A simultaneous confidence band denotes the probability
$$p \big(\hat{f}(x) - w(\hat{f}(x)) \le f(x) \le \hat{f}(x) + w(\hat{f}(x)) \ \ \forall x \big)=1-\alpha$$
where $f$ a function of $x$, $\hat{f}$ and estimator for the value of $f$ at $x$, and $w(\hat{f}(x))$ an estimated interval length at $x$, see also Wikipedia entry.
Is there an existing analogue definition in Bayesian statistics and how is the resulting band estimated? (Using the posterior or posterior predictive distribution.)
Given a prior distribution $\pi$ on a functional space and observations about the function values at some points, or noisy observations of the function itself, the posterior distribution $\pi(\cdot|\mathcal{D})$ can be used to derive an HPD region, $$\left\{f\in\mathcal{F}\,,\ \pi(f |\mathcal{D})\ge k_\alpha\right\}$$ at least in principle since the derivation may prove too complex in a general situation.
For instance,
Breth, M. (1978) Bayesian confidence bands for estimating a function. Annals of Statistics Vol. 6, No. 3, pp. 649-657
seems to address this problem.
