What is the role, if any, of the Central Limit Theorem in Bayesian Inference? As someone who started by studying classical statistics where the Central Limit theorem is key to making inferences, and only later now am studying Bayesian statistics, I was late to realize that the Central Limit Theorem has a much smaller role to play in Bayesian statistics. Does the Central Limit Theorem play any role at all in Bayesian inference/statistics?
Later Addition:
From Bayesian Data Analysis by Gelman et.al. 3rd edition - on the central limit in the Bayesian context.
"This result is often used to justify approximating the posterior  distribution with a normal distribution" (page 35). I went through a graduate course in Bayesian Statistics without encountering an example in which the posterior was approximated with a normal distribution. Under what circumstances is it useful to approximate the posterior with a normal distribution?

Reproduced verbatim from the Wikipedia page:

In Bayesian inference, the Bernstein-von Mises theorem provides the
basis for using Bayesian credible sets for confidence statements in
parametric models. It states that under some conditions, a posterior
distribution converges in the limit of infinite data to a multivariate
normal distribution centered at the maximum likelihood estimator with
covariance matrix given ${\displaystyle
  n^{-1}I(\theta _{0})^{-1}}$, where $\theta _{0}$ is the true
population parameter and ${\displaystyle I(\theta _{0})}$ is
the Fisher information matrix at the true population parameter
value.
The Bernstein-von Mises theorem links Bayesian inference with
frequentist inference. It assumes there is some true probabilistic
process that generates the observations, as in frequentism, and then
studies the quality of Bayesian methods of recovering that process,
and making uncertainty statements about that process. In particular,
it states that Bayesian credible sets of a certain credibility level
$\alpha$ will asymptotically be confidence sets of confidence level
$\alpha$, which allows for the interpretation of Bayesian credible
sets.

With the reference

van der Vaart, A.W. (1998). "10.2 Bernstein–von Mises Theorem".
Asymptotic Statistics. Cambridge University Press.
ISBN 0-521-49603-9.

