Does Central Limit Theorem Apply to Bayesian inference? In reading a Paper on Bayesian estimation, I came across a sentence that had me think:
"Bayesian statistics is not based on large samples (i.e., the central limit theorem) and hence may produce reasonable results even with small to moderate sample sizes, especially when strong and defensible prior knowledge
is available" (p. 240)
So, my question is: Do the authors also mean that the idea of sampling with replacement from a population infinitely many times, and averaging the point estimates to arrive at the original mother population's parameter (Central Limit Theorem) is a Frequentist concept and does not apply to Bayesian statistics?

 A: 
"Bayesian statistics is not based on large samples (i.e., the central limit theorem) and hence may produce reasonable results even with small to moderate sample sizes, especially when strong and defensible prior knowledge is available" (p. 240)
So, my question is: Do the authors also mean that the idea of sampling with replacement from a population infinitely many times, and averaging the point estimate to arrive at the original mother population's parameter (Central Limit Theorem) is a Frequentist concept and does not apply to Bayesian statistics?

The central limit theorem does not depend upon the interpretation of probability.  However, its importance is substantially diminished because of how the different methods build their inference.
The repetition element is important in sampling-based statistics because you are working in the sample space and the concept of optimality is that the estimators minimize some loss and are optimal on average.  Bayesian estimators are optimal, given the actually observed sample.  If you repeat the experiment, then the Bayesian estimates are updated.  Each Frequentist experiment is reported out as separate events, except in meta-analysis.
Using your graphic, the Frequentist conducts each experiment and produces twenty sets of inference.  The Bayesian would use all prior experiments to update the current one, so in a sense, those twenty experiments are one Bayesian sample.
In a Frequentist solution, the concern is with the distribution of $\bar{x}-\mu$ and since $\bar{x}$ is a random variable made up of the sum of many random variables, the sampling distribution of $\bar{x}$ is normal and so the central limit theorem holds and is important.
For a user of Bayesian methods, however, the sample isn't random.  An observation isn't a random variable.  The central limit theorem still holds, but it isn't directly used for inference.  The posterior depends only on the exact sample that was observed and not samples that could have happened but did not happen.
I cannot say what the authors mean as I haven't read the book.  Nonetheless, I do not think that you can infer as much as you are from that statement.  Since Bayesian inference is based on the sample that was observed, there is no direct concept of "power."
Bayesian methods depend on sample size just as null hypothesis methods do to get information, but there is no idea of exploring a sample space in order to perform inference.
I think they are saying something smaller than what you are saying.  Let me give you an example.  Let us imagine that you are about to patent a new species of bean.  A very similar bean has 31 calories per 100 grams, with a variance of 25 calories$^2$ per 100 grams.  You have your first ten beans.  They weight 55 grams.  You have a sample size of ten.  That isn't much.
In doing this, you have basically constrained the search area for $\mu$ to (16,46).  This increases the precision of the test because the method of maximum likelihood is going to treat the entire set of real numbers as the possible number of calories per hundred grams until it sees the sample. Then the confidence interval will be based only one what the sample provides.  This will make for a wider estimate since it lacks the additional information.
I don't think they are saying anything more.
If you did do infinite repetition, then if $\mu$ is a population mean from a density, and the prior is not degenerate, then the marginal posterior density of $\mu$ will converge to the normal distribution.  The central limit theorem has returned, albeit in another form and it isn't the "central limit theorem" anymore, but it is the same idea.  Now $\mu$ is the random variable.  The distinction here, however, is that nobody can collect the data from infinite repetition, but you can collect ten beans.
A: I interpret it to mean that Bayesian inference doesn't require the same assumptions of normality that classical frequentists tests do, ie, the t-test. For the t-test, it is a requirement that the population be normally distributed. This can also be thought of that the means of your data must be normally distributed. (I think it is a misconception that the data themselves must be normally distributed, so long as the departure isn't too strong. See here)
From the linked paper:

First, data do not need
  to be normally distributed in order to apply the t test.
  Only the means need to be, and that property is assured
  by the Central Limit Theorem, even for relatively small
  samples, for all but the most perverse data. This is
  exemplified in Fig. 1, which shows at the upper left a
  very nonnormal (in fact, a uniform) distribution of original
  data. Random samples of size N = 2, 4, and 8
  demonstrate that the distribution of averages based on
  even those small sample sizes rapidly approaches normality.

In Bayesian inference, you aren't running some out-of-the box test. Well, you can in programs like JASP or some R packages where people have constructed them, but by and large, Bayesian inference in modern practice is centered around model building. If your data are distributed funny, use a distribution to describe the data that fits better. For instance, in brms you can use a skew normal distribution or t-distribution when your data are skewed or have outliers as part of your prior distribution. The prior provides information. While one can certainly proceed with flat priors or other uninformative priors, the results asymptotically resemble the frequentist results. Ie, the credible interval will be like the confidence interval. In this case you can lose a lot of benefits the Bayesian approach allows and arguably frequentist concerns more relevant.
Priors allow you to model the data and make your assumptions completely explicit. If you have information about the data from other studies, model it! It can help provide context to your collected data and allow you to make inferences in the context of existing knowledge. 
Also check out this pdf which has slides about frequentist properties of Bayesian inference. One is what they call the Bayesian CLT, which is that with increasing sample size the posterior becomes more similar to the MLE & normally distributed. Frequentist Properties of Bayesian
Methods
