Exact Bayesian and Frequentist Corrections

• Exact Bayesian and Frequentist Corrections are methodologies that reconcile Bayesian posterior inferences to achieve frequentist error controls and coverage matching.
• They leverage p-value based updates and the extremity proposition to align Bayes factors with frequentist tests for consistent decision making.
• The approach extends to complex scenarios such as measurement error and multiplicity corrections, offering precise interval adjustments and robust error rate calibration.

Exact Bayesian and Frequentist Corrections serve as a rigorous bridge between Bayesian and frequentist inferential paradigms, providing methodologies whereby Bayesian inferences can be rendered with frequentist error-control properties and, conversely, frequentist procedures are interpreted as explicit Bayesian updates under specific priors or information sets. Corrections of this kind include exact calibration of posteriors for coverage matching, finite-sample familywise error control under model uncertainty, measurement-error treatment beyond large-sample approximations, and precise mapping between credible and confidence intervals. The following sections synthesize the main frameworks, results, and implications encompassing exact Bayesian and frequentist corrections, with reference to foundational and recent advances.

1. Foundations: Bayesian and Frequentist Paradigms

Bayesian inference is grounded in updating prior beliefs via Bayes’ theorem upon observing data, yielding posterior distributions that quantify uncertainty. Credible intervals and model selection are derived from this posterior. Frequentist inference conceptualizes parameters as fixed and assesses procedures by long-run frequency properties, such as coverage probability of confidence intervals or familywise error rates.

A longstanding issue is that Bayesian credible intervals do not in general coincide with frequentist confidence intervals in finite samples, and Bayesian multiple-testing controls differ from strong frequentist standards (such as FWER). However, certain procedural and prior modifications (“exact corrections”) enable reconciliation between these frameworks, either by calibrating Bayesian outputs to satisfy frequentist guarantees or by identifying cases in which Bayesian and frequentist answers are strictly aligned.

2. Calibration via the Extremity Proposition and p-value-Based Updates

The extremity proposition, as developed in "Bayesian questions with frequentist answers" (Guth et al., 2023), formalizes a Bayesian update based not on the precise value observed but on its extremity with respect to each hypothesis θ. For observed statistic $s$ and hypothesis θ, the extremity region $U_θ(s)$ is defined as those outcomes at least as extreme as $s$ under θ. Updating the prior $p(θ)$ using the probability $p_\mathrm{val,θ}(s) = \int_{U_θ(s)} p(t|θ)\,dt$, the Bayesian posterior is

$$p(θ|E) = \frac{p(θ)\,p_\mathrm{val,θ}(s)}{\int p(θ')\,p_\mathrm{val,θ'}(s)\,dθ'}.$$

The Bayes factor for θ versus θ' is the ratio of the corresponding p-values: $$B_{θ,θ'} = p_\mathrm{val,θ}(s)/p_\mathrm{val,θ'}(s)$$. This is an exact algebraic mapping, requiring no distributional or parametric restrictions beyond the extremity-based update. When σ (measurement precision) is small, this "p-value" update closely approximates the full-data Bayesian update, and in general, it provides an explicit route to interpret reported p-values in terms of Bayesian evidence adjustment for arbitrary priors. The method is applicable to both discrete (e.g., point hypothesis) and continuous parameter spaces.

3. Coverage Matching: Posterior Adjustment for Confidence-Interval Calibration

Applications such as cosmological parameter inference require credible intervals that not only reflect posterior variability but also possess frequentist coverage properties. In "Matching Bayesian and frequentist coverage probabilities when using an approximate data covariance matrix" (Percival et al., 2021), a solution is to inflate the power-law prior on the data covariance Σ beyond the independence-Jeffreys prior. The matching-coverage prior is given by

$$p(θ, Σ) \propto |\Sigma|^{-\frac{m_\mathrm{match}-n_s+n_d+1}{2}},$$

with $m_\mathrm{match}$ chosen such that the expected posterior covariance exactly equals the sampling covariance of the MLEs under repeated realizations:

$$m_\mathrm{match} = n_θ + 2 + \frac{n_s-1 + B(n_d-n_θ)}{1 + B(n_d-n_θ)}, \quad B = \frac{n_s - n_d - 2}{(n_s - n_d - 1)(n_s - n_d - 4)}.$$

Under this prior, posterior credible regions derived from the marginal t-distribution for θ have (approximated) exact coverage, as validated by simulation in complex cosmology-like scenarios. The method is implemented by specifying $n_s$ (number of mocks), $n_d$ (data dimensionality), and $n_θ$ (number of parameters), and constructing the matched prior analytically.

4. Multiplicity Corrections and Model Uncertainty: Joint Bayesian–Frequentist Control

Multiplicity corrections under dependence and uncertainty are handled exactly in several frameworks.

A. Model-averaged hypothesis testing and FWER

The Doublethink methodology (Fryer et al., 2023) uses Bayesian model-averaged posterior odds (PO) as a shortcut closed-testing procedure, constructed over all submodels indexed by binary vectors. The familywise error rate (FWER) under this approach is controlled in the strong sense for any finite sample size, and asymptotically calibrated by chi-squared tail bounds:

$$\Pr[\mathrm{PO}_{\mathcal{A}_v:\mathcal{O}_v} \geq \tau] \lesssim \Pr[\chi^2_1 \geq 2\ln(\tau/(\nu\mu\sqrt{\xi_n}))],$$

where $\tau$ is determined by the desired FWER level and hyperparameters $\mu$, $\xi_n$. This allows simultaneous posterior-odds-based variable selection and frequentist multiple-testing correction, with high simulation accuracy in high-dimensional settings.

B. Multiplicity for mutually exclusive signals

For testing $n$ mutually exclusive signals under dependence, exact FWER control is achievable either by the analytic critical value for the maximum test statistic under the true covariance structure, or by setting the Bayesian posterior-selection threshold $p$ to solve

$$\mathrm{FPP}(p) = \Pr_{M_0}(\max_i P(M_i|X)>p)\le\alpha$$

using analytic or simulation-based inversion (Chang et al., 2016). Under adaptive $\tau^2$, the Bayesian approach—though generally more conservative for fixed $\tau^2$—matches LRT power and maintains FWER control across non-trivial correlation structures.

5. Exact Correction and Alignment of Intervals

A key advance in the alignment of confidence and credible intervals involves mathematical construction of a prior—frequentist prior $\pi_f$—such that the resulting Bayesian credible interval coincides with the Neyman confidence interval for all data (Bitioukov et al., 2012):

$$\pi_f(\lambda; x_\mathrm{obs}) = -\frac{\int_a^{x_\mathrm{obs}} \partial_\lambda f(x|\lambda) dx}{f(x_\mathrm{obs}|\lambda)}$$

(where $f(x|\lambda)$ is the likelihood). For symmetric central intervals in location or scale families, $\pi_f$ reduces to the Jeffreys prior, but in asymmetric or boundary cases, $\pi_f$ depends on $x_\mathrm{obs}$. This construction ensures that, for one-parameter models with suitable Neyman belts, Bayesian credible intervals are exactly frequentist confidence intervals.

An alternative simulation-based correction employs adjustment factors derived from the observed "bias" of the initial interval limits, estimated by repeated sampling under a consistent parameter estimator (Menendez et al., 2012). The corrected interval limits $L_c$, $U_c$ satisfy

$$L_c(x) = L(x) + \hat{\xi}_{\alpha/2}, \quad U_c(x) = U(x) + \hat{\xi}_{1-\alpha/2}$$

where adjustment quantiles $\hat{\xi}_{\cdot}$ are empirically estimated. This framework is model-agnostic and achieves asymptotically exact coverage in both frequentist and Bayesian construction.

6. Handling Covariate Measurement Error and Uncertainty

Exact Bayesian corrections for measurement error model the latent covariate and error-process jointly, integrating over posterior uncertainty via MCMC (Bartlett et al., 2016). Posterior means are consistent and efficient under correct specification, and credible intervals have nominal frequentist coverage in moderate samples, outperforming regression calibration, especially under nonlinearity or low reliability. The approach is computationally tractable given modern software, with flexibility to handle both measurement error and missing data through hierarchical modeling.

Errors on errors—uncertainty in reported variances—are exactly addressed by both Bayesian (gamma prior on precision) and frequentist (chi-square variate for sample variance) approaches, with parameter-by-parameter correspondence (Mishima et al., 10 May 2025). For the $i$th measurement:

• Bayesian: $\omega_i \sim \Gamma(k_i, \lambda_i)$ prior, updated to the posterior; marginal likelihood is Student-$t$-like.
• Frequentist: $w_i \sim \Gamma((n_i-1)/2, (n_i-1)/(2v_i))$ sampling model, with confidence limits based on known quantiles.

The table below summarizes this mapping:

Frequentist ($n_i$)	Bayesian ($k_i,\lambda_i$)
Precision $\Gamma[(n_i-1)/2, \dots]$	Prior $\Gamma(k_i, \lambda_i)$
$k_i = (n_i-1)/2$	$k_i = (n_i-1)/2$
$\lambda_i = (n_i-1)/(2v_i)$	$\lambda_i = (n_i-1)/(2v_i)$

This equivalence enables unified propagation of error–on–error and extraction of exact finite-sample bounds for inference.

7. Practical Implications and Limitations

Exact calibration and correction frameworks facilitate:

• Construction of credible intervals with provable frequentist coverage, particularly in fields requiring conservative claims (cosmology, particle physics).
• Application of Bayesian multiple-testing regimes that adhere to frequentist error guarantees for strong policy control (genomics, high-dimensional inference).
• Propagation and adjustment for structured uncertainties (covariate error, uncertainty in error bars).

Limitations include:

• The necessity, in some approaches, to accept weaker conditioning (e.g., extremity proposition, rather than full data);
• Possible non-invariance under reparameterization for certain matched priors;
• Increased computational demand for simulation-based interval correction;
• Restriction to specific model or belt structures for certain prior constructions;
• Open questions regarding extension to complex multivariate “extremity” definitions and to nonstandard model/nuisance parameter interactions.

Nonetheless, the current literature establishes a rich and operationally meaningful toolkit for exact Bayesian/frequentist correction, offering robust statistical guarantees for a wide range of scientific applications.