Is Frequentist Inference Objective? Bayesian statistics is criticized for being subjective, as it requires a prior distribution encapsulating the subjective befiefs of the observer. Frequentist statistics is commonly advertised as being objective (because it does not require choosing a prior, because its probabilities are defined in an objective way, or whatever reason I'm not really sure about), however Wald proved that most admissible decision rules in Frequentist statistics are each one equivalent to a Bayes rule using some prior (which may be improper among other things, but it exists). One could think of the frequentist rules as being uninformative, but the calculation of uninformative priors may yield results different to the frequentist ones (as in the case of the Bernoulli distribution, for example). My questions are the following:

*

*Is there an objective reason to choose the priors associated to frequentist decision rules over others?

*If not, does that mean that frequentist inference is subjective?

*If yes, what is the personal belief / knowledge that we are injecting when using frequentist inference?

Thanks in advance!
 A: Let me recall (from my book) the precise setting of Wald's characterisation of admissible estimators: first, Stein's theorems for admissible procedures to be limit of Bayes procedures:

Charles Stein (1955) produced a necessary and sufficient condition: if
(i) $f(x|\theta)$ is continuous in $\theta$ and strictly positive on $\Theta$; and
(ii) the loss function $\text{L}(\cdot,\cdot)$ is strictly convex, continuous and, if $E\subset\Theta$ is compact,
$$
  \lim_{\|\delta\|\rightarrow +\infty} \inf_{\theta\in E} \text{L}(\theta,\delta) =+\infty,
$$
then an estimator $\delta$ is admissible if, and only if, there exist (a) a sequence $(F_n)$ of increasing compact sets such that $\Theta=\bigcup_n F_n$, (b) a sequence $(\pi_n)$ of finite measures with support $F_n$, and (c) a sequence
$(\delta_n)$ of Bayes estimators associated with $\pi_n$ such that

*

*there exists a compact set $E_0\subset \Theta$ such that $\inf_n \pi_n(E_0) \ge 1$;

*if $E\subset \Theta$ is compact, $\sup_n \pi_n(E) <+\infty$;

*$\lim_n r(\pi_n,\delta)-r(\pi_n) = 0$; and

*$\lim_n R(\theta,\delta_n)= R(\theta,\delta)$.

Larry Brown (1986) provides an alternative, and quite general,
characterization of admissible estimators. Consider $x\sim
  f(x|\theta)$, and assume $\text{L}$ to be lower semi-continuous and
such that $$   \lim_{||\delta||\rightarrow +\infty}
  \text{L}(\theta,\delta) = +\infty. $$ Brown (1986) shows that, under
these conditions, the closure (for the pointwise convergence) of the
set of all Bayes estimators is a complete class.
Proposition If L is strictly convex, every admissible estimator of $\theta$ is a pointwise limit of Bayes estimators for a sequence of
priors with finite supports.

Second, the generic Wald (1950)'s complete class result:

Theorem Consider the case when $\Theta$ is compact and the risk set $$   \mathcal R = \{(R(\theta,\delta))_{\theta\in\Theta},\
  \delta\in\mathcal D^*\}, $$ is convex (where $\mathcal D^*$ denotes the set of randomised decisions). If all estimators have a
continuous risk function, the Bayes estimators constitute a complete
class.

and a remark about cases when it does not hold:

In the case of distributions with discrete support, the completeness of
generalised Bayes estimators does not always hold and complete classes
involve piecewise-Bayesian procedures (see Berger and Srinivasan
(1978), Brown (1981), and Brown and Farrell (1985)).

These results do not imply that every admissible estimator can be associated with a proper prior or an improper prior. Furthermore, even if this is the case, there are as many "admissible" priors as there are admissible estimators, hence no apparent restriction on the choice of priors. (This is why admissibility is a desirable feature rather than an optimality property per se.)
In the same way that the notion of an "objective", "uninformative", "default" prior does not meet a consensus in the Bayesian community, there is no consensus about a default frequentist procedure that would lead to the notion of a "frequentist prior". Note in addition that associating a prior with a frequentist procedure is depending on the choice of the loss function L, hence varying with the quantity of interest.
A: I discussed your question 2 at some length in appendix E of my Ph.D. thesis (Hatton, 2003, Spin-polarized electron scattering at ferromagnetic interfaces, University of Cambridge).  The position I eventually reached was that true objectivity is achieved when (usually due to having lots of data) key features of the posterior distribution become independent of the choice of prior over some domain of "reasonable" priors.  The frequentist approach, viewed as having a prior in the way you suggest, conceals the prior and renders it immutable, which gets in the way of testing for objectivity, when objectivity is conceived in the way I suggested.
BTW, I didn't know about Wald's proof that 'every admissible decision rule in Frequentist statistics is equivalent to a Bayes rule using an appropiately chosen prior'.  Do you have a specific citation?  (In my thesis (appendix D), I gave my own proof, but mine only works for a certain subset of frequentist significance tests.)
