[_alternator_]

Thank you. The main problem with Bayesian statistics is that if the outcome depends on your priors, your priors, not the data determine the outcome.

Bayesian supporters often like to say they are just using more information by coding them in priors, but if they had data to support their priors, they are frequentists.

[kgwgk]

If they were doing frequentist inference they wouldn’t be using priors at all and there is nothing frequentist in using previous data to construct prior distributions.

[uoaei]

Not true. In frequentist statistics, from the perspective of Bayesians, your prior is a point distribution derived empirically. It doesn't have the same confidence / uncertainty intervals but it does have an unnecessarily overconfident assumption of the nature of the data generating process.

[kgwgk]

Not true. In frequentist statistics, from the perspective of Bayesians and non-Bayesians alike, there are no priors.

—-

Dear ChatGPT, are there priors in frequentist statistics? (Please answer with a single sentence.)

No — unlike Bayesian statistics, frequentist statistics do not use priors, as they treat parameters as fixed and rely solely on the likelihood derived from the observed data.

[zozbot234]

There's always priors, they're just "flat", uniform priors (for maximum likelihood methods). But what "flat" means is determined by the parameterization you pick for your model. which is more or less arbitrary. Bayesians would call this an uninformative prior. And you can most likely account for stronger, more informative priors within frequentist statistics by resorting to so-called "robust" methods.

[_alternator_]

First, there is not such thing as a ‘uninformative’ prior; it’s a misnomer. They can change drastically based on your paramerization (cf change of variables in integration).

Second, I think the nod to robust methods is what’s often called regularization in frequentist statistics. There are cases where regularization and priors lead to the same methodology (cf L1 regularized fits and exponential priors) but the interpretation of the results is different. Bayesian claim they get stronger results but that’s because they make what are ultimately unjustified assumptions. My point is that if they were fully justified, they have to use frequentist methods.

[kgwgk]

One standard way to get uninformative priors is to make them invariant under the transformation groups which are relevant given the symmetries in the problem.

[kgwgk]

It’s not true that “there are always priors”. There are no priors when you calculate the area of a triangle, because priors are not a thing in geometry. Priors are not a thing in frequentist inference either.

You may do a Bayesian calculation that looks similar to a frequentist calculation but it will be conceptually different. The result is not really comparable: a frequentist confidence interval and a Bayesian credible interval are completely different things even if the numerical values of the limits coincide.

[zozbot234]

Frequentist confidence intervals as generally interpreted are not even compatible with the likelihood principle. There's really not much of a proper foundation for that interpretation of the "numerical values".

[kgwgk]

If you want to say that when you do a frequentist analysis which doesn’t include any concept of prior you get a result that has a similar form to the result of a completely different conceptually Bayesian analysis which uses a flat prior (definitely not “a point distribution derived empirically”) that may be correct. It remains true that there is no prior in the frequentist analysis because they are not part of frequentist inference at all.

[uoaei]

Priors are not used in construction of frequentist approaches, but that does not mean that the analyses aren't isomorphic in theory.

Point distribution <=> point estimate as a sample from an initially flat distribution. A priori vs a posteriori perspectives, which are equivocal if we are to take your description of frequentist statistics into account ;)

[kgwgk]

It’s not my description of frequentist statistics. It’s the frequentist statisticians’ description. This is from Wasserman’s All of Statistics:

The statistical methods that we have discussed so far are known as frequentist (or classical) methods. The frequentist point of view is based on the following postulates:

F1 […]

F2 Parameters are fixed, unknown constants. Because they are not fluctuating, no useful probability statements can be made about parameters.

F3 […]