Please clarify Bayesian calibration of the posterior mean In the book Bayesian Data Analysis, the authors state on page 128:

The concept of calibration of the posterior mean [is] the Bayesian analogue to the classical notion of bias.

They define the miscalibration of the posterior mean of some parameter $\theta$ as
$$
\text{miscalibration} = E(\theta|\hat{\theta}) - \hat{\theta}
$$
where
$$
\hat{\theta} = E(\theta|y)
$$
and $y$ is the observed data.
Can someone clarify what this means? How does this $E(\theta|\hat{\theta})$ term work? It's a conditional expectation that's conditional on something that is itself a conditional expectation. How does that work? Can you show that the miscalibration is 0 if the prior distribution is true, i.e. if the data are constructed by first drawing $\theta$ from $p(\theta)$ and then drawing $y$ from $p(y|\theta)$?

It may be confusing that the $\hat \theta = E[\theta \mid y]$ seems to use the chosen prior and the posterior distribution then implied by the observation, while the $E\left[\theta \mid \hat \theta\right]-\hat \theta$ in the miscalibration calculation seems to use the actual (though presumably unknown) distribution for $\theta$.  It is probably easier with an example.
Suppose we have $\theta \in [0,1]$ as the parameter of a Bernoulli random variable $Y$ which is $1$ with probability $\theta$ and $0$ otherwise
If we use a prior $p(\theta)=2\theta$, then

*

*observing $Y=1$ will suggest to us a posterior distribution $p(\theta \mid Y=1)=3\theta^2$ with mean $\hat \theta =\frac34$,

*observing $Y=0$ will suggest to us a posterior distribution $p(\theta \mid Y=0)=6\theta(1-\theta)$ with mean $\hat \theta =\frac12$,

and there is a $1-1$ relationship here between the value of $Y$ and $\hat \theta$.
If the prior is the correct distribution of $\theta$ then

*

*when $\hat \theta =\frac34$, we have $E\left[\theta \mid \hat \theta=\frac34\right] - \hat\theta =\frac34-\frac34=0$


*when $\hat \theta =\frac12$, we have $E\left[\theta \mid \hat \theta=\frac12\right] - \hat\theta =\frac12-\frac12=0$
automatically as asserted, and also

*

*the probability of observing $Y=1$ and estimating $\hat \theta=\frac34$ is $\frac23$

*the probability of observing $Y=0$ and estimating $\hat \theta=\frac12$ is $\frac13$
so $E\left[\hat\theta\right]=\frac23 \times \frac34 +\frac13\times \frac12 =\frac23 = E[\theta]$ and all is well with the world.
But if the prior were wrong and in fact $\theta$ in reality had a density of $2(1-\theta)$ i.e. with $\theta$ and $Y$ more likely to be smaller than the assumed prior suggested, then the actual conditional expectations for $\theta$ would be smaller

*

*when $\hat \theta =\frac34$, we would have $E\left[\theta \mid \hat \theta=\frac34\right] - \hat\theta =\frac12-\frac34=-\frac14$

*when $\hat \theta =\frac12$, we would have $E\left[\theta \mid \hat \theta=\frac12\right] - \hat\theta =\frac14-\frac12=-\frac14$
so a miscalibration of $- \frac14$ (in more complicated cases it does not have to be constant) and

*

*the probability of observing $Y=1$ and estimating $\hat \theta=\frac34$ would have been $\frac13$

*the probability of observing $Y=0$ and estimating $\hat \theta=\frac12$ would have been $\frac23$
making $E[\hat\theta]=\frac13 \times \frac34 +\frac23\times \frac12 =\frac7{12} \not = \frac13 = E[\theta]$, which may be  undesirable.
