How to prove that, in finite sample, Bayesian posterior is more informative than the prior? Suppose there is a space of possible models $\theta \in \Theta$, and that we can generate i.i.d. data $\{x_1, x_2,...\}$ from the true model.
Asymptotically, the Schwartz Theorem shows that Bayesian inference is consistent: the posterior will cluster around the true model.
However, can the same be shown in finite sample? That is, can we show that, in some appropriate sense, the posterior after one data observation $P(\theta | x)$ is (on average) more informative than the prior $P(\theta)$?
When the number of models is finite, a more concrete version of this question is: can we show that $E(P(\theta^\text{true} | x)) > P(\theta^\text{true})$?

You need to define carefully what you mean by 'informative'. I'll give
some examples with binomial experiments.
A common 'noninformative' prior for binomial success probability $\theta$ is
the Jeffreys prior $\mathsf{Beta}(.5,.5).$ The density function is
proportional to $\theta^{.5-1}(1-\theta)^{.5-1}.$
Then suppose you perform one Bernoulli trial, which results in success.
The binomial likelihood is proportional to $\theta^1(1-\theta)^0,$ and by
Bayes' Theorem the posterior distribution is $\mathsf{Beta}(1.5, .5).$
Alternatively, if you perform $n=10$ trials and get $x= 7$ successes, then
the binomial likelihood is proportional to $\theta^7(1-\theta)^3$ and the
posterior distribution is $\mathsf{Beta}(7.5,3.5).$
Plots of Jeffreys prior, the posterior after one trial, and the posterior
after ten trials are shown in the figure below. Roughly speaking, one
might say that beta distributions become increasingly informative as the
sum of the two shape parameters increases.

R code for figure:
par(mfrow=c(1,3))
curve(dbeta(x,.5,.5), 0,1, lwd=2, ylim=c(0,3), col="blue", 
      ylab="Density", xlab="theta", main="BETA(.5,.5)")
 abline(v=0:1, col="green"); abline(h=0, col="green")
curve(dbeta(x,1.5,.5), 0,1, lwd=2, ylim=c(0,3), col="blue", 
      ylab="Density", xlab="theta", main="BETA(1.5,.5)")
 abline(v=0:1, col="green"); abline(h=0, col="green")
curve(dbeta(x,7.5,3.5), 0,1, lwd=2, ylim=c(0,3), col="blue", 
      ylab="Density", xlab="theta", main="BETA(7.5,3.5)")
 abline(v=0:1, col="green"); abline(h=0, col="green")
par(mfrow=c(1,1))

For more detail on how a beta prior and a binomial likelihood yield
a beta posterior, perhaps see this Q&A.
