Bayesian inferential target

Question

In frequentist, i.e., sampling-based statistics, we envision a target population to which inference is made. Notwithstanding the fact that our so-called random samples from this population are usually more convenience-based samples, we try to infer from a sample to the population. For example in a randomized clinical trial we pretend we have a random sample of patients with heart failure and try to make an inference about the mean treatment effect in the world population of heart failure patients.

On the other hand, in Bayesian inference we make probability statements about the unknown mean treatment effect without necessarily speaking about a "population". What is the exact statement of what we are inferring? Is it that the treatment was actually effective in the group of patients we analyzed? Some deeper inference?

Update

It seems that it is safe to phrase the inferential target as a parameter in the underlying data generation process. This is somewhat more general than envisioning a certain human population. This generality allows for one to envision not only making an inference to a population but inference to repeated experiments involving the subjects initially included in the analysis. For example, one might estimate a parameter for the process that generated the observations for a specific set of subjects, were those subjects to be repeatedly studied afresh (as is done in crossover studies with no carryover effects). Such repetitions would observe different random errors for those subjects in measuring their outcomes.

There may be yet a more general way to phrase the target.

Answer 1

"What is the exact statement of what we are inferring? Is it that the treatment was actually effective in the group of patients we analyzed? Some deeper inference?"

I think you are confusing the terms of art with the discussion. One of the challenges of talking about things in multiple paradigms is that the different paradigms may use the same words to define different things, or they may not directly discuss something that is of critical importance to one paradigm, but not the other. Both Frequentists and Bayesians, for example, have a concept called an "expectation," but they both define it in a manner that is nonsensical in the other paradigm.

I think this is what is happening here. Sampling statistics have to concern themselves with the "population," precisely because they work in the sample space. It isn't that a Bayesian does not care, it is that it doesn't impact their calculation on anything as directly.

A second problem is that Bayesian statistics isn't one field as there are multiple axiomatic structures you could use. How you discuss reality may change if you use de Finetti's axioms instead of Cox's. It also could depend upon whether you are an objectivist Bayesian who believes as Frequentists do that population parameters are fixed points, but whose location is unknown, versus subjectivist Bayesians who believe that the population parameter is a distribution that nature draws from and not a fixed point.

Someone like Jaynes, who uses Cox's postulates, would create hypothesis in terms of logical assertions. For example, hypothesis one could be that a drug is non-harmful. Hypothesis two would be that it is harmful. Implicitly, this is a universal statement and hence a population statement. The population is never mentioned.

Both methods depend upon the sample for inference, but a Bayesian can have an infinite number of hypothesis. It is more important for a Bayesian to be clear in what they are asserting and why.

There is one other difference that is important. When you use a Frequentist method you are concerned with the sampling distribution of the statistic and not the sampling distribution of the data. Infinitely many distributions will have a population mean and they will all use either a t-test or z-test. The Bayesian is concerned with the sampling distribution of the data, but not the parameters.

Consider a set of independent events that map to a probability over the set [0,1] in $d$ dimensions. It will be approximately multivariate normal as the sample size becomes large enough. Now let us assume that although the events are independent, the components that make up the dimensions are not. They are part of a system. Let us also assume they share a common variance, $\sigma^2_i=\sigma^2_j,\forall{i,j}\in{D}$, and that information about any one mean exists in the other means.

The Bayesian posterior for the set $\mu_i,i\in{1\dots{d}}$ for independent dimensions with independent variances and no shared information on the means would look very different from one where you assume a common variance $\sigma^2$ and shared information. The Frequentist tests would be no different but the Bayesian posteriors would be.

Bayesian methods are concerned about the population through the likelihood function because it models how the data is generated in the first place in nature. That is why Bayesian model selection methods are so important, because you may not know the true model in nature that the population uses.

Answer 2

This question seems to be based on a misconception. The difference between Bayesian and frequentist inferential frameworks is not that one is "sampling-based" and the other is not. It is resampling methods that yield results with the limited direct scope of inference like the statement "Is ... the treatment ... actually effective in the group of patients we analyzed?".

As I see it, there are several differences between the frequentist and Bayesian frameworks that assume greater or lesser importance depending on context.

One important difference is that frequentist approaches can only yield probabilities of the frequency of events flavour whereas the Bayesian approaches can yield probabilities of the fractional belief flavour. Thus whereas a Bayesian method might yield a result in the form of a rational belief about the effect of a treatment on a set of patients, a frequentist one will yield a statement about how often such a treatment effect might be observed in the long run given a specified value of the parameter of interest. Any connection between the frequentist result and belief is non-statistical.

Frequentist methods treat parameters as fixed where Bayesian methods treat them as variable. That leads to a difference in the treatment of evidence. Evidence provided by observations usually enters Bayesian methods in the form of a likelihood function which has at least one axis (dimension) corresponding to the parameter(s) of interest. Frequentist methods often focus on a single (null) value for the parameter of interest and therefore rarely utilise the whole likelihood function. Frequentist approaches therefore do not weigh the evidential support of all possible parameter values and the data are taken to either support the null (fail to un-support it?) or support an unspecified point within a diffuse non-null alternative.

The final notable difference that I will mention is one that is often described as being the primary difference: Bayesian methods can explicitly take existing information or opinion into account in the statistical calculations whereas frequentist methods cannot. If a non-misleading prior is available then using it seems to be a good thing, but many people are worried by the possibility of subjectivity in statistical inference.

Answer 3

A paper that greatly clarified these issues for me was "Models for Nonresponse in Sample Surveys" by Rod Little.

It discusses models which include as inferential targets both 1. summary quantities for actually-existing populations, and 2. parameters specifying hypothetical "superpopulations". It is also exceptionally clear.