For my mathematics bachelor-thesis at the Statistics Netherlands, i became acquainted with frequentist and Bayesian statistics. I had set up a simulation-study, and I am not sure if the risk I conceived is faulty somewhow.

You see, I do not like that others in their simulation-studies only chose one parameter, since the behaviour of the estimators (in our context) is surely to change depending on the true parameter. However, I still like the idea of a true parameter and being able to sample data using the parameter. So here is an overview of different risk formulations i came up with:
1. Frequentist risk with chosen parameter: $R(L,\delta,\theta) = \int_X L(\theta,\delta(x))f(x|\theta)dx$
2. Bayesian posterior risk: $R(L,\delta,x) = \int_\Theta L(\theta,\delta(x))p(\theta|x)d\theta$

Notice that (1) is perfectly fine for the sample mean, i.e. The MSE for $\bar{X}$ is the same for any $\theta$ if the underlying distribution is normal. However, not every distribution gives rise to this invariance property.
Bayesian risk has the 'problem' that it does not consider data from the long run, but that is something I am not sure about*.

So hereby, I present you, pragmatic risk: $R(L,\delta,p)=\int_\Theta L(\Theta,\delta(x))f(x|\theta)p(\theta)d\theta$

If my supervisor and I were to choose an objective flat-prior, then we can choose to approximate a 'Grand mean squared error' in the simulation-study for multiple estimators. And my interpretation of such a risk is that if the risk of two estimators differ, one of the estimators is safer in the long run, regardless of the true parameter.


*Does the Bayesian framework disregard 'the long run', or does it implicitly consider it in its computations, because the data-distribution-generating $\theta$'s are considered/weighted?

Is my interpretation correct?


This question is related to (if not a duplicate of) one XV question I answered a few days ago on the various definitions of Bayes risk. It is correct to note that the frequentist risk $\mathrm{R}(\theta,\delta)$ is dependent on the value of the parameter $\theta$ for most models and most estimators $\delta(\cdot)$. While the integrated Bayes risk $$\mathrm{R}(p,\delta)=\int_{\cal{X}}\int_\Theta \mathrm{L}(\Theta,\delta(x))f(x|\theta)p(\theta)\,\text{d}\theta\,\text{d} x$$ is a constant real number for a given prior $p(\cdot)$ and a given estimator $\delta(\cdot)$. However it does depend on the choice of the prior $p(\cdot)$, which is unacceptable for non-Bayesians. For instance, F. Samaniego examines the frequentist properties of Bayesian estimators in his 2010 book, A comparison of the Bayesian and frequentist approaches to estimation, by using several priors. Which is not an approach I particularly recommend. The "objective" Bayes solution of relying on a flat prior is similarly rejected by non-Bayesians (and some Bayesians), for reasons ranging from the issue of the choice of the dominating measure (lack of invariance by change of parameterisation), to the flat prior being outside probability distributions, to the possibility of an infinite Bayes risk.

However, the Bayesian framework does account for the frequentist distribution of the data, through the posterior, which means that most Bayes estimators are (a) admissible and (b) consistent ("in the long run"). See for instance complete class theorems (mentioned on XV here and here and there).


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