Can "cross-validation" be used to choose a prior? To be clear, I doubt I am using the term "cross-validation" correctly here; what I am suggesting also seems similar to "boot-strapping" and "hyperparameter tuning". Terminology is not my strength.
Let's say we have a data set with $20$ observations, $D_1, \dots, D_{20}$. We don't know what prior to use for the data set, so we decide to use the maximum entropy prior given the population mean and variance, i.e. a normal prior. (This of course assumes the population distribution has finite second moment. I am not convinced that this assumption is innocuous, but it is common.)
But of course we don't know the population mean and population variance, so we estimate them. We can't use all of the data to estimate them, because then there wouldn't be any data left to do our inference on. So let's say we use observations $D_1, \dots, D_{15}$ to get an estimate $\hat{\mu}$ for the population mean $\mu$ and an estimate $\hat{\sigma}^2$ for the population variance $\sigma^2$. Then we choose $\mathscr{N}(\hat{\mu}, \hat{\sigma}^2)$ as our prior and then use the remaining $5$ observations $D_{16}, \dots, D_{20}$ to do inference with our prior.
No one would like this situation because we are no longer using all of our data for inference. So:

Question: In this situation would it make sense to:

*

*Calculate priors $\mathscr{N}(\hat{\mu}_1, \hat{\sigma}^2_1)$, $\mathscr{N}(\hat{\mu}_2, \hat{\sigma}_2^2)$, $\mathscr{N}(\hat{\mu}_3, \hat{\sigma}_3^2)$, $\mathscr{N}(\hat{\mu}_4, \hat{\sigma}_4^2)$, the first using the exact same procedure above, the second using an analogous procedure with $D_{11}, \dots, D_{15}$ as the "holdout set", the third using an analogous procedure with $D_{6}, \dots, D_{10}$ as the "holdout set", and the fourth using an analogous procedure with $D_1, \dots, D_5$ as the "holdout set",


*Choose as our prior either (a) the convex combination of these above four priors, which would be a Gaussian mixture model I guess, or (b) take as our prior $\mathscr{N}(\tilde{\mu}, \tilde{\sigma}^2)$, where $$\tilde{\mu} := \frac{1}{4}(\hat{\mu}_1 + \hat{\mu}_2 +  \hat{\mu}_3 + \hat{\mu}_4 ) \,, \quad \tilde{\sigma}^2 := \frac{1}{4}(\hat{\sigma}^2_1 + \hat{\sigma}_2^2 + \hat{\sigma}_3^2 + \hat{\sigma}_4^2) \,?$$

The above example generalizes readily, of course (for example I didn't even specify a particular method to get our estimates $\hat{\mu}$ and $\hat{\sigma}^2$), but I thought I would use this concrete example because I don't think I could explain myself clearly in full generality.
(Actually I even doubt that the above concrete example is explained clearly.)

Additional questions: Does something similar to the above procedure already have an established name? And is there any literature either showing its lack of optimality properties or otherwise analyzing it theoretically?

I think this question is different from this related question because that question has both an internal and external source of data. In this example, our parameter estimation and inference are both "competing" for the same data, so we use "cross-validation" or repeated sub-sampling ("bootstrapping") to accomplish the required "hyperparameter tuning" for the prior.
It is also different from the method suggested here which Andrew Gelman argued (probably convincingly, I don't understand the argument to be honest) does not work well. But that method is suggesting use of an "M-estimation" approach with cross-validation to get the prior, i.e. to select the "best-performing" prior from $\mathscr{N}(\hat{\mu}_1, \hat{\sigma}^2_1)$, $\mathscr{N}(\hat{\mu}_2, \hat{\sigma}_2^2)$, $\mathscr{N}(\hat{\mu}_3, \hat{\sigma}_3^2)$, $\mathscr{N}(\hat{\mu}_4, \hat{\sigma}_4^2)$, whereas I am suggesting using some combination of them. This makes more sense to me than the other method, in order to both (a) avoid "overfitting"  and (b) to use more of the data to inform the choice of prior.
 A: In this paper Andrew Gelman used cross validation on a corpus of datasets to propose a weakly informative prior, intended for routine use. That seems a sensible approach!
Gelman, Andrew, et al. "A weakly informative default prior distribution for logistic and other regression models." The annals of applied statistics 2.4 (2008): 1360-1383.
A: Since prior etymologically signifies before: 

prior
adjective
existing or coming before in time, order, or importance.
"he has a prior engagement this evening"
synonyms:   earlier, previous, preceding, foregoing, antecedent, advance, preparatory, preliminary, initial.

using the data to build the prior is not correct within a Bayesian perspective. It is found however in the "empirical Bayes" methodology, initiated by Robbins (1955) and defended by Efron, which uses first the data to estimate the parameters in a prior, like your Normal example, and a second time to run a pseudo-Bayesian analysis as if the prior was a true prior. Some versions of this approach enjoy convergence properties, for instance in semi- and non-parametric settings.
The question however seems to shy away from this solution by making a single use of the data and separating it into learning and inference parts. This is connected with the construction of intrinsic Bayes factors in the 1990's, by Jim Berger and co-authors, where a fraction of the data is used to make a flat (or otherwise improper) prior into a proper posterior, and use the remaining fraction to compute a Bayes factor (and run a test decision). In order to avoid the choice of the partition impacting the final result, all possible permutations are considered and one form of average (among arithmetic, geometric, harmonic, median) is computed. A much more elegant alternative is O'Hagan's (1995) fractional Bayes factor where the likelihood $L(\theta)$ is replaced with a fractional power $L^\alpha(\theta)$, which is used to create a posterior and this posterior is then used as a prior for the remainder of the likelihood $L^{1-\alpha}(\theta)$. The difficulty with these approaches is in determining the "right" amount of partitioning, e.g., the value of $\alpha$.
A: Firstly, I completely understand the temptation to do this. You want your model assumptions to be defensible and your model to be valid out-of-sample.
The issue is: this is not Bayesian inference. This is regularization using Bayes theorem. That isn't "good" or "bad." The answer to your titular question is yes absolutely. The only thing I would say is that there are probably better regularization methods that would be easier and equally valid under cross-validation.
Bayesian priors are not validated, they are solicited. A prior reflects your preexisting beliefs. If you have to validate it, you didn't believe it in the first place and therefore it is not a genuine prior in the true sense of the term.
The related technique of using secondary data sources to generate a prior that has been brought up is what is refereed to as "empirical" Bayes method, using a second separate date set to generate a prior. This works well for hierarchical models and for situations in which you have empirical prior knowledge you can call on.
I don't see how this makes sense if you are taking one data set and splitting it into two specifically for that purpose. It begs the question why not perform inference on the whole data set with a subjective prior? If you're going to infer your prior from a subset of the data using Bayesian inference, that requires a non-empirical prior which you could just use on the entire data set (in other words, you're just splitting Bayesian inference into two unnecessary steps). If you are not using Bayesian inference to generate a prior, then you don't really believe in using Bayesian inference to solve your given problem in the first place. In that case, just use the method you really believe in on the entire data set.
TLDR: if you don't have a prior belief you feel comfortable using, don't use Bayesian inference; just skip to the method you would use to generate the prior and use it across the board.
