Does MLE give a distribution of the possible parameter values?

Question

I don’t think so, but I’m not sure.

I think MLE selects the best parameter assuming that we have, for each parameter value, the possibility of the current dataset given that value. (But when we don’t actually have all those values, we can still infer the best parameter subject to this criterion from other clues.) These possibilities each belong to a different conditional distribution and don’t even add up to one.

Am I correct?

Is there any possibility that we can get a distribution of possible parameters like in the Bayesian estimation case?

If you're looking for a distribution of probable parameters, you're asking for a Bayesian credible interval. Sometimes, but only sometimes, credible intervals coincide with frequentist confidence intervals, which would allow you to interpret them as probability distributions of parameters. — Durden, May 29 '23 at 22:24
An interval is not a distribution. A (possibly marginal) posterior is a distribution. — Glen_b, May 29 '23 at 23:06

score 8 · Accepted Answer · answered May 29 '23 at 19:07

8

A maximum likelihood estimator (MLE) gives a point estimate of a parameter. In that regard, no, you do not get a distribution.

What might be reasonable to regard as a frequentist analogue of a Bayesian posterior distribution is the sampling distribution, but you get a sampling distribution from any frequentist estimation technique, not just MLE. Ways of obtaining sampling distributions include calculations based on theory to bootstrapping the data and calculating an estimate for each bootstrap sample.

I would not want to interpret a sampling distribution as a posterior distribution, however, as parameters in frequentist statistics have fixed (unknown) values instead of distributions like they do in Bayesian statistics.

answered May 29 '23 at 19:07

Dave

62,186

So the sampling distribution represents the distribution of “from-sample” estimation probabilities of a fixed value, while the Bayesian posterior distribution models the possibilities of an unknown value. Am I correct? – Yan Yang May 29 '23 at 19:26
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There are also confidence distributions which are related to fiducial stuff and kind of make a Bayesian omelette without breaking the egg; see Schweder & Hjort "Confidence, Likelihood, Probability" (2016). – Richard Hardy May 30 '23 at 10:22

Does MLE give a distribution of the possible parameter values?

1 Answers1