omega$ then for $M\times D$ matrix $\Psi$ we have $\Psi \Omega \Psi^T \sim W^{-1}(\upsilon, \Psi S \Psi^T)$.

   it seems that there is a switch from $\gamma$ to $\upsilon$ that isn't acknowledged/explained - is that a typo?

[jsilve24]

Wonderful!
(1) you are correct
(2) I think you are write calling this \upsilon would be better. I had distinguished gamma and upsilon only because they are different priors and so I should have really added upsilon = gamma.

Will fix these in a few hours.