Talk:Invariant estimator

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Equivariant?

Can someone say where the term "equivariant" has been used ... it is not in any of my dictionaries of maths or stats. Melcombe (talk) 09:32, 12 May 2008 (UTC)[reply]

It appears, for example, in Lehmann and Casella, Theory of Point Estimation. --Zvika (talk) 18:17, 12 May 2008 (UTC)[reply]

non-Bayesian?

Should such prominence be given to "non-Bayesian"? After all ideas of invariance can be applied to Bayesian estimation just as well. Consider for example HPD (highest posterior density) estimation (either point or interval estimates), which is not invariant to transformation of the parameters. Melcombe (talk) 09:14, 19 May 2008 (UTC)[reply]

Can you provide a source dealing with Bayesian equivariant estimators? I haven't encountered one. --Zvika (talk) 18:33, 19 May 2008 (UTC)[reply]

I am not sure that it is important to actually find examples of invariant Bayesian estimators, just that the idea of invariance can be applied to estimation in a Bayesian setting, where this would go beyond the impractical definition given in Bayesian estimation since contexts with prescribed loss functions are rare. Thus things like Maximum a posteriori need to be included as Bayesian estimates, and also the expectation of the posterior distribution, both in contexts of no-loss-function. Obviously HPD estimates are not invariant to transformations of the parameter space, and nor are expected values of the posterior, and it is important to be able to say this. Melcombe (talk) 09:41, 30 May 2008 (UTC)[reply]

Hi again Melcombe, nice to hear from you. You raise two separate points.

As I understand it, there is a difference between Bayesian estimation and Bayes estimator. The fact that currently one redirects to the other is simply a result of the fact that we don't yet have an article on Bayesian estimation. Bayesian estimation refers to the general approach of using of prior information (as opposed to frequentist or classical point estimation); this includes MAP, MMSE, etc. By contrast, a Bayes estimator is an estimator which is optimal under the "minimum Bayes risk" criterion (see, e.g., Lehmann and Casella, p.225).
So, as far as this issue is concerned, I think the best solution is to create a new article Bayesian estimation which would discuss the various Bayesian approaches to estimation and provide links to Bayes estimator, minimum mean square error, maximum a posteriori, etc. I would be happy to work with you on such an article. I do not think MAP should be included in Bayes estimator because it is not a Bayes estimator under this definition.
Concerning the relation between invariance/equivariance and Bayes methods: As I said above, the critical issue is to find a reliable source dealing with this question. Bayes estimators might or might not be invariant, but unless you can find a source saying so, including such statements in Wikipedia constitutes original research. I have not myself looked specifically for such a source, so it may well be present in Bayesian textbooks like Berger's. I urge you to go ahead and find such a source; it will definitely improve the article if you do.

Zvika (talk) 17:52, 30 May 2008 (UTC)[reply]