Limit Bayesian Posterior In the Bayesian setting, we update prior distribution $\pi(\theta)$ to posterior distribution $\pi(\theta | x)$ given data $x$. So data $x$ provides an operator $T_x$ on the set of distributions on the parameter space.
$$T_x: Dist(\Theta) \to Dist(\Theta)$$
In many cases, people argue on which prior distribution to take. One popular option is the Jeffreys prior (essentially the Fisher information) because it is invariant under reparametrization.
As what actually matters is the posterior distribution, I wonder why don't we take the fixed points of $T_x$, i.e. $\pi_x(\theta)$ such that
$$\pi_x(\theta | x) = \pi_x(\theta)$$
as the posterior once $x$ is observed? A heuristic way to construct such $\pi_x$ is to take $\lim_{n \to \infty}T^n_x(\pi)$ for any $\pi$ if exists.
Questions

*

*Do people take such fixed points as posterior distribution, and make inferences from that?

*Do such limit always exist? If not, is there always some $\pi$ such that the limit exists?


The condition $\pi(\theta|x)=\pi(\theta)$ means that
$$\pi(\theta)=\frac{\pi(\theta)f(x|\theta)}{\int_\Theta \pi(\theta)f(x|\theta)\,\text d\theta}$$
i.e.,
$$f(x|\theta) = \int_\Theta \pi(\theta)f(x|\theta)\,\text d\theta$$
which can only hold when the density of $X$ at this realisation $x$ does not depend on $\theta$.
Looking at limits of posteriors to define "objective" or "non-informative"  priors is found in the theory of reference priors, see e.g. Berger, Bernardo, & Sun (2009).
While using the posterior as prior and iterating the action leads to the notion of prior feedback for deriving maximum likelihood estimators. I developed this method (later revamped as data cloning (Lele, 2007) and MCMC maximum likelihood (Jacquier et al., 2007)).
