Why there is no uniform prior for Box-Cox Power Transformed Normal Models

Question

I am trying to get intuition why uniform prior like below will not work for the box-cox model.

Box-cox model: $y^{(\phi)}_i \sim N(\mu, \sigma^{2})$ where

$y^{(\phi)}_i = (y^{\phi}_i-1)/\phi $ if $\phi \ne 0$ Or $ln$ $y_i$ otherwise

For the above transformation $P(\mu,\sigma,\phi) \propto 1/\sigma $ is not an uniform prior. Why? I have a vague sense that if original $y_i$ is scaled, then the prior is no longer uniform. I tried to read original Box-Cox(1964) paper but failed to get correct intuition.

Answer 1

The problem with that "model" is that it is'nt really, really a model! This was discussed at length in McCullaghs (2000) paper "What is a statistical model?" http://projecteuclid.org/euclid.aos/1035844977 which I am not going to review here. (He finds that to answer the question, he needs quite advanced mathematics, specifically, category theory). So I will try to give a simpler explanation of that (which basically is the reason given by Box and Cox themselves).

When you vary the transform parameter $\phi$, then you change the measurement scale of $y$ itself. Lets say that the original data $y$ (corresponding to transform parameter $\phi=1$) is measured in meter (m). Then, when $\phi=1/2$, $y$ is square root of meter (whatever that may be!) and if $\phi=2$ y is now square meter! So, not only the scale, but the meaning of the two other parameters $\mu, \sigma$ change when $\phi$ is changing (That is, their unit of measurement change). This argument could be seen as an application of dimensional analysis, an underused tool in statistics.

So, maybe we should say that the "boxcox model" is not a model, but a family of models indexed by $\phi$. If you want a Bayesian treatment of this (probably not a very good idea?) you must give separate prior distributions for $\mu, \sigma$ for each value of $\phi$ (they should of course be correctly related). Say you have a prior of the form $\propto 1/\sigma$ when $\phi=1$, then for other values of $\phi$ the prior distribution could be calculated (try it!) and it would be quite different.

If that is worth it, or practical, I do not know, but it could be interesting to try.