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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 $\hat{\theta}$.

Since I'm more familiar with frequentist statistics, I have some doubts about this.
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?
Is the miscalibration of the posterior mean the same concept of bias of a bayesian estimator?
I've researched a bit in the internet but I've found out that the miscalibration of the posterior mean doesn't seem to be a common concept in bayesian statistics.
Thank you

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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: $$ \text{miscalibration} = \text{(posterior mean using inferential prior)} - \text{(posterior mean using true prior)} $$

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    $\begingroup$ This angle is helpful, it definitely seems to be what Gelman was implying. But I am still confused because it means $E(\theta | \hat{\theta},y)$ is supposed to be the posterior mean using the inferential prior. I thought "inferential prior" meant the prior actually used in the model. Wouldn't that mean the posterior mean with the inferential prior is just $\hat{\theta} = E(\theta | y)$? $\endgroup$
    – landau
    Commented Jul 3, 2021 at 3:23

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