[toth]

You make a valid point, but I feel there is something in the direction the article is gesturing at...

The mean of the n-dimensional gaussian is an element of R^n, an unbounded space. There's no uninformed prior over this space, so there is always a choice of origin implicit in some way...

As you say, you can shrink towards any point and you get a valid James-Steiner estimator that is strictly better than the naive estimator. But if you send the point you are shrinking towards to infinity you get the naive estimator again. So it feels like the fact you are implicitly selecting a finite chunk of R^n around an origin plays a role in the paradox...

[kgwgk]

> But if you send the point you are shrinking towards to infinity you get the naive estimator again.

You get close to it but strictly speaking wouldn’t it always be better than the naive estimator?

[toth]

Right, it's a limit at infinity

[rssoconnor]

> There's no uninformed prior over this space, so there is always a choice of origin implicit in some way...

You could use an uninformed improper prior.

[kgwgk]

You would just need to come up with a way to pick a point at random uniformly from an unbounded space.

[rssoconnor]

You can just use the function that is constantly 1 everywhere as your improper prior.

Improper priors are not distributions so they don't need to integrate to 1. You cannot sample from them. However, you can still apply Bayes' rule using improper priors and you usually get a posterior distribution that is proper.

[kgwgk]

Sure.

The point is that you wrote that « you can pick any point […] » and when toth pointed out that « there is always a choice of origin implicit in some way » you replied that « you could use an uninformed improper prior. »

However, it seems that we agree that you cannot pick a point using an uninformed improper prior - and in any method for picking a point there will be an implicit departure from that (improper) uniform distribution.

[rssoconnor]

Oh.

When I said "you can pick any point P", I meant universal quantification, i.e "for all points P", rather than a randomly chosen P.

I did say "choose P", which was pretty bad phrasing on my part.