# Can a frequentist 'sampling distribution' be interpretted as a bayesian 'prior'?

#### gnutrol

##### New Member
I've read quite a few posts comparing/contrasting bayesian and frequentist statistics. But there's one thing that I'm still struggling to understand. Based on the following premises (which I think are correct):

Premise 1: A 'sampling distribution' of a statistic (e.g., a sample mean) is a piece of knowledge that tells us what we should expect a statistic to be (given some null hypothesis)
Premise 2: A bayesian 'prior distribution' is a piece of knowledge that we use to describe our degree of belief in a particular parameter of some model/hypothesis (e.g., a particular mean of our data?).

Is it incorrect to assume that a frequentist sampling distribution is just a prior on the mean value (or any particular statistic) of the null model?

If this isn't correct, what makes the sampling distribution different from a bayesian prior distribution?

#### noetsi

##### No cake for spunky
I am not an expert to answer this, but my understanding about frequentist and bayesians is they have different understandings of reality so their approaches are really not compatible. Since most data types are not true believers in such things most do use both approaches.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I am sure this will be insufficientor clear. Frequentists believe the target parameter is fixed and data are random. While Bayesian believe the flip, that data are fixed and parameter is random. Since the parameter varies or the belief in its value, the prior is just the existing info on its possible value and its variability incorporated into Bayes rule.

Side comment, I typically use published frequencies results as my Bayesian priors.

This thread as some nuggets of information:
https://stats.stackexchange.com/que...st-interpretations-of-probability#:~:text=The frequentists view is that the data is,event as the number of trials approaches infinity%29.

#### gnutrol

##### New Member
I am not an expert to answer this, but my understanding about frequentist and bayesians is they have different understandings of reality so their approaches are really not compatible. Since most data types are not true believers in such things most do use both approaches.
good point. though, I'm curious if -- disregarding their fundamental beliefs about the philosophical definition of "probability"-- there is a way to demonstrate frequency nhst tests as a special case of bayesian analysis with a particular choice of prior (the sampling distribution).

#### gnutrol

##### New Member
I am sure this will be insufficientor clear. Frequentists believe the target parameter is fixed and data are random. While Bayesian believe the flip, that data are fixed and parameter is random. Since the parameter varies or the belief in its value, the prior is just the existing info on its possible value and its variability incorporated into Bayes rule.
That explanation makes sense. So is there a "class" of analyses that supersedes both Bayesian and Frequentist methods?

Side comment, I typically use published frequencies results as my Bayesian priors.
Interesting! Is that standard practice? It seems like an advantage of frequentists sampling distribution is that it is empirically motivated <-- but if priors were empirically motivated as well, that seems like a solution to the "subjectivity" criticism I've seen leveraged against bayesian analysis.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
On your second statement, yes this help defend selections. Also people typically run the model also with flat prior as a form of sensitivity analysis - to show the influence of the informative priors on posterior.

#### gnutrol

##### New Member
On your second statement, yes this help defend selections. Also people typically run the model also with flat prior as a form of sensitivity analysis - to show the influence of the informative priors on posterior.
intresting; so like a, "did it even matter in the first place" demonstration