## From “Mathematical simplicity is not always the same as conceptual simplicity” to scale-free parameterization and its connection to hierarchical models

I sent the following message to John Cook:

This post popped up, and I realized that the point that I make (“Mathematical simplicity is not always the same as conceptual simplicity. A (somewhat) complicated mathematical expression can give some clarity, as the reader can see how each part of the formula corresponds to a different aspect of the problem being modeled.”) is the kind of thing that you might say!

Cook replied:

On a related note, I [Cook] am intrigued by dimensional analysis, type theory, etc. It seems alternately trivial and profound.

Angles in radians are ratios of lengths, so they’re technically dimensionless. And yet arcsin(x) is an angle, and so in some sense it’s a better answer.

I’m interested in sort of artificially injecting dimensions where the math doesn’t require them, e.g. distinguishing probabilities from other dimensionless numbers.

To which I responded:

That’s interesting. It relates to some issues in Bayesian computation. More and more I think that the scale should be an attribute of any parameter. For example, suppose you have a pharma model with a parameter theta that’s in micrograms per liter, with a typical value such as 200. Then I think we should parameterize theta relative to some scale: theta = alpha*phi, where alpha is the scale and phi is the scale-free parameter. This becomes clearer if you think of there being many thetas, thus theta_j = alpha * phi_j, for j=1,…,J. The scale factor alpha could be set a priori (for example, alpha = 200 micrograms/L) or it could itself be estimated from the data. This is redundant parameterization but it can make sense from a scientific perspective.