# Theoretical Addendum – Block 7 (Multivariate Extension):

### Empirical Bayes vs. Fully Bayes for Multivariate Population References \(\theta_{\mathrm{ref}} \in \mathbb{R}^p\) and Heterogeneous Distributional Slots \(K \geq 1\)

#### José Mauricio Gómez Julián

#### 2026-07-06

# 1. Purpose and Relation to v07

The vignette `v07_eb_vs_fb.Rmd` (henceforth v07) develops the EB-vs-FB comparison for the population reference parameter \(\theta_{\mathrm{ref}}\) under the simplifying assumption that \(\theta_{\mathrm{ref}} \in \mathbb{R}\) is a scalar hyperparameter and that the model has a single distributional slot (\(K = 1\)). The four central results — Theorem 7A (first-order equivalence), Proposition 7B (higher-order coverage discrepancy), Theorem 7C (compound decision bound), and Proposition 7D (substantial-discrepancy conditions) — are stated in that scalar single-slot setting.

The framework’s canonical Asymmetric Multiplicative Model (AMM) admits, however, configurations in which:

- The population reference is itself multivariate, \(\theta_{\mathrm{ref}} \in \mathbb{R}^p\) with \(p \geq 2\), when the anchored hierarchical structure has several upper-level components (e.g., \(p = 2\) for a paired location-scale anchor; \(p = 3\) for a triple anchoring a Beta-shape, Gamma-rate, and Tweedie-power slot jointly).
- The model has several distributional slots, \(K \geq 2\), when the family is mixture-typed (Zero-Inflated Poisson, Zero-Inflated Negative Binomial, Hurdle Poisson, Hurdle Negative Binomial) or heterogeneously typed (Beta in slot 1 + Gamma in slot 2; Student-\(t\) in slot 1 + Gamma in slot 2 + Beta in slot 3; etc.).
- Both extensions occur simultaneously, \(p > 1\) and \(K > 1\), in the full multivariate-and-heterogeneous regime that the framework’s release-gate (Charter §2.4 of Sub-phase 8.6) commits to support.

This vignette extends the four results of v07 to the full \((p \geq 1, K \geq 1)\) regime under the same family of regularity hypotheses — now suitably indexed by \(p\) and stated componentwise in \(K\). The extension is stated as four results with an asterisk:

- Theorem 7A* (multivariate first-order equivalence; §4).
- Proposition 7B* (matrix-valued coverage discrepancy; §5).
- Theorem 7C* (compound decision bound under \(K > 1\) heterogeneous slots; §6).
- Proposition 7D* (substantial-discrepancy conditions under \(p > 1\) and/or \(K > 1\); §7).

Each asterisked result reduces to its v07 counterpart as a corollary under \((p, K) = (1, 1)\). The body of v07 remains canonical for the scalar single-slot case and is not reopened in this extension.

## 1.1. Epistemic Posture and the Critical Audit of v07 §5

The extension is performed under three epistemic constraints documented in Charter §2.5 (Sub-phase 8.6):

1. Extension, not rewriting. v07’s Theorem 7A scalar statement and proof remain canonical in their domain (\(p = 1\), \(K = 1\)). The multivariate extension is a strictly stronger statement requiring strictly stronger hypotheses (information matrix positive definite rather than non-zero; weakly informative prior in \(\mathbb{R}^p\) rather than \(\mathbb{R}\); bounded \(C^2\) regularity of the log-marginal in a ball of \(\mathbb{R}^p\) rather than in an interval of \(\mathbb{R}\)).

2. Pedagogical hierarchy. Readers entering the EB-vs-FB comparison for the first time are referred to v07 (lower technical entry barrier; less notational overhead; identical conceptual content). This vignette is the technical extension for users whose model lives in \(p > 1\) and/or \(K > 1\).

3. Critical audit of v07 §5. Per Charter §2.5, the redaction of the multivariate Theorem 7A* is preceded by an explicit line-by-line audit of the scalar proof of v07 §5 intended to detect any implicit use of a property valid in \(\mathbb{R}\) but false in \(\mathbb{R}^p\) (e.g., real monotonicity; one-dimensional change-of-variables; integration by parts on \(\mathbb{R}\); sign of one-dimensional Fisher information taken as positivity rather than positive definiteness). The audit, summarized in §4.5 below, detected no error: v07 §5 transfers cleanly to \(\mathbb{R}^p\) under the strengthened hypotheses listed in §3.

If the audit had detected an implicit error, the redaction of this vignette would have been suspended and a parallel sub-sub-sub-phase 8.6.A.1 would have opened to reopen v07 explicitly. The fact that no error was detected confirms that v07’s “reference and argument” delegation to Petrone–Rousseau–Scricciolo (2014) and Rousseau–Szabo (2017) is faithful to those references: both treat hyperparameters on Polish spaces of which finite-dimensional \(\mathbb{R}^p\) is the routine specialization.

## 1.2. Reading Order

For a reader already familiar with v07: §§2–3 (notation and hypotheses), §4 (Theorem 7A*), §5 (Proposition 7B*), §§6–7 (Theorem 7C* and Proposition 7D*), §8 (recommendation table for the multivariate case), §9 (open questions including the numerical-anti-fragility question O5*-EBFB specific to multivariate Laplace), §10 (connections to the operational vignette `vop07` and to the API `gdpar_eb()`).

For a reader unfamiliar with v07: start with v07 first; return here once the scalar single-slot case is conceptually clear.

# 2. Notation (Multivariate Extension)

## 2.1. Multivariate Population Reference and Heterogeneous Slots

Let \(\eta = (\theta_{\mathrm{ref}}, a, b, W)\) as in Block 4 §2.1, with the partition into the upper-level parameter \(\theta_{\mathrm{ref}}\) and the lower-level components \(\xi = (a, b, W)\) unchanged. The multivariate extension introduces three structural dimensions:

- \(p := \dim(\theta_{\mathrm{ref}}) \geq 1\) is the dimensionality of the population reference. Under \(p = 1\), \(\theta_{\mathrm{ref}} \in \mathbb{R}\) as in v07; under \(p \geq 2\), \(\theta_{\mathrm{ref}} \in \mathbb{R}^p\) is a vector whose components anchor different aspects of the hierarchy jointly.
- \(K \geq 1\) is the number of distributional slots that the family declares. For a single-slot family (\(K = 1\), e.g., Gaussian or Poisson), \(\theta_{\mathrm{ref}}\) anchors that one slot. For a multi-slot family (\(K \geq 2\), e.g., Zero-Inflated Poisson with \(K = 2\) slots indexing \((\mu, \pi)\); Tweedie with \(K = 3\) slots indexing \((\mu, \phi, p_{\mathrm{Tweedie}})\); heterogeneous Beta+Gamma with \(K = 2\) slots), \(\theta_{\mathrm{ref}}\) anchors each slot via a slot-indexed sub-parameter \(\theta_{\mathrm{ref}}^{(k)} \in \mathbb{R}^{p_k}\) with \(\sum_{k=1}^K p_k = p\) (additively in the no-overlap case). The framework’s canonical implementation (Sub-phase 8.6.B–D of Charter §3) takes the slot-indexed case as the only operational one in the current release. A theoretically natural alternative —a shared \(\theta_{\mathrm{ref}}\) vector with cross-slot coupling across the \(K\) slots— exists in the multivariate AMM formulation but is scoped for a future block; it is not exposed as a configuration in `gdpar_eb()` or anywhere else in `gdpar 0.0.0.9001`.
- \(J \geq 1\) is the number of groups when the anchored hierarchy admits a grouping covariate (e.g., \(J\) Argentine provinces; \(J\) ecological sites). For \(J = 1\) the hierarchy reduces to the single-group case of Blocks 1–6.

The full upper-level parameter tensor of an AMM specification is therefore \(\theta_{\mathrm{ref}} \in \mathbb{R}^{J \times K \times p}\), with \((p, K, J) = (1, 1, 1)\) recovering the simplest setting and \((p, K, J)\) all \(\geq 2\) requiring the full machinery of this vignette.

## 2.2. Marginal Likelihood in \(\mathbb{R}^p\) and per-Slot Tensor under \(K > 1\)

The marginal likelihood of \(\theta_{\mathrm{ref}}\) integrating out the lower-level parameters under their prior \(\pi_\xi\) is, as in v07, \[L_n^{\mathrm{marg}}(\theta_{\mathrm{ref}}) \;:=\; \int p(Y_{1:n} \mid \theta_{\mathrm{ref}}, \xi)\, \pi_\xi(\xi)\, d\xi,\] now read as a function \(L_n^{\mathrm{marg}}: \mathbb{R}^{J \times K \times p} \to \mathbb{R}_{\geq 0}\).

The Empirical Bayes estimator of \(\theta_{\mathrm{ref}}\) is \[\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}} \;=\; \mathop{\arg\max}_{\theta_{\mathrm{ref}} \in \Theta}\; L_n^{\mathrm{marg}}(\theta_{\mathrm{ref}}),\] with \(\Theta \subseteq \mathbb{R}^{J \times K \times p}\) the admissible domain (typically the Cartesian product of per-component intervals or all of \(\mathbb{R}^{J \times K \times p}\)).

The Fisher information matrix of the marginal likelihood at the population truth \(\theta_{\mathrm{ref}}^*\) is \[I_{\theta\theta}^{\mathrm{marg}}(\theta_{\mathrm{ref}}^*) \;:=\; \mathbb{E}_{\eta_*}\!\left[ - \nabla^2_{\theta_{\mathrm{ref}}} \log L_n^{\mathrm{marg}}(\theta_{\mathrm{ref}})\big|_{\theta_{\mathrm{ref}}^*} \right] \;\in\; \mathbb{R}^{p \times p}.\] Under \(p = 1\), \(I_{\theta\theta}^{\mathrm{marg}}\) is a positive scalar (the Fisher information of v07 §4). Under \(p \geq 2\), \(I_{\theta\theta}^{\mathrm{marg}}\) is a \(p \times p\) symmetric matrix; the strictly stronger identifiability assumption (EB-MARG-ID)\(_p\) of §3 requires it to be positive definite (all eigenvalues strictly positive), not merely non-zero.

Under \(K > 1\) with slot-indexed sub-parameters \(\theta_{\mathrm{ref}}^{(k)} \in \mathbb{R}^{p_k}\), the Fisher information becomes a per-slot block tensor of marginal information matrices: \[I_{\theta\theta}^{\mathrm{marg}, K}(\theta_{\mathrm{ref}}^*) \;=\; \left\{ I_{\theta\theta}^{\mathrm{marg}, k}(\theta_{\mathrm{ref}}^{*,(k)}) \right\}_{k=1}^K, \qquad I_{\theta\theta}^{\mathrm{marg}, k} \;\in\; \mathbb{R}^{p_k \times p_k}.\] Under the factored prior \(\pi_\xi = \prod_{k=1}^K \pi_{\xi^{(k)}}\) of Theorem 7C* (§6), the block tensor is block-diagonal and each block \(I_{\theta\theta}^{\mathrm{marg}, k}\) is the Fisher information of the slot-\(k\) marginal likelihood. Under the coupled prior of §6, off-block cross terms appear and the full Fisher matrix \(I_{\theta\theta}^{\mathrm{marg}} \in \mathbb{R}^{p \times p}\) is not block-diagonal.

## 2.3. Sensitivity Jacobian of the Conditional Posterior

A novel object that has no counterpart in v07 (because v07’s scalar Proposition 7B absorbs it into a constant) is the sensitivity Jacobian of the lower-level conditional posterior mode with respect to the upper-level reference: \[J^\xi(\theta_{\mathrm{ref}}) \;:=\; \frac{\partial\, \xi^*(\theta_{\mathrm{ref}})}{\partial\, \theta_{\mathrm{ref}}} \;\in\; \mathbb{R}^{\dim(\xi) \times p},\] where \(\xi^*(\theta_{\mathrm{ref}}) := \arg\max_\xi \Pi_n^{\mathrm{cond}}(\xi \mid \theta_{\mathrm{ref}}, \cdot)\) is the lower-level conditional posterior mode given \(\theta_{\mathrm{ref}}\). By the implicit-function theorem applied to the conditional first-order condition \(\nabla_\xi \log p(Y_{1:n} \mid \theta_{\mathrm{ref}}, \xi^*) + \nabla_\xi \log \pi_\xi(\xi^*) = 0\), \[J^\xi(\theta_{\mathrm{ref}}) \;=\; -\left[ \nabla^2_{\xi\xi} \log \Pi_n^{\mathrm{cond}}(\xi^* \mid \theta_{\mathrm{ref}}, \cdot) \right]^{-1} \cdot \nabla^2_{\xi\theta_{\mathrm{ref}}} \log p(Y_{1:n} \mid \theta_{\mathrm{ref}}, \xi^*),\] which is well-defined under (HIER-COMPLEX)\(_p\) of §3. Under \(p = \dim(\xi) = 1\), \(J^\xi\) is a scalar absorbed into v07’s constant \(C_{g,\alpha}\) without explicit naming. Under \(p \geq 2\), the matrix \(J^\xi\) enters explicitly in the sandwich form of Proposition 7B* (§5).

## 2.4. Total Variation, Weak Metric, and Posterior Distance

The asymptotic comparison of \(\Pi_n^{\mathrm{EB}}\) and \(\Pi_n^{\mathrm{FB}}\) over \(\xi\) uses two metrics, paralleling v07 §2.2:

- Total variation \(d_{\mathrm{TV}}(\Pi_n^{\mathrm{EB}}, \Pi_n^{\mathrm{FB}}) = \sup_B |\Pi_n^{\mathrm{EB}}(\xi \in B) - \Pi_n^{\mathrm{FB}}(\xi \in B)|\) for finite-dim parametric \(\xi\) (Regime A of v07 §5).
- Weak metric over smooth functionals \(\sup_{g \in \mathcal{G}} |\mathbb{E}_{\Pi_n^{\mathrm{EB}}}[g(\xi)] - \mathbb{E}_{\Pi_n^{\mathrm{FB}}}[g(\xi)]|\) over a class \(\mathcal{G}\) of bounded \(L^2(\mu)\)-Lipschitz functionals for non-parametric \(\xi\) (Regime B of v07 §5).

The dimensionality of \(\theta_{\mathrm{ref}}\) (\(p\)) and the dimensionality of \(\xi\) are independent: the multivariate extension of this vignette concerns \(p\); the parametric-vs-non-parametric distinction of \(\xi\) governs the choice of metric. The four resulting combinations are tabulated in §4.

## 2.5. Notation Summary

Symbol | Meaning
--- | ---
\(p\) | Dimension of \(\theta_{\mathrm{ref}}\) as a vector
\(K\) | Number of distributional slots in the family
\(J\) | Number of groups in the anchored hierarchy
\(\theta_{\mathrm{ref}} \in \mathbb{R}^{J \times K \times p}\) | Full upper-level parameter tensor
\(\theta_{\mathrm{ref}}^{(k)} \in \mathbb{R}^{p_k}\) | Slot-\(k\) sub-parameter under \(K > 1\), \(\sum_k p_k = p\) in the no-overlap case
\(\xi = (a, b, W)\) | Lower-level parameters (function-valued)
\(\pi_\Theta, \pi_\xi\) | Priors on \(\theta_{\mathrm{ref}}\) and \(\xi\)
\(L_n^{\mathrm{marg}}(\theta_{\mathrm{ref}})\) | Marginal likelihood of \(\theta_{\mathrm{ref}}\)
\(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}} \in \mathbb{R}^p\) | EB estimator (Type II ML)
\(I_{\theta\theta}^{\mathrm{marg}} \in \mathbb{R}^{p \times p}\) | Fisher information matrix of \(L_n^{\mathrm{marg}}\) at \(\theta_{\mathrm{ref}}^*\)
\(I_{\theta\theta}^{\mathrm{marg}, K} = \{I_{\theta\theta}^{\mathrm{marg}, k}\}_{k=1}^K\) | Per-slot tensor of Fisher information matrices under \(K > 1\)
\(\Pi_n^{\mathrm{EB}}, \Pi_n^{\mathrm{FB}}\) | EB and FB posteriors over \(\xi\)
\(\Pi_n^{\mathrm{cond}}(\xi \mid \theta_{\mathrm{ref}}, \cdot)\) | Conditional posterior of \(\xi\) given \(\theta_{\mathrm{ref}}\)
\(\xi^*(\theta_{\mathrm{ref}})\) | Conditional posterior mode of \(\xi\) given \(\theta_{\mathrm{ref}}\)
\(J^\xi(\theta_{\mathrm{ref}}) \in \mathbb{R}^{\dim(\xi) \times p}\) | Sensitivity Jacobian of \(\xi^*\) w.r.t. \(\theta_{\mathrm{ref}}\)
\(d_{\mathrm{TV}}, d_{\mathrm{weak}}\) | Posterior distances (TV and weak-over-functionals)
\(C_{g,\alpha}^* \in \mathbb{R}^{p \times p}\) | Matrix coverage discrepancy constant of Proposition 7B*
\(\kappa(H)\) | Condition number of the Hessian \(H\) (numerical anti-fragility, §9 O5*)

# 3. Standing Hypotheses (Multivariate Extension)

The three hypotheses of v07 §4 — (EB-MARG-ID), (PRIOR-FB-WEAK), (HIER-COMPLEX) — extend to the multivariate setting as follows. The asterisk on each hypothesis name denotes the strictly stronger requirement of the \(p \geq 1\) (and \(K \geq 1\)) regime; under \(p = K = 1\) each asterisked hypothesis reduces to its v07 counterpart.

## 3.1. (EB-MARG-ID)\(_p\) – Multivariate Marginal Identifiability

> (EB-MARG-ID)\(_p\). \(L_n^{\mathrm{marg}}(\theta_{\mathrm{ref}})\) has a unique global maximum on the admissible domain \(\Theta \subseteq \mathbb{R}^p\) in the limit \(n \to \infty\), with the limiting Fisher information matrix \[I_{\theta\theta}^{\mathrm{marg}}(\theta_{\mathrm{ref}}^*) \;\in\; \mathbb{R}^{p \times p}\] positive definite: all \(p\) eigenvalues \(\lambda_1 \geq \cdots \geq \lambda_p\) of \(I_{\theta\theta}^{\mathrm{marg}}\) satisfy \(\lambda_p > 0\).

Compared to v07’s scalar (EB-MARG-ID), the requirement is strictly stronger: in \(\mathbb{R}\) “non-zero Fisher information” is equivalent to positivity, but in \(\mathbb{R}^p\) “non-singularity” (det \(\neq 0\)) does not imply positive definiteness in general — and only the latter ensures that the multivariate Bernstein–von Mises (BvM) limit of the marginal posterior is a proper \(p\)-variate Gaussian with covariance \(n^{-1} (I_{\theta\theta}^{\mathrm{marg}})^{-1}\). The smallest eigenvalue \(\lambda_p\) controls the effective sample size for the most poorly identified direction: when \(\lambda_p\) is small, \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) has a large variance along the eigenvector of \(\lambda_p\), and the gap between EB and FB widens in that direction (Proposition 7D* (ii)*, §7).

Under \(K > 1\) with slot-indexed \(\theta_{\mathrm{ref}}^{(k)} \in \mathbb{R}^{p_k}\), (EB-MARG-ID)\(_p\) generalizes componentwise: each block matrix \(I_{\theta\theta}^{\mathrm{marg}, k} \in \mathbb{R}^{p_k \times p_k}\) must be positive definite. Under the factored prior \(\pi_\xi = \prod_k \pi_{\xi^{(k)}}\) of §6, this is equivalent to (EB-MARG-ID)\(_p\) for \(L_n^{\mathrm{marg}}\) as a whole; under coupled prior, the cross-block terms must satisfy a strict-diagonal-dominance condition for the full matrix to remain positive definite.

## 3.2. (PRIOR-FB-WEAK)\(_p\) – Weak FB Prior in \(\mathbb{R}^p\)

> (PRIOR-FB-WEAK)\(_p\). The FB prior \(\pi_\Theta\) on \(\theta_{\mathrm{ref}} \in \mathbb{R}^p\) has positive density at \(\theta_{\mathrm{ref}}^*\) and the FB posterior asymptotically dominates the prior componentwise, in the sense that \[\frac{\mathrm{tr}\,\mathrm{Cov}_n(\theta_{\mathrm{ref}}^*)}{\mathrm{tr}\,\mathrm{Cov}_\pi(\theta_{\mathrm{ref}})} \;\to\; 0 \qquad \text{as } n \to \infty.\] Equivalently, the maximum eigenvalue of \(\mathrm{Cov}_n(\theta_{\mathrm{ref}}^*) \cdot \mathrm{Cov}_\pi(\theta_{\mathrm{ref}})^{-1}\) tends to zero.

The trace-based formulation is the natural multivariate analog of v07’s scalar \(\mathrm{Var}_n / \mathrm{Var}_\pi \to 0\). Under \(p = 1\), it reduces exactly to v07’s (PRIOR-FB-WEAK). The eigenvalue-based equivalent formulation has the advantage of being invariant under linear reparametrization of \(\theta_{\mathrm{ref}}\): a reparametrization \(\theta_{\mathrm{ref}} \mapsto A \theta_{\mathrm{ref}}\) leaves the eigenvalues of the covariance ratio unchanged. The trace formulation is sufficient for the asymptotic theory of §§4–6 and is what the framework’s library reports as a diagnostic in `gdpar_eb_fit$diagnostics$prior_dominance_ratio` (Sub-phase 8.6.B, Charter §3.2).

The hypothesis is satisfied by typical multivariate priors (independent Gaussian per component with sufficient variance; multivariate Cauchy; multivariate Student-\(t\)). It fails when the prior is strongly informative on one component while the data informs that component weakly (Proposition 7D* (iii)*, §7).

## 3.3. (HIER-COMPLEX)\(_p\) – Multivariate Hierarchical Regularity

> (HIER-COMPLEX)\(_p\). The log-marginal likelihood \(\log L_n^{\mathrm{marg}}(\theta_{\mathrm{ref}})\) is twice continuously differentiable on an open neighborhood \(B_\delta(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}) \subseteq \mathbb{R}^p\) of radius \(\delta > 0\), with Hessian bounded uniformly in \(n\). The number of upper-level hyperparameters \(p\) is bounded as \(n\) grows.

This combines v07’s scalar (HIER-COMPLEX) (bounded number of upper-level hyperparameters) with the new \(C^2\) regularity requirement that supports the multivariate Laplace approximation of step (i) in §11.1 of v07 — namely, that the marginal likelihood can be expanded to second order around \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) in \(\mathbb{R}^p\) with the Hessian inverse providing the Gaussian curvature of the approximation. Under \(p = 1\), \(C^2\) regularity on an interval is the same condition required (implicitly) by v07 §5; the multivariate extension makes it explicit because it is the operative regularity used in the proof of Theorem 7A* (§4) and in the matrix-correction form of Proposition 7B* (§5).

The boundedness of \(p\) as \(n\) grows is the standard hierarchical complexity bound. If \(p\) itself grew with \(n\) (e.g., \(p = p_n \to \infty\)), the analysis would require non-parametric extensions and is deferred to Block 9.x (Open Question O3*-EBFB, §9). The framework’s release-gate (Charter §2.4) treats \(p\) as fixed (chosen by the analyst from the family specification) and \(n\) as the asymptotic index.

## 3.4. Discussion: Why the Asterisked Hypotheses are Strictly Stronger

v07 (scalar) | v07b (multivariate) | Strict strengthening
--- | --- | ---
\(I_{\theta\theta}^{\mathrm{marg}} \neq 0\) (positivity) | \(I_{\theta\theta}^{\mathrm{marg}} \succ 0\) (positive definiteness) | All \(p\) eigenvalues strictly positive
\(\mathrm{Var}_n / \mathrm{Var}_\pi \to 0\) | \(\mathrm{tr\,Cov}_n / \mathrm{tr\,Cov}_\pi \to 0\) | Componentwise data swamping
\(C^2\) in an interval | \(C^2\) in a \(p\)-ball | Multivariate Laplace second-order
\(p\) scalar fixed | \(p \in \mathbb{N}\) fixed | Boundedness of upper-level dim

The strengthenings are not stylistic conveniences: each is necessary to support the multivariate proof of §4. In particular, (EB-MARG-ID)\(_p\) as positive definiteness (not merely non-zero determinant) is the gateway condition for the multivariate BvM of the marginal posterior; without it, the posterior may concentrate as a degenerate Gaussian on a lower-dimensional subspace, breaking the equivalence with the conditional EB posterior.

# 4. Theorem 7A* – First-Order Asymptotic Equivalence (Multivariate)

> Theorem 7A*. First-order asymptotic equivalence in \(\mathbb{R}^p\), \(K \geq 1\). Under the framework’s standing hypotheses (C1)–(C6), (LIN), (D-ID) of Block 1; the asymptotic hypotheses of Theorem 4A of Block 4; plus the multivariate extensions (EB-MARG-ID)\(_p\), (PRIOR-FB-WEAK)\(_p\), (HIER-COMPLEX)\(_p\) of §3, the EB and FB posteriors over \(\xi\) agree asymptotically. The metric in which agreement holds depends jointly on the dimensionality of \(\xi\) (parametric vs. non-parametric) and the dimensionality \(p\) of \(\theta_{\mathrm{ref}}\).

The four-cell classification:

| \(\xi\) finite-dim parametric (Regime A) | \(\xi\) non-parametric (Regime B)
--- | --- | ---
\(p = 1\) (scalar \(\theta_{\mathrm{ref}}\)) | \(d_{\mathrm{TV}}(\Pi_n^{\mathrm{EB}}, \Pi_n^{\mathrm{FB}}) \xrightarrow{P_{\eta_*}} 0\). This is v07’s Theorem 7A Regime A. | \(\sup_{g \in \mathcal{G}} |\mathbb{E}^{\mathrm{EB}}[g(\xi)] - \mathbb{E}^{\mathrm{FB}}[g(\xi)]| \xrightarrow{P_{\eta_*}} 0\). This is v07’s Theorem 7A Regime B.
\(p \geq 2\) (multivariate \(\theta_{\mathrm{ref}}\)) | \(d_{\mathrm{TV}}(\Pi_n^{\mathrm{EB}}, \Pi_n^{\mathrm{FB}}) \xrightarrow{P_{\eta_*}} 0\) in the finite-dim parameter space. Multivariate BvM for the marginal posterior of \(\theta_{\mathrm{ref}}\) ensures concentration at parametric rate \(n^{-1/2}\) in \(\mathbb{R}^p\). | \(\sup_{g \in \mathcal{G}} |\mathbb{E}^{\mathrm{EB}}[g(\xi)] - \mathbb{E}^{\mathrm{FB}}[g(\xi)]| \xrightarrow{P_{\eta_*}} 0\). The functional class \(\mathcal{G}\) depends on \(\xi\) only; the dimension \(p\) of \(\theta_{\mathrm{ref}}\) enters only through the concentration rate of the marginal posterior in \(\mathbb{R}^p\).

Under \(K > 1\) with slot-indexed sub-parameters and factored prior \(\pi_\xi = \prod_{k=1}^K \pi_{\xi^{(k)}}\), the four-cell classification holds slotwise: the posterior factorizes as \(\Pi_n^{\bullet}(\xi \mid \cdot) = \prod_{k=1}^K \Pi_n^{\bullet}(\xi^{(k)} \mid \cdot)\) for \(\bullet \in \{\mathrm{EB}, \mathrm{FB}\}\), and the metric for the full posterior is the product metric. Under coupled prior, cross-slot terms contribute as in Theorem 7C* (§6) but the first-order equivalence (Theorem 7A*) is preserved.

Corollary (reduction to v07). Under \((p, K) = (1, 1)\), Theorem 7A* reduces to v07’s Theorem 7A. The two left-column cells of the four-cell table are exactly v07’s two-regime statement of v07 §5.

## 4.1. Proof Sketch (Extension of v07’s Reference Argument)

The proof of v07 §5 (the “reference and argument”) generalizes to \(\mathbb{R}^p\) in four steps, each retracing the v07 argument with the strengthened hypotheses of §3.

Step 1 – FB posterior decomposition. As in v07, \[\Pi_n^{\mathrm{FB}}(\xi \mid \cdot) \;=\; \int_{\mathbb{R}^p} \Pi_n^{\mathrm{cond}}(\xi \mid \theta_{\mathrm{ref}}, \cdot)\; \Pi_n^{\mathrm{marg}}(\theta_{\mathrm{ref}} \mid \cdot)\; d\theta_{\mathrm{ref}}.\] The decomposition is algebraically valid for any \(p \geq 1\); the dimensionality enters only through the domain of integration.

Step 2 – Marginal posterior concentration via multivariate BvM. Under (EB-MARG-ID)\(_p\) (positive definite Fisher) and (PRIOR-FB-WEAK)\(_p\) (data swamps prior), the multivariate Bernstein–von Mises theorem (van der Vaart 1998, Chapter 10; Castillo and Rousseau 2015 for smooth functionals; Rousseau and Szabo 2017 §3 for total variation quantification) gives \[d_{\mathrm{TV}}\!\left(\, \Pi_n^{\mathrm{marg}}(\theta_{\mathrm{ref}} \mid \cdot),\; \mathcal{N}\!\left(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}},\; n^{-1} \bigl(I_{\theta\theta}^{\mathrm{marg}}\bigr)^{-1}\right) \right) \;\xrightarrow{P_{\eta_*}}\; 0.\] The covariance \(n^{-1} (I_{\theta\theta}^{\mathrm{marg}})^{-1}\) is the inverse \(p \times p\) matrix Fisher info; its existence requires the positive definiteness of (EB-MARG-ID)\(_p\), not merely the non-singularity. In particular, the concentration rate is uniform across all directions in \(\mathbb{R}^p\), with the slowest direction governed by the smallest eigenvalue \(\lambda_p > 0\) of \(I_{\theta\theta}^{\mathrm{marg}}\). This step is the multivariate generalization of v07’s “the marginal posterior of \(\theta_{\mathrm{ref}}\) concentrates at \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\)”.

Step 3 – Identification of MML maximizer and posterior mode. Under (PRIOR-FB-WEAK)\(_p\), the gradient \(\nabla_{\theta_{\mathrm{ref}}} \log \pi_\Theta(\theta_{\mathrm{ref}}) / n \to 0\) componentwise (the prior’s logarithmic-derivative scaled by \(n^{-1}\) vanishes), so the maximizer of \(L_n^{\mathrm{marg}} \cdot \pi_\Theta\) (posterior mode) and the maximizer of \(L_n^{\mathrm{marg}}\) alone (\(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\)) differ by \(O_P(n^{-1})\). The two coincide to leading order, and the Gaussian centering in Step 2 can be taken at either.

Step 4 – Lipschitz/smoothness of \(\theta_{\mathrm{ref}} \mapsto \Pi_n^{\mathrm{cond}}(\xi \mid \theta_{\mathrm{ref}}, \cdot)\). Under (HIER-COMPLEX)\(_p\) (\(C^2\) in a ball), the conditional posterior depends on \(\theta_{\mathrm{ref}}\) smoothly: for any \(h \in \mathbb{R}^p\) small, \[\left\| \Pi_n^{\mathrm{cond}}(\xi \mid \widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}} + h, \cdot) - \Pi_n^{\mathrm{cond}}(\xi \mid \widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}, \cdot) \right\|_{\bullet} \;\leq\; L\, \|h\|,\] where \(\|\cdot\|_\bullet\) is \(d_{\mathrm{TV}}\) in Regime A and the weak-over-\(\mathcal{G}\) metric in Regime B, \(\|h\|\) is the Euclidean norm in \(\mathbb{R}^p\), and \(L > 0\) is the Lipschitz constant given by the maximum eigenvalue of the conditional Hessian (existence of \(L\) follows from (HIER-COMPLEX)\(_p\)). Combined with the \(O_P(n^{-1/2})\) concentration of Step 2, the difference \[\left| \int \Pi_n^{\mathrm{cond}}(\xi \mid \theta_{\mathrm{ref}}, \cdot)\, \Pi_n^{\mathrm{marg}}(\theta_{\mathrm{ref}} \mid \cdot)\, d\theta_{\mathrm{ref}} \;-\; \Pi_n^{\mathrm{cond}}(\xi \mid \widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}, \cdot) \right|\] in the relevant metric is \(O_P(L \cdot n^{-1/2}) \to 0\). This identifies the FB posterior with \(\Pi_n^{\mathrm{EB}}(\xi \mid \cdot) = \Pi_n^{\mathrm{cond}}(\xi \mid \widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}, \cdot)\) asymptotically. \(\square\)

The four steps reproduce v07’s “reference and argument” with \(\mathbb{R}\) replaced by \(\mathbb{R}^p\) throughout. Each step is the routine multivariate generalization established in the cited literature (Petrone–Rousseau–Scricciolo 2014; Rousseau–Szabo 2017; Castillo–Rousseau 2015; van der Vaart 1998).

## 4.2. Why Total Variation is Restricted to Regime A (Multivariate Refinement)

The v07 §5 caveat that TV is restricted to Regime A is independent of \(p\): it depends on the dimensionality of \(\xi\), not of \(\theta_{\mathrm{ref}}\). In non-parametric Bayesian inference, the posterior over an infinite-dim \(\xi\) does not converge in TV but in Hellinger or \(L^2(\mu)\) or in the weak-over-\(\mathcal{G}\) metric (van der Vaart and van Zanten 2008; Ghosal–van der Vaart 2017). The multivariate dimension of \(\theta_{\mathrm{ref}}\) does not change this conclusion: the only modification is the rate of concentration of the marginal posterior (which is \(n^{-1/2}\) in each of the \(p\) coordinates).

A subtlety specific to the multivariate setting: even in Regime A (finite-dim parametric \(\xi\)), TV convergence under \(p \geq 2\) requires the additional regularity that the Laplace approximation in \(\mathbb{R}^p\) is uniformly accurate over the relevant neighborhood. This is guaranteed by (HIER-COMPLEX)\(_p\) but is worth noting because the multivariate Laplace approximation can fail numerically (not theoretically) when \(I_{\theta\theta}^{\mathrm{marg}}\) has condition number \(\kappa = \lambda_1 / \lambda_p\) large — see §9 O5*-EBFB.

## 4.3. Practical Implication of Theorem 7A*

For large \(n\), weakly informative priors on \(\theta_{\mathrm{ref}}\), and well-conditioned \(I_{\theta\theta}^{\mathrm{marg}}\), EB and FB give essentially the same posterior over \(\xi\) in the metric appropriate to the dimensionality of \(\xi\). The dimensionality \(p\) of \(\theta_{\mathrm{ref}}\) and the number of slots \(K\) enter only through (i) the strength of the regularity hypotheses (which scale with \(p\) and \(K\)), (ii) the rate at which the smallest eigenvalue \(\lambda_p\) of \(I_{\theta\theta}^{\mathrm{marg}}\) governs identifiability, and (iii) the numerical difficulty of the multivariate Laplace step (§9 O5*-EBFB).

The choice between EB and FB therefore remains, to first order, primarily computational and methodological: EB is faster (Sub-phase 8.6.B’s `cmdstanr::laplace()` call typically completes in \(\sim 10\%\) of the FB sampling time on the same model); FB integrates over \(\theta_{\mathrm{ref}}\) honestly (does not require choosing \(\pi_\Theta\)). Higher-order coverage and finite-sample considerations (§§5–6) refine this picture.

## 4.4. Reduction to v07’s Theorem 7A under \((p, K) = (1, 1)\)

Under \(p = 1\) and \(K = 1\):

- (EB-MARG-ID)\(_1\) = (EB-MARG-ID) of v07 §4 (positivity of scalar Fisher information).
- (PRIOR-FB-WEAK)\(_1\) = (PRIOR-FB-WEAK) of v07 §4 (scalar variance ratio).
- (HIER-COMPLEX)\(_1\) = (HIER-COMPLEX) of v07 §4 augmented with explicit \(C^2\) on an interval (the implicit regularity of v07’s “reference and argument”).
- The four-cell table reduces to its two left-column cells: Regime A (TV) and Regime B (weak-over-\(\mathcal{G}\)).
- The proof sketch (§4.1) reduces to v07’s Steps 1–4 with \(\mathbb{R}\) in place of \(\mathbb{R}^p\) and scalars in place of matrices.

Theorem 7A* therefore strictly contains Theorem 7A of v07: every assertion of v07 §5 is an instance of Theorem 7A* under \((p, K) = (1, 1)\).

## 4.5. Audit Trail: v07 §5 Reviewed Line by Line

The redaction of this vignette was preceded by a line-by-line audit of v07 §5 intended to detect implicit use of any property valid in \(\mathbb{R}\) but false in \(\mathbb{R}^p\). The four audit-relevant assertions, in the order they appear in v07 §5, are:

1. “The argument decomposes the FB posterior as the average of the conditional posterior given \(\theta_{\mathrm{ref}}\) over the marginal posterior of \(\theta_{\mathrm{ref}}\).” Algebraic identity (mixture decomposition) valid in any dimension. No scalar-specific property used.
2. “Under (PRIOR-FB-WEAK), the marginal posterior of \(\theta_{\mathrm{ref}}\) concentrates at \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\).” Multivariate BvM in \(\mathbb{R}^p\) requires positive definite \(I_{\theta\theta}^{\mathrm{marg}}\); the scalar v07 version requires positivity. The strengthening is explicit in (EB-MARG-ID)\(_p\) of §3. No implicit scalar-specific assumption.
3. “The conditional posterior at the concentration point coincides with the EB posterior.” Definition of \(\Pi_n^{\mathrm{EB}}\); valid in any dimension.
4. “The difference between FB and EB is thus the difference between integrating with respect to a concentrated distribution and evaluating at its mode, which goes to zero in the appropriate metric for the dimension of \(\xi\).” Requires Lipschitz/smoothness of \(\theta_{\mathrm{ref}} \mapsto \Pi_n^{\mathrm{cond}}(\xi \mid \theta_{\mathrm{ref}}, \cdot)\) in the appropriate metric. In \(\mathbb{R}\) this is \(|h|\)-Lipschitz; in \(\mathbb{R}^p\) it is \(\|h\|\)-Lipschitz with Euclidean norm. Norm equivalence in finite dimension ensures transfer. No scalar-specific argument.

Audit conclusion. v07 §5 transfers cleanly to \(\mathbb{R}^p\) under the strictly stronger hypotheses of §3. No implicit error detected. The “reference and argument” delegation of v07 §5 to Petrone–Rousseau–Scricciolo (2014) and Rousseau–Szabo (2017) is faithful: both treat hyperparameters on Polish spaces, of which finite-dimensional \(\mathbb{R}^p\) is the routine specialization.

A minor pedagogical opacity in v07 §6 (not §5) is addressed in §5 of this vignette: v07’s scalar formula \(C_{g,\alpha} \approx (g'(\xi^*))^2 / I_{\theta\theta}^{\mathrm{marg}} \cdot \kappa(\alpha)\) implicitly absorbs a sensitivity factor \(\partial \xi^* / \partial \theta_{\mathrm{ref}}\) into the leading coefficient. The multivariate Proposition 7B* makes this Jacobian (called \(J^\xi\) in §2.3) explicit as part of a sandwich quadratic form, recovering the scalar v07 §6 formula as the \(p = \dim(\xi) = 1\) specialization.

# 5. Proposition 7B* – Higher-Order Coverage Discrepancy (Matrix Form)

> Proposition 7B*. Higher-order coverage discrepancy in \(\mathbb{R}^p\), \(K \geq 1\). Under the hypotheses of Theorem 7A* and the additional regularity required by standard higher-order expansions in \(\mathbb{R}^p\) (smoothness of the functional \(g: \mathbb{R}^{\dim(\xi)} \to \mathbb{R}\) at \(\xi^*\) with three or more bounded derivatives; well-conditioned \(I_{\theta\theta}^{\mathrm{marg}}\) at the truth; the multivariate Edgeworth-type expansion conditions of Bickel and Ghosh 1990 generalized to \(\mathbb{R}^p\)), the EB credible interval for the smooth functional \(g(\xi)\) has coverage that differs from the nominal level by an amount of order \(n^{-1}\): \[\mathbb{P}_{\eta_*}\!\bigl(\, g(\xi^*) \in \mathrm{CI}_n^{\mathrm{EB}, \alpha} \bigr) \;=\; (1 - \alpha)\; -\; \mathrm{tr}\!\bigl(C_{g, \alpha}^*\bigr) \cdot n^{-1}\; +\; o(n^{-1}),\] where the matrix coverage discrepancy constant \(C_{g, \alpha}^* \in \mathbb{R}^{p \times p}\) has the explicit sandwich form \[C_{g, \alpha}^* \;=\; \kappa(\alpha) \cdot \bigl[J^\xi(\theta_{\mathrm{ref}}^*)\bigr]^\top \cdot \nabla_\xi g(\xi^*) \cdot \nabla_\xi g(\xi^*)^\top \cdot J^\xi(\theta_{\mathrm{ref}}^*) \cdot \bigl(I_{\theta\theta}^{\mathrm{marg}}\bigr)^{-1},\] with \(J^\xi \in \mathbb{R}^{\dim(\xi) \times p}\) the sensitivity Jacobian of §2.3 and \(\kappa(\alpha)\) the standard-normal level constant (\(\kappa(0.05) \approx 1.92\)). The FB credible interval has coverage of nominal level to first order: \[\mathbb{P}_{\eta_*}\!\bigl(\, g(\xi^*) \in \mathrm{CI}_n^{\mathrm{FB}, \alpha} \bigr) \;=\; (1 - \alpha) + o(n^{-1/2}).\] Under the stated standard expansion conditions, EB credible intervals therefore systematically under-cover relative to FB at the \(n^{-1}\) rate.

## 5.1. Derivation Sketch

The under-coverage arises from the fact that EB conditions on a point estimate \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) and ignores the variability in this estimator, whereas FB integrates over the posterior of \(\theta_{\mathrm{ref}}\). Under the multivariate BvM of Step 2 of §4.1, \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) has approximate covariance \(n^{-1} (I_{\theta\theta}^{\mathrm{marg}})^{-1}\). The EB-side variance of the plug-in estimator \(g(\xi^*(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}))\), by the multivariate delta method, includes the propagated contribution \[\mathrm{Var}_{\mathrm{EB-FB}}\!\bigl[ g(\xi^*) \bigr] \;\approx\; \bigl[\nabla_\xi g(\xi^*)^\top \cdot J^\xi \cdot \mathrm{Cov}(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}) \cdot (J^\xi)^\top \cdot \nabla_\xi g(\xi^*)\bigr] \;\approx\; \frac{1}{n} \nabla_\xi g(\xi^*)^\top J^\xi (I_{\theta\theta}^{\mathrm{marg}})^{-1} (J^\xi)^\top \nabla_\xi g(\xi^*).\] The EB credible interval, computed from \(\Pi_n^{\mathrm{EB}}(\xi \mid \widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}})\) alone, omits this contribution; FB’s posterior includes it via the marginalization over \(\theta_{\mathrm{ref}}\). The coverage deficit at level \(1 - \alpha\) is therefore proportional to the trace of the omitted covariance contribution scaled by \(\kappa(\alpha)\), which is exactly \(\mathrm{tr}(C_{g,\alpha}^*) / n\). The Edgeworth-type expansion conditions of Bickel–Ghosh (1990) generalized to \(\mathbb{R}^p\) ensure the \(o(n^{-1})\) remainder.

## 5.2. Reduction to v07’s Scalar Proposition 7B under \(p = \dim(\xi) = 1\)

When \(p = 1\) and \(\dim(\xi) = 1\):

- \(J^\xi \in \mathbb{R}^{1 \times 1}\) is the scalar sensitivity \(\partial \xi^* / \partial \theta_{\mathrm{ref}}\).
- \(\nabla_\xi g(\xi^*) \in \mathbb{R}^{1 \times 1}\) is the scalar derivative \(g'(\xi^*)\).
- \((I_{\theta\theta}^{\mathrm{marg}})^{-1} \in \mathbb{R}^{1 \times 1}\) is the scalar inverse Fisher information \(1 / I_{\theta\theta}^{\mathrm{marg}}\).
- \(C_{g,\alpha}^*\) degenerates to a scalar: \(C_{g,\alpha}^* = \kappa(\alpha) \cdot (g'(\xi^*))^2 \cdot (J^\xi)^2 / I_{\theta\theta}^{\mathrm{marg}}\).

The scalar v07 §6 formula \(C_{g,\alpha} \approx (g'(\xi^*))^2 / I_{\theta\theta}^{\mathrm{marg}} \cdot \kappa(\alpha)\) absorbs \((J^\xi)^2\) into the proportionality constant. Under the convention that \(J^\xi = 1\) for a one-dimensional smooth identity \(\xi \mapsto \xi\) (Carlin–Gelfand 1990 §3), the formulas coincide exactly. In the multivariate extension this absorption is no longer harmless: \(J^\xi\) depends explicitly on the hierarchy’s coupling between \(\theta_{\mathrm{ref}}\) and \(\xi\), and a value \(\|J^\xi\| \gg 1\) amplifies the under-coverage proportionally.

## 5.3. Effective Sensitivity Diagonalization (Practical Form)

For a practical formula independent of explicit Jacobian computation, diagonalize the sandwich: \[\mathrm{tr}\!\bigl(C_{g,\alpha}^*\bigr) \;=\; \kappa(\alpha) \cdot \nabla_\xi g(\xi^*)^\top \cdot J^\xi (I_{\theta\theta}^{\mathrm{marg}})^{-1} (J^\xi)^\top \cdot \nabla_\xi g(\xi^*).\] Let \(\Sigma_{\mathrm{EB-FB}} := J^\xi (I_{\theta\theta}^{\mathrm{marg}})^{-1} (J^\xi)^\top \in \mathbb{R}^{\dim(\xi) \times \dim(\xi)}\) be the EB-vs-FB covariance gap matrix, computable post-hoc from the EB fit (via finite differences on \(\theta_{\mathrm{ref}} \mapsto \xi^*(\theta_{\mathrm{ref}})\) for \(J^\xi\) and from the Laplace Hessian for \(I_{\theta\theta}^{\mathrm{marg}}\)). Then \[\mathrm{tr}\!\bigl(C_{g,\alpha}^*\bigr) \;=\; \kappa(\alpha) \cdot \nabla_\xi g(\xi^*)^\top \cdot \Sigma_{\mathrm{EB-FB}} \cdot \nabla_\xi g(\xi^*),\] and the EB credible interval for \(g(\xi^*)\) is corrected by inflating its width by a factor approximately \(\sqrt{1 + \mathrm{tr}(C_{g,\alpha}^*)/(n - q)}\) to recover nominal coverage, where \(q\) is the effective degree-of-freedom adjustment (typically the dimension of \(\xi\) relevant to the conditional posterior of \(g(\xi)\)).

This is the form implemented by `eb_correction = TRUE` (the default in `gdpar_eb()`, per Charter §2.6 of Sub-phase 8.6).

## 5.4. Operational Note and the `eb_correction` Argument

Per Charter §2.6 (Sub-phase 8.6) and per the operative description of v07 §11.1, the framework’s library applies the Proposition 7B* correction by default (`eb_correction = TRUE`):

- Under \(p = 1\), the scalar correction of v07 §6 is applied.
- Under \(p \geq 2\), the matrix correction \(C_{g,\alpha}^* \in \mathbb{R}^{p \times p}\) of this section is applied via its diagonalized form (§5.3).
- The user can set `eb_correction = FALSE` to retrieve nominal (un-corrected) intervals, in which case the library emits a `gdpar_diagnostic_warning` advising of the expected \(O(n^{-1})\) under-cover (per Sub-phase 8.6.B’s implementation).

Caveat: the correction is approximate. As in v07 §6, the correction depends on standard Edgeworth-type expansion conditions and on smoothness of \(g\) at \(\xi^*\). Under non-smooth functionals, near-singular \(I_{\theta\theta}^{\mathrm{marg}}\), or distributions for which Edgeworth expansions fail (e.g., heavy-tailed Student-\(t\) slots; mixture families with near-modes-of-non-identifiability), the correction’s accuracy degrades. The framework recommends FB whenever the standard expansion conditions are in doubt; the EB correction is offered as a partial correction for cases where the user has chosen EB for computational reasons.

# 6. Theorem 7C* – Compound Decision Bound (Multi-Slot \(K > 1\))

> Theorem 7C*. Compound decision bound under \(K \geq 1\) heterogeneous slots. Suppose the hierarchical model has \(K \geq 1\) distributional slots indexed by \(k = 1, \ldots, K\), with slot-\(k\) parameter \(\xi^{(k)}\) drawn (under exchangeability across \(J\) groups within slot) from a per-slot prior \(\pi_{\xi^{(k)}}\) depending on a slot-specific hyperparameter \(\theta_{\mathrm{ref}}^{(k)} \in \mathbb{R}^{p_k}\). Let \(\widehat{\xi}^{(k), \mathrm{EB}}\) denote the slot-\(k\) EB estimator with \(\widehat{\theta}_{\mathrm{ref}}^{(k), \mathrm{EB}}\) plugged in, and \(\widehat{\xi}^{(k), \mathrm{FB}}\) the slot-\(k\) FB estimator (joint posterior mean). Then:

> (Case A) Factored prior \(\pi_\xi = \prod_{k=1}^K \pi_{\xi^{(k)}}\). Slotwise compound decision factorizes: \[\frac{1}{K} \sum_{k=1}^K \mathbb{E}\!\bigl[ (\widehat{\xi}^{(k), \mathrm{EB}} - \xi^{(k)*})^2 \bigr] \;\leq\; \frac{1}{K} \sum_{k=1}^K \mathbb{E}\!\bigl[ (\widehat{\xi}^{(k), \mathrm{FB}} - \xi^{(k)*})^2 \bigr] + \frac{1}{K} \sum_{k=1}^K B_{J, k},\] where each per-slot gap \(B_{J, k} \leq C_1^{(k)} J^{-1} \mathbb{E}\bigl[ \|\widehat{\theta}_{\mathrm{ref}}^{(k), \mathrm{EB}} - \theta_{\mathrm{ref}}^{(k)*}\|^2 \bigr] \to 0\) as \(J \to \infty\), for a constant \(C_1^{(k)} > 0\) depending on the sensitivity of the slot-\(k\) conditional posterior mean to \(\theta_{\mathrm{ref}}^{(k)}\).

> (Case B) Coupled prior \(\pi_\xi\) not factored across slots. The bound includes an explicit cross-slot interaction term: \[\frac{1}{K} \sum_{k=1}^K \mathbb{E}\!\bigl[ (\widehat{\xi}^{(k), \mathrm{EB}} - \xi^{(k)*})^2 \bigr] \;\leq\; \frac{1}{K} \sum_{k=1}^K \mathbb{E}\!\bigl[ (\widehat{\xi}^{(k), \mathrm{FB}} - \xi^{(k)*})^2 \bigr] + \frac{1}{K} \sum_{k=1}^K B_{J, k}\; +\; \mathcal{B}_{\mathrm{coupling}},\] where the coupling residual \(\mathcal{B}_{\mathrm{coupling}}\) is bounded explicitly by \[\mathcal{B}_{\mathrm{coupling}} \;\leq\; \frac{1}{K^2} \sum_{k \neq k'} C_{\mathrm{cross}}^{(k, k')} \cdot J^{-1} \cdot \mathbb{E}\!\bigl[ \|\widehat{\theta}_{\mathrm{ref}}^{(k), \mathrm{EB}} - \theta_{\mathrm{ref}}^{(k)*}\| \cdot \|\widehat{\theta}_{\mathrm{ref}}^{(k'), \mathrm{EB}} - \theta_{\mathrm{ref}}^{(k')*}\| \bigr],\] with \(C_{\mathrm{cross}}^{(k, k')} > 0\) derived from the cross-slot block of the coupled prior’s Hessian. \(\mathcal{B}_{\mathrm{coupling}} \to 0\) as \(J \to \infty\) under (HIER-COMPLEX)\(_p\).

## 6.1. Reduction to v07’s Theorem 7C under \(K = 1\)

Under \(K = 1\): Case A collapses to v07’s Theorem 7C with \(K\) replaced by \(J\) (in v07’s notation, the symbol \(K\) stood for the number of exchangeable units within the single slot; in this vignette, \(K\) stands for the number of distributional slots and \(J\) stands for the number of groups within each slot). The relabeling does not change content: the gap \(B_{J, 1}\) tends to zero as the number of within-slot exchangeable units grows, reproducing v07’s \(B_K \to 0\) as \(K \to \infty\) verbatim. Case B is vacuous under \(K = 1\) (no cross-slot coupling exists).

## 6.2. Two Regimes of Practical Use (Multi-Slot Refinement)

- \(J\) large within slots, \(K\) moderate: per-slot EB risk approaches per-slot FB risk; EB is the practical choice slot-by-slot. The aggregate bound is dominated by the per-slot gaps.
- \(J\) small, \(K\) small: the gap \(B_{J, k}\) is non-negligible per slot, and under coupled prior the residual \(\mathcal{B}_{\mathrm{coupling}}\) adds an explicit cross-slot penalty. FB is preferred to capture the full covariance.
- \(K\) large, \(J\) large: per-slot bound tightens uniformly; if the prior is factored (Case A), EB is asymptotically efficient. If the prior is coupled (Case B), the cross-slot interaction term decays at \(J^{-1}\) but adds a constant overhead at any fixed \(J\), advising caution.

## 6.3. Caveat (Squared-Error Loss; Coverage Picture Unchanged)

As in v07 §7, Theorem 7C* is for squared-error loss on point estimates. For interval coverage, Proposition 7B* (§5) shows that EB systematically under-covers regardless of \(K\) or \(J\); the choice between EB and FB therefore depends on whether the user prioritizes point estimate accuracy under squared-error loss (EB acceptable for large \(J\), possibly large \(K\) under Case A) or coverage calibration (FB preferred uniformly).

## 6.4. Reference

This is the standard compound decision result of Robbins (1956) extended to \(K \geq 1\) slots. The per-slot bound of Case A is the routine slotwise application of Brown (2008) and Efron (2019, Theorem 1.1). The cross-slot residual of Case B is derived from the second-order Taylor expansion of the coupled prior’s log-density at the truth and is, to the author’s knowledge, not previously stated in this form in the literature; it is canonized here as the AMM specialization of the multi-hyperparameter compound decision setting.

# 7. Proposition 7D* – Substantial Discrepancy Conditions (Multivariate)

> Proposition 7D*. EB and FB produce substantially different posteriors in any of the following regimes:

> (i)* Multimodal marginal likelihood for \(\theta_{\mathrm{ref}}\) in \(\mathbb{R}^p\). When \(L_n^{\mathrm{marg}}: \mathbb{R}^p \to \mathbb{R}\) has multiple local maxima, the EB estimator may converge to a non-global mode; FB integrates over all modes weighted by the prior. The probability of multimodality grows with \(p\) under generic non-conjugacy: in \(\mathbb{R}^p\), the number of saddle-points and local maxima of a generic smooth function scales (in the worst case) as \(O(c^p)\) for some \(c > 1\) (Morse-theoretic genericity). For \(p \geq 3\) in non-conjugate settings, multimodality is the default rather than the exception, and the EB estimator’s stability depends crucially on the multi-start strategy of §9 O5*-EBFB.

> (ii)* Near-singular Fisher information matrix \(I_{\theta\theta}^{\mathrm{marg}}\). When the smallest eigenvalue \(\lambda_p\) of \(I_{\theta\theta}^{\mathrm{marg}}\) is small (formally, when the condition number \(\kappa(I_{\theta\theta}^{\mathrm{marg}}) = \lambda_1 / \lambda_p \geq \kappa_{\mathrm{threshold}}\)), the variance of \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) along the eigenvector of \(\lambda_p\) is large, and FB’s integration over this variance produces a substantially wider posterior on \(\xi\) along the same direction. The framework’s library reports \(\kappa(I_{\theta\theta}^{\mathrm{marg}})\) as a diagnostic and aborts with a `gdpar_eb_numerical_error` when \(\kappa > \kappa_{\mathrm{threshold}}\) (default \(10^{10}\), per Charter §2.8).

> (iii)* Strongly informative prior \(\pi_\Theta\) in \(\mathbb{R}^p\). When the prior \(\pi_\Theta\) has narrow support around \(\theta_{\mathrm{ref}}^*\) in any direction (formally, the KL divergence from \(\pi_\Theta\) to a flat prior on the same support is at least \(\epsilon > 0\) in some coordinate), FB benefits from the prior information whereas EB ignores it; the FB posterior is concentrated more tightly on \(\xi\) along the prior-informed direction.

> (iv)* Hierarchy with \(\geq 3\) levels under \(p > 1\) or \(K > 1\). When the model has three or more hierarchical levels (e.g., \(\theta_{\mathrm{ref}}\) depends on a meta-hyperparameter \(\theta_{\mathrm{meta}}\) which in turn depends on another), the EB-style point estimation cascades errors across levels. Under \(p = K = 1\) the cascade is benign; under \(p > 1\) or \(K > 1\) the cascade compounds: the per-level Fisher information becomes a tensor and the multivariate Laplace approximation must be applied recursively at each level, with each application contributing \(O(n^{-1})\) approximation error that accumulates additively across levels.

> In each regime, the framework recommends FB by default, with EB only when justified by computational considerations and accompanied by the under-coverage correction of Proposition 7B* (§5).

## 7.1. Reduction to v07’s Proposition 7D under \((p, K) = (1, 1)\)

Under \(p = 1\) and \(K = 1\), the four asterisked conditions reduce to v07’s Proposition 7D conditions (i)–(iv) with the following correspondences:

v07 (scalar) | v07b (multivariate) | Remark
--- | --- | ---
(i) Small effective \(n\) for \(\theta_{\mathrm{ref}}\) (small \(I_{\theta\theta}^{\mathrm{marg}}\)) | (ii)* Near-singular \(I_{\theta\theta}^{\mathrm{marg}}\) (small \(\lambda_p\)) | v07 (i) becomes v07b (ii)* with eigenvalue interpretation.
(ii) Strongly informative prior | (iii)* Strongly informative prior in \(\mathbb{R}^p\) | KL-based directional generalization.
(iii) Multimodal \(L_n^{\mathrm{marg}}\) | (i)* Multimodal in \(\mathbb{R}^p\) | Genericity of multimodality scales with \(p\).
(iv) Misspecified lower-level prior \(\pi_\xi\) | Subsumed by Proposition 7D* (iv)* and Open Question O3*-EBFB | Misspecification is treated within the general hierarchical-cascade frame of (iv)*.

The renumbering (v07’s (iii) becomes v07b’s (i)*) reflects the ordering of practical importance in \(\mathbb{R}^p\): multimodality is the most frequent failure mode of multivariate EB and is treated first.

# 8. Recommendation by Scenario (Multivariate Extension)

The framework’s recommendation table of v07 §9 extends to the multivariate setting with explicit rows for \(p\), \(K\), and the product \(K \cdot p\). The recommendation depends jointly on the sample size \(n\), the number of within-slot exchangeable units \(J\), the number of distributional slots \(K\), the dimensionality \(p\) of \(\theta_{\mathrm{ref}}\), the smallest eigenvalue \(\lambda_p(I_{\theta\theta}^{\mathrm{marg}})\), and the computational budget.

Scenario | \(n\) | \(J\) | \(K\) | \(p\) | \(\lambda_p\) | Compute | Recommended | Reason
--- | --- | --- | --- | --- | --- | --- | --- | ---
Large \(n\), large \(J\), \(K = 1\), \(p = 1\), well-identified | Large | Large | 1 | 1 | Large | Tight | EB | v07 §9 row 1; Theorems 7A*, 7C* hold cleanly.
Same but coverage critical | Large | Large | 1 | 1 | Large | Tight | FB | Proposition 7B* coverage discrepancy.
Large \(n\), large \(J\), \(K \geq 2\) factored prior, \(p\) small | Large | Large | \(\geq 2\) | Small | Large | Tight | EB | Theorem 7C* Case A slotwise compound decision.
Large \(n\), \(K \cdot p\) moderate, well-conditioned \(I_{\theta\theta}^{\mathrm{marg}}\) | Large | Large | \(\geq 2\) | \(\geq 2\) | Large | Tight | EB | Multivariate Theorem 7A*; numerical regime well-conditioned.
Large \(n\), \(K \cdot p\) large, near-singular \(I_{\theta\theta}^{\mathrm{marg}}\) | Large | Large | Large | Large | Small | Any | FB | Proposition 7D* (ii)*; multivariate Laplace fails or under-covers severely.
Moderate \(n\), \(K \geq 2\) coupled prior | Moderate | Moderate | \(\geq 2\) | Any | Moderate | Moderate | FB | Theorem 7C* Case B cross-slot residual non-negligible.
Small \(J\), small \(K\), small \(p\) | Small | Small | 1–2 | 1–2 | Small | Any | FB | Multiple conditions of Proposition 7D* may apply jointly.
Multimodal \(L_n^{\mathrm{marg}}\) in \(\mathbb{R}^p\) (\(p \geq 3\)) | Any | Any | Any | \(\geq 3\) | Any | Any | FB | Proposition 7D* (i)*; multimodality default in \(\mathbb{R}^{p \geq 3}\).
Strong prior knowledge on \(\theta_{\mathrm{ref}}\) in some direction | Any | Any | Any | Any | Any | Any | FB | Proposition 7D* (iii)*.
Highly hierarchical (3+ levels) | Any | Any | Any | \(\geq 2\) | Any | Any | FB | Proposition 7D* (iv)* cascade compounds.

The framework’s default remains FB for all configurations. EB is offered as an explicit configuration option via `gdpar_eb()` (Charter §2.2) for cases where the analyst has determined that EB is preferred (computational constraints, methodological avoidance of priors on \(\theta_{\mathrm{ref}}\), or large-\(J\) slotwise compound decision settings under Case A of §6).

Multivariate-specific operational guidance (not present in v07 §9):

1. When \(p \geq 3\) and no prior knowledge of multimodality: always run `gdpar_eb()` with `laplace_control$multi_start_M = 10` (overriding the default of 5) to mitigate Proposition 7D* (i)*. The framework’s library reports the inter-init dispersion in `gdpar_eb_fit$diagnostics$multi_start_dispersion`; a coefficient of variation \(> 0.05\) across inits signals likely multimodality and warrants switching to FB.

2. When \(K \geq 3\) with heterogeneous families (e.g., Beta + Gamma + Student-\(t\)): compute the condition number of each per-slot block of \(I_{\theta\theta}^{\mathrm{marg}}\) independently; if any per-slot condition number exceeds \(10^8\), abandon EB for that slot and fall back to FB on the full joint model.

3. When the user has computational budget for both: run `gdpar_eb()` first to obtain \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) and use it as the initialization for FB sampling (`gdpar(init = ...)`), reducing the FB sampling time substantially. This is the recommended EB-warm-start FB workflow for \(p \geq 2\).

# 9. Open Questions

Four open questions of v07 §10 (O1-EBFB through O4-EBFB) extend to the multivariate setting with the asterisked versions below, plus one new open question O5*-EBFB specific to the numerical anti-fragility of multivariate Laplace under non-conjugacy.

## 9.1. (O1*-EBFB) Adaptive Choice between EB and FB Based on Data (Multivariate)

v07’s O1-EBFB is the data-driven adaptive selection between EB and FB based on a diagnostic. The multivariate extension requires a diagnostic that handles the multi-dimensional Fisher information, the per-slot block structure under \(K > 1\), and the inter-init dispersion of multi-start Laplace. A candidate diagnostic is the ratio of EB-side coverage gap \(\mathrm{tr}(C_{g,\alpha}^*)/n\) (computable post-hoc from `gdpar_eb_fit`) to the FB-side variance of \(g(\xi^*)\) obtained from a short pilot FB run; the framework will release a candidate diagnostic in Block 9.x as `gdpar_eb_diagnostic()` after sufficient empirical calibration. Partial work: Petrone–Rousseau–Scricciolo (2014, §7) discuss adaptive empirical Bayes in the abstract; the AMM specialization in \(\mathbb{R}^p\) with heterogeneous slots is open.

## 9.2. (O2*-EBFB) Higher-Order Coverage Correction for Multivariate EB

v07’s O2-EBFB is the search for higher-order corrections (Edgeworth-type, bootstrap-based) beyond the first-order Proposition 7B correction. The multivariate extension requires corrections that handle (i) the sandwich matrix form of \(C_{g,\alpha}^*\), (ii) the per-slot tensor under \(K > 1\), and (iii) the propagated uncertainty from the multi-start Laplace inter-init dispersion. Multivariate Edgeworth expansions are technically more delicate (Bhattacharya and Ghosh 1978; Hall 1992 Chapter 5) and bootstrap calibration in \(\mathbb{R}^p\) requires more replications to control simulation noise. The framework’s current implementation uses the first-order matrix correction (§5); higher-order corrections are deferred to Block 9.x.

## 9.3. (O3*-EBFB) Behavior under Model Misspecification (Multivariate)

v07’s O3-EBFB is the question of which of EB or FB degrades more gracefully under misspecification. Under multivariate hyperparameter and heterogeneous slots, the question further refines: does misspecification of the per-slot prior \(\pi_{\xi^{(k)}}\) in one slot affect the EB estimator of \(\theta_{\mathrm{ref}}^{(k)}\) in only that slot (a slot-local insulation property of Case A of Theorem 7C*) or across slots through the coupled prior of Case B? Partial empirical observation from the AMM regression tests (`tests/testthat/test-eb_misspecification.R` to be written in Sub-phase 8.6.D) will inform; full theoretical treatment is deferred.

## 9.4. (O4*-EBFB) Computational Competitiveness of FB at Moderate \(n\) under \(K \cdot p\) Large

v07’s O4-EBFB is the question of when modern HMC narrows the computational gap enough to make FB the default even for very large \(n\). The multivariate extension asks: how does the FB sampling time scale with the joint dimensionality \(K \cdot p\)? Empirical observation from Bloque 7 benches (5 species × 4 NE-USA sub-regions, single-slot \(K = 1\), \(p = 1\)): the framework’s FB sampling took \(\sim 6600\) s vs. mgcv’s \(\sim 0.7\) s; under \(K \cdot p \geq 4\) the gap may grow super-linearly. The threshold at which FB becomes infeasible per \(K \cdot p\) is problem-dependent and remains open.

## 9.5. (O5*-EBFB) Numerical Anti-Fragility of Multivariate Laplace under Non-Conjugacy (New)

The most important new open question raised by the multivariate extension is the formal characterization of when `cmdstanr::laplace()` fails numerically in \(\mathbb{R}^p\) under non-conjugate family + prior combinations. The failure modes are:

- Singular or ill-conditioned Hessian of the marginal log-likelihood at the candidate \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\). The condition number \(\kappa(H) = \lambda_1(H) / \lambda_p(H)\) controls the relative error of the inverse Hessian used in the Laplace approximation; when \(\kappa(H) \gtrsim 10^{10}\), the Gaussian approximation around \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\) is unreliable and the EB credible intervals are invalid.
- Non-positive-definite Hessian when the L-BFGS optimization converges to a saddle-point or to a local maximum of \(L_n^{\mathrm{marg}}\) that is not a true global maximum.
- Multiple disconnected modes when \(L_n^{\mathrm{marg}}\) has more than one local maximum in \(\mathbb{R}^p\) (recall §7 Proposition 7D* (i)*: multimodality is generic for \(p \geq 3\) under non-conjugacy).
- Slow gradient decay along ridge directions when one or more eigenvalues of the marginal Hessian are near zero, causing L-BFGS to oscillate rather than converge.

The framework’s anti-fragility strategy (canonized in Charter §2.8 of Sub-phase 8.6) addresses these four failure modes with four operational components:

1. Preventive detection of ill-conditioned Hessian. Before accepting the output of `cmdstanr::laplace()`, the library computes the condition number \(\kappa(H)\) of the marginal Hessian at the candidate \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\). If \(\kappa(H) > \kappa_{\mathrm{threshold}}\) (default \(10^{10}\), configurable via `laplace_control$kappa_threshold`), the EB fit aborts with class `gdpar_eb_numerical_error` and an explicit diagnostic message identifying the most poorly conditioned direction (via the eigenvector of \(\lambda_p\)).
2. Adaptive Levenberg-Marquardt ridge. If \(|\det(H)| < \epsilon_{\mathrm{LM}}\) (Hessian effectively singular) or \(H\) is not positive-definite (a true Newton step would diverge), the library substitutes \(H \leftarrow H + \lambda I\) with an adaptive ridge \(\lambda\) chosen via the standard L-M heuristic (start with \(\lambda_0 = \mathrm{tr}(H) \cdot 10^{-3}\); geometrically increase by 10 until \(H + \lambda I\) is positive-definite with condition number below threshold; report the used \(\lambda\) in `gdpar_eb_fit$diagnostics$lm_perturbation`).
3. Multiple random restarts. The L-BFGS optimization is restarted from \(M\) independent random inits within a bounding box derived from the prior \(\pi_\Theta\) (default \(M = 5\); configurable via `laplace_control$multi_start_M$`). The library retains the init that maximizes the marginal log-likelihood and reports the inter-init dispersion (coefficient of variation of marginal log-likelihood at each converged init) in `gdpar_eb_fit$diagnostics$multi_start_dispersion`. A coefficient of variation \(> 0.05\) across inits signals likely multimodality and the library issues a `gdpar_diagnostic_warning` advising the user to consider FB.
4. Documented fallback. If all \(M\) inits fail (all produce singular Hessian or all diverge), `gdpar_eb()` aborts with class `gdpar_unsupported_feature_error` and a diagnostic message recommending the user to use `gdpar()` (FB) instead. The library never returns a silently invalid estimator.

The formal characterization of the conditions under which `cmdstanr::laplace()` fails (independent of the empirical heuristics above) is open. A candidate framework is to express failure as a function of two intrinsic quantities of the AMM specification: (a) the condition number \(\kappa(I_{\theta\theta}^{\mathrm{marg}})\) of the population Fisher matrix, and (b) the Morse index spread of \(L_n^{\mathrm{marg}}\) in \(\mathbb{R}^p\) (the variance of the number of negative eigenvalues of the Hessian across the interior of the admissible domain \(\Theta\)). A precise theorem of the form “if \(\kappa \leq \kappa_*(p)\) and Morse spread \(\leq \mu_*(p)\), then Laplace succeeds with probability \(\geq 1 - \delta_n(p)\) where \(\delta_n(p) = O(n^{-1/p})\)” is the open form; the constants \(\kappa_*(p)\), \(\mu_*(p)\), and \(\delta_n(p)\) are unknown for general non-conjugate AMM specifications and require systematic empirical-and-theoretical investigation. This is the primary open theoretical question that motivates the anti-fragility strategy as a working substitute pending formal resolution.

# 10. Connections to Subsequent Blocks

The multivariate extension of this vignette feeds into the operational implementation in five concrete artifacts of Sub-phase 8.6 (Charter §3.2–§3.5):

- `gdpar_eb()` (Sub-phase 8.6.B): the orchestrator function exposing the EB workflow to the user. The function signature is documented in Charter §2.2 and includes the `eb_correction` argument (defaulting to `TRUE`, implementing the correction of §5 of this vignette) and the `laplace_control` argument (implementing the anti-fragility strategy of §9 O5*-EBFB).

- `vop07_eb_workflow.Rmd` (Sub-phase 8.6.E): the operational companion vignette to this theoretical extension. Whereas the present vignette establishes the theoretical foundations of multivariate EB, `vop07` walks the user through the operational workflow (data preparation, `gdpar_eb()` call, interpretation of diagnostics, EB-vs-FB comparison) with executable code examples. The two vignettes are read together: theory in v07b, practice in vop07.

- `gdpar_compare_eb_fb()` (Sub-phase 8.6.E): the comparator function that takes an EB fit and an FB fit and reports the empirical total variation distance between \(\Pi_n^{\mathrm{EB}}\) and \(\Pi_n^{\mathrm{FB}}\) over \(\xi\), the empirical coverage discrepancy of nominal credible intervals (verifying Proposition 7B* of §5), and a side-by-side diagnostic table. Symmetric with `gdpar_compare_meta_learners` of Sub-phase 8.5.B (canonized in v08c).

- `inst/stan/amm_eb_marginal.stan` and `inst/stan/amm_eb_conditional.stan` (Sub-phase 8.6.B): two new Stan templates dedicated to EB, written multivariate-capable from the start per Charter §2.7. The marginal template treats \(\theta_{\mathrm{ref}}\) as a free parameter (vector or full tensor) and computes the marginal log-likelihood via `cmdstanr::laplace()`; the conditional template treats \(\theta_{\mathrm{ref}}\) as data (the plug-in \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\)) and samples \(\xi\) via HMC. The templates extend the existing FB templates (originally `amm_main.stan` and `amm_distrib_multi.stan`, plus `amm_distrib_K.stan`) without replacing or unifying them, preserving the bit-exactness of the FB goldens established in Sub-phases 8.3.X (goldens that are not re-bootstrapped in Sub-phase 8.6). Sub-sub-fase 9.3.a (Block 9, Session B9.3, 2026-05-27) subsequently unified the two K = 1 FB templates `amm_main.stan` and `amm_distrib_multi.stan` at the R-side codegen layer (lateral decision B.iv) via the dispatcher `.gdpar_emit_canonical_stan()`, relocating the byte-identical pieces to `inst/stan/_canonical_pieces/`; the EB-side templates remain at `inst/stan/` root pending the parallel cascade in Block 9.5+.

- Block 9.x (post-release): the formal characterization of O5*-EBFB (numerical anti-fragility) is deferred to Block 9.x as an open research direction. Block 9.x also receives the operational diagnostic `gdpar_eb_diagnostic()` from O1*-EBFB (adaptive choice) and the higher-order coverage corrections from O2*-EBFB.

The reader is referred to `vop07_eb_workflow.Rmd` (Sub-phase 8.6.E) for the operational reading of the present theoretical results and to the Sub-phase 8.6 charter (`CHARTER_SUBFASE_8_6.md`) for the full implementation roadmap.

# 11. Joint Kernel Stein Discrepancy: the `gdpar_ksd_joint()` Helper

Sub-bloque 9.3.c (Bloque 9, Sesión B9.4, 2026-05-27) operationalizes the open question raised in the Roxygen of `gdpar_compare_eb_fb()`: the marginal total-variation distance returned by that comparator is only a coarse proxy of the distributional discrepancy between \(\Pi_n^{\mathrm{EB}}\) and \(\Pi_n^{\mathrm{FB}}\), since it does not detect deviations in the joint dependence structure of \(\xi = (a, b, W, \mathrm{dispersion})\). The kernel Stein discrepancy (KSD) of Gorham and Mackey (2017) and Liu, Lee, and Jordan (2016) provides the canonical density-free spectral metric on a high-dimensional joint posterior.

## 11.1. The Stein Operator and the Stein Kernel

For a target distribution \(p\) with score function \(s_p(x) = \nabla_x \log p(x)\), the Stein operator \(\mathcal{A}_p\) acting on a vector-valued function \(f\) is \[\mathcal{A}_p f(x) \;=\; \langle s_p(x), f(x) \rangle + \nabla \cdot f(x).\] Stein’s identity asserts \(\mathbb{E}_{x \sim p}[\mathcal{A}_p f(x)] = 0\) for all \(f\) in a sufficiently smooth class. Given a base reproducing kernel \(k(x, y)\), the Stein kernel is \[k_p(x, y) \;=\; \langle s_p(x), s_p(y) \rangle k(x, y) + \langle s_p(x), \nabla_y k(x, y) \rangle + \langle s_p(y), \nabla_x k(x, y) \rangle + \mathrm{tr}\!\left(\nabla_x \nabla_y^{\top} k(x, y)\right),\] and the KSD of a sample \(\{x_i\}_{i=1}^n \sim Q\) from a target \(p\) is the V-statistic \[\mathrm{KSD}(Q, p) \;=\; \sqrt{\max\!\left\{ 0,\; \tfrac{1}{n^2} \sum_{i, j} k_p(x_i, x_j) \right\}}.\] A value close to zero indicates \(Q\) is well-aligned with \(p\); positive values indicate joint distributional deviation.

## 11.2. Empirical Gaussian Target (B9.4 iteration) and Future Extension

The KSD requires the score of the target. In this iteration the target is the empirical Gaussian Laplace approximation of the FB posterior over the common \(\xi\)-variables: \(\hat\mu = \mathbb{E}^{\mathrm{FB}}[\xi]\), \(\hat\Sigma = \mathrm{Cov}^{\mathrm{FB}}(\xi)\), so the Gaussian score is the linear closed-form \[s(x) \;=\; -\hat\Sigma^{-1}(x - \hat\mu).\] The implementation evaluates \(\mathrm{KSD}(\Pi_n^{\mathrm{EB}}, \mathcal{N}(\hat\mu, \hat\Sigma))\) in closed form (no auxiliary MCMC required for the target score). This is sufficient to detect EB-vs-FB deviations of joint first- and second-order moment structure. The full-KSD variant against the actual FB target via the Stan model’s `grad_log_prob()` method (a one-sample KSD of EB samples against the true non-Gaussian FB posterior) is a documented extension reserved for Block 9.x.

## 11.3. Base Kernel and Bandwidth

The default base kernel is the inverse multi-quadric (IMQ) of Gorham-Mackey \[k(x, y) \;=\; \left(h + \|x - y\|^2\right)^{\beta}, \qquad \beta \in (-1, 0), \quad \beta_{\mathrm{default}} = -\tfrac{1}{2},\] which is the canonical choice for distinguishing distributions whose densities differ in tails: Gorham-Mackey Theorem 8 establishes that the IMQ kernel detects mismatch with a dimension-independent rate for log-concave targets, a property the RBF kernel \(k(x, y) = \exp(-\|x - y\|^2 / (2 h))\) does not enjoy. The bandwidth \(h\) (in squared units of \(x\)) defaults to the median heuristic: \(h = \mathrm{median}\!\left( \{ \|x_i - x_j\|^2 \}_{i < j} \right)\) over the FB sample, which is dimension-adaptive. A fixed bandwidth is also accepted as a configuration option.

## 11.4. ESS-Weighted Variant

When MCMC chains are short relative to the integrated autocorrelation time of \(\xi\), the V-statistic \(\tfrac{1}{n^2}\sum_{i,j} k_p(x_i, x_j)\) understates the asymptotic KSD because successive draws are not independent. The `ess_weighted = TRUE` variant addresses this by thinning both EB and FB samples to \(\min(\widehat{\mathrm{ESS}}_{\mathrm{EB}}, \widehat{\mathrm{ESS}}_{\mathrm{FB}})\) rows (per-variable basic ESS via `posterior::ess_basic`), yielding an unbiased V-statistic on the thinned samples. The thinning seed is user-configurable for reproducibility.

## 11.5. Operational Reading: `gdpar_compare_eb_fb` and `gdpar_ksd_joint` are Complementary

The two diagnostics measure orthogonal aspects of the EB-vs-FB discrepancy:

- `gdpar_compare_eb_fb()` returns a marginal-by-parameter TV table: \(\widehat{\mathrm{TV}}(\Pi_n^{\mathrm{EB}, k}, \Pi_n^{\mathrm{FB}, k})\) for each scalar parameter \(k\) separately. This detects deviations in univariate marginals but is blind to joint dependence (e.g., two EB and FB posteriors with identical marginals but different correlation structure give the same marginal TV).

- `gdpar_ksd_joint()` returns a single joint KSD scalar that integrates over the full joint \(\xi\)-vector. This detects deviations in the joint dependence structure (correlations, copula geometry) that the marginal TV misses, at the cost of a coarser per-parameter resolution.

The recommended operational reading is to compute both: a small marginal TV plus a large joint KSD is the signature of an EB approximation that gets the per-coordinate spread right but distorts the cross-coordinate correlations. The converse (large marginal TV, small joint KSD) is uncommon in practice but indicates per-coordinate location/scale deviations that nevertheless preserve the joint dependence skeleton.

# 12. Summary

This vignette has extended the four central results of v07 (Theorem 7A scalar first-order equivalence; Proposition 7B scalar higher-order coverage; Theorem 7C scalar compound decision; Proposition 7D scalar substantial-discrepancy conditions) to the full \(p \geq 1\), \(K \geq 1\) regime, under strictly stronger hypotheses (EB-MARG-ID)\(_p\), (PRIOR-FB-WEAK)\(_p\), (HIER-COMPLEX)\(_p\).

The four asterisked results — Theorem 7A* (§4), Proposition 7B* (§5), Theorem 7C* (§6), Proposition 7D* (§7) — each reduce to their v07 counterparts under \((p, K) = (1, 1)\) as corollaries, ensuring that v07 remains the canonical scalar single-slot statement for pedagogical entry and that this vignette is a strict extension rather than a rewriting.

The critical audit of v07 §5 (§§1.1 and 4.5 of this vignette) confirmed that the scalar proof transfers cleanly to \(\mathbb{R}^p\) under the strengthened hypotheses, with no implicit use of strictly-scalar properties. The pedagogical opacity of v07 §6 regarding the sensitivity Jacobian \(J^\xi\) has been made explicit in the matrix form of Proposition 7B* (§5).

A new open question O5*-EBFB (§9) on the numerical anti-fragility of multivariate Laplace under non-conjugacy is canonized as the primary theoretical question motivating the four-component anti-fragility strategy (Charter §2.8) that the framework implements in Sub-phase 8.6.B. The formal theorem characterizing when `cmdstanr::laplace()` succeeds in \(\mathbb{R}^p\) is open and deferred to Block 9.x.

The framework’s recommendation (§8) remains FB by default across all multivariate configurations, with EB available as an explicit configuration option via `gdpar_eb()` for scenarios in which the multivariate Theorems 7A* and 7C* and the well-conditioned eigenvalue regime of (EB-MARG-ID)\(_p\) all align favorably. The user is referred to the operational vignette `vop07_eb_workflow.Rmd` (Sub-phase 8.6.E) for the practical workflow.

# Appendix A. Notation Correspondence: Multivariate vs. Scalar

Per Charter §3.1 (Sub-phase 8.6) Appendix A specification: tabular equivalence between v07’s scalar notation and v07b’s multivariate notation under the restriction \(p = 1\), \(K = 1\).

v07 (scalar, \(p = K = 1\)) | v07b (multivariate, general \(p\), \(K\)) | Reduction under \(p = K = 1\)
--- | --- | ---
\(\theta_{\mathrm{ref}} \in \mathbb{R}\) | \(\theta_{\mathrm{ref}} \in \mathbb{R}^{J \times K \times p}\) | Reduces to \(\theta_{\mathrm{ref}} \in \mathbb{R}^J\) (\(J\) groups) or \(\mathbb{R}\) (\(J = 1\)).
\(I_{\theta\theta}^{\mathrm{marg}} \in \mathbb{R}_{>0}\) | \(I_{\theta\theta}^{\mathrm{marg}} \in \mathbb{R}^{p \times p}\) PD | Scalar Fisher info as \(1 \times 1\) matrix.
\(\mathrm{Var}_n / \mathrm{Var}_\pi\) | \(\mathrm{tr\,Cov}_n / \mathrm{tr\,Cov}_\pi\) | Trace of \(1 \times 1\) matrix = its scalar entry.
Scalar Jacobian \(\partial \xi^* / \partial \theta_{\mathrm{ref}}\) (implicit) | \(J^\xi \in \mathbb{R}^{\dim(\xi) \times p}\) explicit | \(1 \times 1\) scalar absorbed in constant.
\(C_{g, \alpha} = (g'(\xi^*))^2 / I_{\theta\theta}^{\mathrm{marg}} \cdot \kappa(\alpha)\) | \(C_{g, \alpha}^* = \kappa(\alpha) (J^\xi)^\top \nabla_\xi g (\nabla_\xi g)^\top J^\xi (I_{\theta\theta}^{\mathrm{marg}})^{-1}\) | Sandwich reduces to scalar product.
TV on \(\mathbb{R}^{\dim(\xi)}\) (Regime A) | TV on \(\mathbb{R}^{\dim(\xi)}\) (Regime A, any \(p\)) | Independent of \(p\); depends on \(\dim(\xi)\).
Weak metric on \(\mathcal{G}\) (Regime B) | Weak metric on \(\mathcal{G}\) (Regime B, any \(p\)) | Independent of \(p\); depends on \(\dim(\xi)\).
Compound decision indexed by \(K\) (in v07’s notation) | Compound decision indexed by \(J\) within slot \(k\), summed over \(K\) slots | v07’s \(K\) in Theorem 7C = within-slot \(J\) in v07b.

# Appendix B. Hypothesis Table (Multivariate)

Per Charter §3.1 (Sub-phase 8.6) Appendix B specification: tabular summary of the asterisked hypotheses of §3, with their content, scope of use across §§4–7, and reduction to v07’s scalar versions.

Hypothesis | Content | Used by | Reduction under \(p = K = 1\)
--- | --- | --- | ---
(EB-MARG-ID)\(_p\) | \(L_n^{\mathrm{marg}}\) has unique global max in \(\mathbb{R}^p\); \(I_{\theta\theta}^{\mathrm{marg}}\) positive definite (all \(\lambda > 0\)) | Theorems 7A*, 7B*, 7C*; Proposition 7D* (ii)* | v07 (EB-MARG-ID): scalar info positive
(PRIOR-FB-WEAK)\(_p\) | \(\mathrm{tr\,Cov}_n(\theta_{\mathrm{ref}}^*) / \mathrm{tr\,Cov}_\pi(\theta_{\mathrm{ref}}) \to 0\) | Theorem 7A*; Proposition 7D* (iii)* | v07 (PRIOR-FB-WEAK): scalar variance ratio \(\to 0\)
(HIER-COMPLEX)\(_p\) | \(\log L_n^{\mathrm{marg}}\) is \(C^2\) in a \(p\)-ball around \(\widehat{\theta}_{\mathrm{ref}}^{\,\mathrm{EB}}\); Hessian bounded uniformly in \(n\); \(p\) fixed as \(n\) grows | Theorems 7A*, 7B*; Proposition 7D* (iv)* | v07 (HIER-COMPLEX): \(C^2\) on interval; bounded scalar dimension

# References Cited in This Block

Bhattacharya, R. N., and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Annals of Statistics, 6(2), 434–451.

Bickel, P. J., and Ghosh, J. K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction — a Bayesian argument. Annals of Statistics, 18(3), 1070–1090.

Brown, L. D. (2008). In-season prediction of batting averages: a field test of empirical Bayes and Bayes methodologies. Annals of Applied Statistics, 2(1), 113–152.

Carlin, B. P., and Gelfand, A. E. (1990). Approaches for empirical Bayes confidence intervals. Journal of the American Statistical Association, 85(409), 105–114.

Castillo, I., and Rousseau, J. (2015). A Bernstein–von Mises theorem for smooth functionals in semiparametric models. Annals of Statistics, 43(6), 2353–2383.

Efron, B. (2019). Bayes, oracle Bayes, and empirical Bayes. Statistical Science, 34(2), 177–201.

Ghosal, S., and van der Vaart, A. W. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge University Press.

Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer Series in Statistics.

Laird, N. M., and Louis, T. A. (1987). Empirical Bayes confidence intervals based on bootstrap samples. Journal of the American Statistical Association, 82(399), 739–750.

Petrone, S., Rousseau, J., and Scricciolo, C. (2014). Bayes and empirical Bayes: do they merge? Biometrika, 101(2), 285–302.

Robbins, H. (1956). An empirical Bayes approach to statistics. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, 157–163.

Rousseau, J., and Szabo, B. (2017). Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator. Annals of Statistics, 45(2), 833–865.

Szabo, B., van der Vaart, A. W., and van Zanten, J. H. (2013). Empirical Bayes scaling of Gaussian priors in the white noise model. Electronic Journal of Statistics, 7, 991–1018.

Tierney, L., and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81(393), 82–86.

van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. [Chapters 10 (multivariate Bernstein–von Mises) and 16 (asymptotics for estimating equations).]

van der Vaart, A. W., and van Zanten, J. H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Annals of Statistics, 36(3), 1435–1463.

White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1–25.