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The frequentist and Bayesian analyses give different answers to the central question. Which one is more correct?

by te 13y ago

thearn4 13y ago
I would guess that it mostly depends on the quality of the prior.
- deleted 13y ago
  [deleted]
_delirium 13y ago
There isn't really 'a' correct frequentist or Bayesian answer to the problem; it's more two different ways of thinking about the problem, which could well get you the same numerical results (though they might not). The frequentist way of thinking about it is to ask what you mean by "more correct", i.e. what properties do you want an optimal estimator to have? Another way of putting this is: if you were to set up a simulation where the real answer is known and data is sampled, and then you judge estimators by how close they get (according to some penalty function scoring closeness) when you run this simulation 10,000 times, which estimator would score the best? The estimator with the minimum variance of all unbiased estimators (the MVUE) will do optimally under some definitions of optimal; the MLE is another one that is optimal for other definitions. Note that they're both frequentist and give different answers. The Bayesian analysis of the situation is that it basically comes down to your choice of prior: the observed information is not in itself sufficient to produce a single "best" estimate, but rather you combine it with your prior distribution to produce an estimate. The Bayesian could end up producing exactly the same estimate as the estimators this article labels "frequentist". The Bayesian argument would be that what these estimators are really doing under the language of MVUE/MLE/etc. is implicitly choosing priors, whereas the Bayesian would explicitly choose one. The Bayesian would also probably not really like the simulation-experiment idea (which is a pretty directly frequentist thought experiment).
- bazzargh 13y ago
  The Bayesian argument would be that what these estimators are really doing under the language of MVUE/MLE/etc. is implicitly choosing priors I'm curious, which choice of priors corresponds to the frequentist answer? It looks like it comes down to the distribution (n|k)? Seems like something that must've been studied.

The frequentist and Bayesian analyses give different answers to the central question. Which one is more correct?

by te 13y ago

The frequentist and Bayesian analyses give different answers to the central question. Which one is more correct?

thearn4 13y ago

I would guess that it mostly depends on the quality of the prior.

deleted 13y ago
[deleted]

_delirium 13y ago

There isn't really 'a' correct frequentist or Bayesian answer to the problem; it's more two different ways of thinking about the problem, which could well get you the same numerical results (though they might not). The frequentist way of thinking about it is to ask what you mean by "more correct", i.e. what properties do you want an optimal estimator to have? Another way of putting this is: if you were to set up a simulation where the real answer is known and data is sampled, and then you judge estimators by how close they get (according to some penalty function scoring closeness) when you run this simulation 10,000 times, which estimator would score the best? The estimator with the minimum variance of all unbiased estimators (the MVUE) will do optimally under some definitions of optimal; the MLE is another one that is optimal for other definitions. Note that they're both frequentist and give different answers. The Bayesian analysis of the situation is that it basically comes down to your choice of prior: the observed information is not in itself sufficient to produce a single "best" estimate, but rather you combine it with your prior distribution to produce an estimate. The Bayesian could end up producing exactly the same estimate as the estimators this article labels "frequentist". The Bayesian argument would be that what these estimators are really doing under the language of MVUE/MLE/etc. is implicitly choosing priors, whereas the Bayesian would explicitly choose one. The Bayesian would also probably not really like the simulation-experiment idea (which is a pretty directly frequentist thought experiment).

bazzargh 13y ago
The Bayesian argument would be that what these estimators are really doing under the language of MVUE/MLE/etc. is implicitly choosing priors I'm curious, which choice of priors corresponds to the frequentist answer? It looks like it comes down to the distribution (n|k)? Seems like something that must've been studied.