# Question about posterior mean calibration

I'm reading the article "Prior distributions for variance parameters in hierarchical models" by Andrew Gelman(link). This is an extract that I don't understand very well:

Posterior inferences can be evaluated using the concept of calibration of the posterior mean, the Bayesian analogue to the classical notion of “bias ”. For any parameter $\theta$, we label the posterior mean as $\hat{\theta}=E(\theta|y)$ and define the miscalibration of the posterior mean as $E(\theta|\hat{\theta},y)-\hat{\theta}$, for any value of $\hat{\theta}$.

Can I consider $\hat{\theta}$ as the true posterior mean?
What is exactly $E(\theta|\hat{\theta},y)$? The notation confuses me...can I consider it as the expected value of a estimator as in frequentist statistics?
Later in that section, there is an example where the posterior mean using the inferential prior is larger than the posterior mean using the true prior, and this is said to be an example of positive miscalibration. Therefore I think the intended definition of miscalibration is: $$\text{miscalibration} = \text{(posterior mean using inferential prior)} - \text{(posterior mean using true prior)}$$