Post date: Jul 07, 2015 6:24:59 PM
We might think that the mean and the standard deviation are THE parameters of a normal distribution, but this is ridiculous of course. Given any non-degenerate family of distributions, there are generally infinitely many ways to index individual distributions from the family. Instead of mean and standard deviation, we could use the first quartile and the variance. In the case of uniforms, we can index all distributions in terms of their minimum and maximum, but we could also index them in terms of the midpoint and width, or with mean and standard deviation. Everyone who has to work with gamma distributions is perhaps already aware that there are multiple ways to specify each distribution, and that these different ways have different utilities in different contexts.
So this suggests an obvious question about any estimation process which traffics in a particular parameterization. Certaintly different parameterizations would lead to different c-boxes (because we're estimating different parameters), but are the results subtly inconsistent between different parameterizations? Would they, for instance, lead to different predictive distributions? If they do, is this just another manifestation of the non-uniqueness of c-boxes?