Credible sets are posterior probability statements; confidence sets are repeated-sampling statements. The two agree only under additional conditions.
Credibility and coverage
The same random set can be evaluated under two probability measures.
Definition. A set \(C_n(X)\) has posterior credibility \(1-\alpha\) if \(\Pi(C_n(X)\mid X)\ge1-\alpha\). It has frequentist coverage \(1-\alpha\) at \(\theta_0\) if \[ P_{\theta_0}\{\theta_0\in C_n(X)\}\to1-\alpha. \]
Remark. Posterior probability conditions on the observed data. Coverage averages over repeated data generated at a fixed parameter.
Coverage from BvM
In regular parametric models, BvM supplies the bridge between the two notions.
Theorem. Under BvM, credible ellipsoids based on the efficient center and information matrix have asymptotic frequentist coverage equal to their posterior credibility.
Proof. The local posterior radius converges to the Gaussian quantile, while the sampling distribution of the efficient center has the same Gaussian covariance. Thus the event that \(\theta_0\) lies in the credible ellipsoid has the same limiting probability as the posterior Gaussian ball.
Marginal credible intervals
Marginal intervals can have good first-order coverage even when joint sets require more care.
Proposition. If a scalar functional \(\psi(\theta)\) is differentiable and BvM holds for \(\theta\), then the marginal posterior of \(\sqrt n(\psi(\theta)-\psi(\hat\theta_n))\) is asymptotically normal with variance \(\dot\psi^\top I^{-1}\dot\psi\).
Proof. Apply the delta method to the local posterior variable and to the sampling distribution of \(\hat\theta_n\).
Nonparametric credible sets
In infinite-dimensional models, credible balls may undercover unless inflated or carefully centered.
Remark. Posterior contraction alone does not guarantee coverage. Bias, polished-tail conditions, self-similarity assumptions, or hierarchical adaptation can determine whether credible sets have honest frequentist coverage.
Credible ellipsoids under BvM
Theorem. Under the regular BvM theorem, let \[ C_n =\left\{ \theta: n(\theta-\widehat\theta_n)^\top I_{\widehat\theta_n} (\theta-\widehat\theta_n) \le\chi^2_{d,1-\alpha} \right\}. \] Then \[ \Pi(C_n\mid X^{(n)}) \xrightarrow{P_{\theta_0}}1-\alpha \] and \[ P_{\theta_0}(\theta_0\in C_n) \to1-\alpha. \]
Proof. BvM and continuity of information make the posterior quadratic form converge to \(\chi_d^2\), giving credibility. For coverage, \[ \sqrt n(\widehat\theta_n-\theta_0) \Rightarrow N(0,I_{\theta_0}^{-1}), \] so \[ n(\widehat\theta_n-\theta_0)^\top I_{\theta_0} (\widehat\theta_n-\theta_0) \Rightarrow\chi_d^2. \] Slutsky's theorem replaces \(I_{\theta_0}\) by \(I_{\widehat\theta_n}\).
Equal-tail and highest-density intervals
Definition. A scalar equal-tail interval is \[ [q_{\alpha/2},q_{1-\alpha/2}], \] where \(q_u\) is a posterior quantile. A highest posterior density set is \[ \{\theta:\pi(\theta\mid x)\ge c_\alpha\}, \] with threshold chosen to contain probability \(1-\alpha\).
For a symmetric unimodal normal posterior they coincide. In a skewed or multimodal posterior, an HPD set can be asymmetric or disconnected, while an equal-tail interval is connected and transformation-equivariant under monotone maps. HPD sets are not generally invariant under reparameterization because density height changes by a Jacobian.
Proposition. If \(\phi=g(\theta)\) is strictly increasing, the image of a \(\theta\)-equal-tail interval is the equal-tail interval for \(\phi\).
Proof. Posterior quantiles transform as \[ q_u^\phi=g(q_u^\theta) \] under a strictly increasing map.
One-sided credible bounds
Proposition. If the posterior \(1-\alpha\) upper bound is \[ U_n=\widehat\theta_n+ \frac{z_{1-\alpha}\widehat\sigma}{\sqrt n} +o_P(n^{-1/2}) \] and \[ \sqrt n(\widehat\theta_n-\theta_0)/\widehat\sigma \Rightarrow N(0,1), \] then \[ P_{\theta_0}(\theta_0\le U_n)\to1-\alpha. \]
Proof. Rearranging the coverage event gives \[ \frac{\sqrt n(\widehat\theta_n-\theta_0)} {\widehat\sigma} \ge-z_{1-\alpha}+o_P(1). \] The normal limit has probability \(1-\alpha\).
Probability matching priors
In regular scalar models, certain priors improve frequentist coverage beyond first order: \[ P_{\theta_0}\{\theta_0\le q_{1-\alpha}(X)\} =1-\alpha+O(n^{-1}) \] for first-order matching, and sometimes \(O(n^{-3/2})\) error under stronger matching equations. Jeffreys prior is first-order probability matching for many scalar regular models. In multiparameter problems, matching depends on the functional and nuisance geometry; no universal prior matches every target.
Bias versus posterior radius
Proposition. Suppose a scalar posterior is approximately \[ N(\theta_0+b_n,r_n^2) \] in repeated samples, with a credible interval centered at its posterior mean and radius \(z_{1-\alpha/2}r_n\). If \[ \frac{|b_n|}{r_n}\to\infty, \] then frequentist coverage tends to zero.
Proof. The interval contains \(\theta_0\) only if the stochastic centering error cancels a deterministic displacement much larger than its radius. Under tight normalized fluctuations, this probability vanishes.
This is the central nonparametric coverage problem: posterior spread can contract at the minimax stochastic rate while regularization bias is of the same or larger order.
Honesty and adaptation
Definition. A confidence set \(C_n\) is honest over a class \(\mathcal F\) if \[ \liminf_n\inf_{f\in\mathcal F} P_f(f\in C_n)\ge1-\alpha. \] It is adaptive over subclasses \(\mathcal F_s\) if its diameter achieves the corresponding rate on each \(\mathcal F_s\).
Fully honest adaptive confidence balls are impossible over many nested smoothness scales without excluding difficult functions. Self-similarity or polished-tail conditions rule out signals whose energy hides at unresolved frequencies and can restore coverage for empirical or hierarchical Bayes sets after inflation.
Weak and multiscale topologies
A nonparametric BvM theorem may hold in a weak norm even when it fails in \(L^2\) or supremum norm. Credible sets built in that weak topology can have honest coverage but may be too large or difficult to interpret pointwise. Bands for \(f(t)\), balls for \(f\), and intervals for a smooth functional \(\int\varphi f\) are different inferential objects and require separate bias analyses.
Calibration workflow
For any reported credible set, identify:
- the parameter or functional and topology;
- the posterior center and radius;
- the contraction rate and regularization bias;
- whether a BvM or matching theorem applies;
- the parameter class over which coverage is claimed;
- whether inflation, undersmoothing, or bias correction is required.
Simulation over representative fixed truths can reveal finite-sample undercoverage, but it cannot establish uniform honesty over an infinite class.