The anthropic principle is often suggested as essential in linking observations to theories, such that without the proper anthropic assumption, we cannot do science. (For an example of such discussions see Nick Bostrom's ‘Anthropic Bias’ Chapter 3&5)
Say you are conducting an experiment, e.g. measuring the spin of an electron, to evaluate two competing theories. According to theory #1, the probability should be 99% spin up, 1% spin down. For theory #2, 1% spin up and 99% spin down. Let’s say you got the result of spin up. This is strong evidence favoring theory #1. If you were indifferent towards the two theories before, after the experiment you should be 99% sure that theory #1 is the correct one. All this is pretty straightforward.
However many anthropic thinkers would say this reasoning is simple-minded and technically incorrect. Without proper anthropic reasoning, the result only means there is a spin-up electron in the universe. Given the vast size of the universe, a spin-up electron is (practically) guaranteed to exist somewhere according to both theories. You could be a “freak observer” who sees a spin-up in a theory#2 universe. If the universe is infinite, even when the experiment is interpreted using all information available, like defining you as an observer with all specifiable attributes who find the electron spin-up, the outcome would still occur with probability 1 regardless of which theory is correct. So there is no way to evaluate any hypothesis at all.
Their solution is this: you have to regard yourself as a random sample from all observers. What all observers include is another matter of debate, i.e. maybe it should include potentially existing observers or maybe only include actually existing observers, but that part is not relevant for this topic. With this assumption, even though all experiment result happens with a probability of 1 in the universe, they have different probabilities happening to you. So in the end you can still use observations to evaluate theories.
To say the very least, this anthropic consideration seems unnecessary. Good scientific theories should be able to predict our observations, using experiments to evaluate them directly reflects that. The anthropic view above cuts off this tie by “zooming out” and taking a god’s eye view of the entire universe first. By doing so, how to locate “your observation” in the universe becomes a problem that needs to be addressed. Then they propose the solution: consider yourself as a random sample. It enables us to do science just like we did before.
So before anthropics was widely discussed, when people focused on experiment results without minding the entire universe or its possible infinity, we were selecting theories based on the wrong reasoning. And how convenient that the correct reasoning with the proper anthropic assumption gives the same effect as before. This is too coincidental and unparsimonious.
It is sensible to suspect our reasoning before was right all along. If so, what does this mean for anthropics? Maybe starting from a god’s eye view then assuming sampling to explain who you are and your location in the universe is wrong. Instead of the god’s-eye view, you should keep to your perspective, treat who you are as given so we can directly use observations to evaluate competing theories. And that is what Perspective-Based Reasoning suggests in contrast to the common zoom-out-and-sampling approach (Like SSA and SIA).
(A relating topic is how to understand scientific objectivity, which has been discussed here. )
Is anthropic reasoning, as used in this example, an attempt to provide frequentist foundations for Bayesian reasoning?
No. Anthropics is primarily about how to treat indexical information (such as "I", "here" and "now") in reasoning. Most camps regard Bayesianism as an independent probability interpretation that requires no justification from frequentism. I actually think most anthropic camps do not pay enough attention to the frequentist interpretation.
You could make the exact same argument about quantum mechanics.
We favor quantum mechanics because it can explain/predict some experiment observartions while classical mechanics cannot. This reasoning is exactly what I am arguing for.
Anthropics however argue without regarding "I" as a random sample there is no way to use our observations to evaluate theories. Because no matter how unlikely, any observation possible would have happened in the entire universe.
Frankly, I don't see any parallel here.