Equivariance

The text I'm prepping from states the Equivariance Principal as a data reduction technique. While formally true, I'm not sure I'm buying that characterization of it. Not because I think the math is bogus (I don't), but simply because it does not fit my subjective view of data reduction. Equivariance essentially reduces the number of conclusions, not the actual data. Splitting hairs? Perhaps.

Anyway, it's actually two principals. One is measurement equivariance. This means that if you measure in inches and then form your estimator in inches, you should come up with the same answer as you measured in meters and formed your estimator in meters and then converted that to inches. Just about everybody agrees on that part, though it's never absolutely true in real life. The measurement system will impact both the precision and accuracy of the measurements. However, if we except that we are getting the same answers within the limits of our precision, that's good enough.

The second one is a little funky, but it's where the reduction comes from. This is the principal of Formal Invariance. It basically says that if the model for your sample space and distribution are the same, the means of reaching a conclusion should be the same, regardless of what the model is being applied to. So, for example, if you choose to estimate the probability of a coin coming up heads by applying some function to a series of coin tosses, you should be able to apply that same function to the inverse (where "success" is now getting tails) and wind up with the inverse of your estimator (that is, 1 - your estimator for heads)

Mathematically, a set of transformations from a sample space onto itself that obeys these two properties must form a group. That is, they are closed under inverse and composition. That's generally a pretty easy thing to prove.

OK, so where's the rub? Well, just because the math works out doesn't make it so. Models are just that, models. They aren't the real thing. Conclusions that work for some parameters make no sense for others, even when the model used is identical. You could model the height of five-year-old's as a normal random variable. It's not a perfect fit, but it's not terrible either. But, just because the left side of the tail is chopped (it can't possibly go below 10 inches, even though the normal distribution would assign some very small, but positive, probability to that), doesn't mean that some other variable modeled as a normal random variable also has a chopped left tail. The limiting of the set of conclusions based on other conclusions from similar models only works if the model exactly describes the reality. And, that's never true.

So, I'm calling shenanigans on this one.