Cholesky Posterior Recalibration for Survey-Weighted Inference

In survey-weighted Bayesian models the pseudo-posterior covariance $\Sigma_{\mathrm{MCMC}}$ typically over-covers because the prior contributes substantially to the curvature (Prior Inflation ratios of 600–6500 are common for fixed effects). The Cholesky correction constructs an affine map $$ \theta^{*(m)} = \hat\theta + A\bigl(\theta^{(m)} - \hat\theta\bigr), \quad A = L_{\mathrm{sand}}\, L_{\mathrm{MCMC}}^{-1}, $$ where $L_{\mathrm{sand}}$ and $L_{\mathrm{MCMC}}$ are the lower-triangular Cholesky factors of the sandwich variance $V_{\mathrm{sand}}$ and the MCMC covariance $\Sigma_{\mathrm{MCMC}}$, respectively.

Usage

cholesky_correct(fit, sandwich, level = 0.95)

Arguments

fit: An object of class "hbb_fit" returned by hbb or similar model-fitting function. Must contain a CmdStanR fit accessible via fit$fit$draws().
sandwich: An object of class "hbb_sandwich" returned by sandwich_variance. Must contain V_sand, Sigma_MCMC, H_obs_inv, DER, and param_labels.
level: Numeric scalar in $(0,1)$. Confidence level for interval construction. Default is 0.95.

Value

An S3 object of class "hbb_cholesky" containing:

theta_corrected: Numeric matrix of dimension M by D containing the Cholesky-corrected posterior draws.
theta_hat: Numeric vector of length D: posterior means (column means of the original draws).
A: The D by D affine transformation matrix $A = L_{\mathrm{sand}}\,L_{\mathrm{MCMC}}^{-1}$.
L_MCMC: Lower-triangular Cholesky factor of the MCMC posterior covariance.
L_sand: Lower-triangular Cholesky factor of the sandwich variance.
comparison_table: Data frame with one row per parameter containing naive, corrected, and Wald confidence intervals, width ratios, DER, DER-vs-MCMC, prior inflation, and the square root of DER.
verification: Named list of numerical verification checks: mean preservation error, relative variance error, A algebraic identity error, and logical pass/fail flags.
level: The confidence level used.
D: Integer: number of parameters (2P + 1).
M: Integer: number of MCMC draws.
P: Integer: number of covariates per margin.
pd_corrections: List of PD correction details for each matrix (whether ridge or nearPD was applied, condition numbers).

Details

Applies the affine Cholesky correction of Williams and Savitsky (2021, Theorem 4.1) to MCMC posterior draws, replacing the prior-dominated posterior covariance with the design-consistent sandwich variance while preserving the posterior mean.

Mathematical properties

The transformation satisfies three exact algebraic identities:

Mean preservation. $E[\theta^*] = \hat\theta$ because the affine map is centred at the posterior mean.
Covariance recovery. $$ \mathrm{Cov}(\theta^*) = A\,\Sigma_{\mathrm{MCMC}}\,A^\top = L_{\mathrm{sand}}\,L_{\mathrm{MCMC}}^{-1}\, \Sigma_{\mathrm{MCMC}}\, L_{\mathrm{MCMC}}^{-\top}\,L_{\mathrm{sand}}^\top = V_{\mathrm{sand}}. $$ The middle cancellation uses $\Sigma_{\mathrm{MCMC}} = L_{\mathrm{MCMC}}\,L_{\mathrm{MCMC}}^\top$.
Affine equivariance. Quantile ordering is preserved under monotone marginal transformations, so credible intervals from the corrected draws are valid frequentist confidence intervals with correct nominal coverage.

Prior domination and shrinkage

For parameters where the prior dominates the likelihood (Prior Inflation $\mathrm{PI}_p = \Sigma_{\mathrm{MCMC},pp} / H_{\mathrm{obs},pp}^{-1} \gg 1$), the diagonal of $A$ is much less than unity (typically 0.01–0.06). This shrinks the over-dispersed MCMC draws towards $\hat\theta$, replacing prior-inflated credible intervals with data-driven confidence intervals. This is the expected and correct behaviour: the sandwich variance $V_{\mathrm{sand}}$ is smaller than both $\Sigma_{\mathrm{MCMC}}$ and $H_{\mathrm{obs}}^{-1}$ when design effects are moderate.

For parameters where the prior contributes little (e.g., log_kappa with $\mathrm{PI} \approx 2.3$), $A_{pp} \approx 1.06$, producing slight inflation consistent with design-effect adjustment.

Design Effect Ratio

The DER is the key diagnostic for the correction magnitude: $$ \mathrm{DER}_p = \frac{V_{\mathrm{sand},pp}}{H_{\mathrm{obs},pp}^{-1}}. $$ Values in the range 1–5 are typical for complex survey designs.

Relationship to Wald inference

Wald confidence intervals $\hat\theta_p \pm z_{1-\alpha/2}\,\sqrt{V_{\mathrm{sand},pp}}$ are algebraically equivalent to the marginal quantiles of the corrected draws in large samples. They are recommended as the primary reporting device because they do not depend on MCMC sampling variability.

References

Williams, M. R. and Savitsky, T. D. (2021). Uncertainty estimation for pseudo-Bayesian inference under complex sampling. International Statistical Review, 89(1), 72–107. doi:10.1111/insr.12376

Ghosal, R., Ghosh, S. K., and Maiti, T. (2020). Two-part regression models for longitudinal zero-inflated count data. Journal of the Royal Statistical Society: Series A, 183(4), 1603–1626.

Examples

if (FALSE) { # \dontrun{
# After fitting and computing sandwich variance:
sand <- sandwich_variance(fit)
chol_obj <- cholesky_correct(fit, sand, level = 0.95)
print(chol_obj)

# Compare interval widths
chol_obj$comparison_table[, c("parameter", "naive_width",
                              "corrected_width", "wald_width")]
} # }