How would a bayesian estimate a mean from a large sample? What would a bayesian do if she wanted to do inference for the mean with a large sample but has no idea of the underlying distributions?
A frequentist statitician would use the sample mean as a point estimate and CLT for the distribution of the estimator. All she has to assume is finite variance.
I have found Bernstein Von-Mises Theorem which states that given a random sample $X_1, \ldots, X_n$ and

*

*a prior $p(\mu)$ for the mean

*a distribution for the sample given the parameter $f_{x | \mu}$
Then if the sample size is sufficiently large the posterior distribution of the mean given the sample is approximately normal, i.e
$$ \mu_{|_{X_1, \ldots X_n}} \approx \mathcal{N}\left( \hat{\mu}_{ml}\, ; \, \frac{1}{I(\mu_0) n}\right)$$
Where the mean is the maximum likelihood estimator and $I(\mu_0)$ is Fisher information number for the true mean.
In the case where we have no knowledge of $f_{x | \mu}$ there is the problem that  maximum likelihood nor fisher information number can be calculated.  But what can be done? Becuase even if we do not know it the distribution exists and it will be approximately normal.

Intentions:
Some buisness woman has to take an action based on the true sign of the mean. A decision has to be made so she will use the sign of the sample mean but she needs some measure of the potential cost that arise in case of making the wrong decistion.
So, if the sample mean is positive things like
$$ P\left( \mu < 0 | X_1 \ldots X_n \right) = \int_{-\infty}^0 f_{\mu |x_1 \ldots x_n}( t) \, dt.$$  or
$$ E \left( \mu I_{(-\infty, 0)}| X_1 \ldots X_n \right)  = \int_{-\infty}^0 t f_{\mu |x_1 \ldots x_n}( t) \, dt.$$
would make a lot of (buisness) sense for her.
In the frequentist context she is stuck with affirmations of the kind:
With $95\%$ confidence the mean is greater than a certain value.

There are various flavours of Bayesian statistics. One of them is subjectivist (e.g., according to de Finetti). Subjectivist Bayesians hold that probability applies to an individual's state of belief and information but not to underlying data generating processes, which can never be infinitely repeated, which would be necessary to define a true frequentist probability. For this reason (and potentially some others that are harder to discuss), according to a subjectivist, there is no such thing as a true underlying distribution. So the job of the subjective Bayesian in this problem is not to guess the underlying distribution, but rather to specify a distribution that summarises her belief and knowledge about the expected distribution of the data given $\mu$. Not only $p(\mu)$ is a prior choice, also what you call $f_{x|\mu}$!
In fact, this is even the case in what many call "objectivist Bayes", as long as the probabilities are epistemic, i.e., do refer to a state of knowledge rather than really existing underlying data generating processes. The objectivist also will have to choose an $f_{x|\mu}$ that expresses all existing information about the expected distribution of the data given $\mu$ (except that subjective belief is not supposed to play a role here; although in reality it is often hard to bring existing information into a suitable formal form without any subjective choices).
These are the major streams of traditional Bayesian philosophy. In the present, much of Bayesian data analysis is based on an implicit assumption that there is a true underlying distribution, which we have called "falsificationist Bayes" here:
https://rss.onlinelibrary.wiley.com/doi/10.1111/rssa.12276
Even here (as in frequentism), the task would be to specify a model that makes sense from a subject matter perspective, and that can then be checked, for example by comparing data generated from it with your actual data, as in so-called posterior predictive checks (hence "falsificationist").
There is also the field of Bayesian nonparametrics, which is about very  large models with potentially infinite-dimensional parameters covering large sets of the model space in case you don't want to commit to a specific simple one. This may be relevant regardless of whether your probability model is interpreted in an epistemic or frequentist (underlying data generating process) sense.
