Draft. This pass carries the full outline, the definitions needed to state the main results, and the statements of Theorems 1 and 2. Supporting lemmas, closed forms, and all proofs are left as stubs in §4–§7 and the Appendix, to be filled in next.
We study a two-block Gibbs sampler for the posterior of a Bayesian hierarchical generalized linear mixed model (GLMM), and ask how quickly its distribution after \(n\) sweeps approaches the target \(\pi\) in total variation. The sampler alternates an exact draw of the random effects \(\beta=(\beta_j)_{j\le J}\) given the population parameter \(\gamma\), and an exact (or near-exact) draw of \(\gamma\) given \(\beta\). The \(\gamma\)-marginal of this chain is itself Markov, and it is on this marginal chain that we establish geometric ergodicity.
The argument has two independent parts. Part I (§2) is a general fact about any Markov chain that satisfies a Doeblin minorization condition on a subset \(C_d\) of its state space: the chain, run on \(C_d\), converges geometrically to the restriction of \(\pi\) to \(C_d\), and this transfers to a bound on distance from the full target \(\pi\) at the cost of the mass \(\pi(C_d^c)\) left outside. Part II (§3–§6) is specific to the hierarchical GLMM: it constructs a set \(C_d\) and a minorization constant explicitly, for every choice of tail-mass budget \(\delta\), under hypotheses satisfied by the package’s supported GLM families.
Theorem 1 is an application of Rosenthal (1995), Proposition 2
(uniform ergodicity from a Doeblin condition on the whole restricted
state space); see minor.pdf, p. 5. What is specific to this
note is Theorem 2: an explicit, checkable construction of the minorizing
set and constant for a hierarchical GLMM posterior, rather than an
existence argument alone.
Definition 1 (state space, restricted kernel). Let \((\gamma,\beta)\in\mathcal X\) and let \(\pi\) be the two-block Gibbs target on \(\mathcal X\). Fix \(C_d\subseteq \mathcal X\) with \(\pi(C_d)>0\). Write
\[ \pi(A\mid C_d):=\frac{\pi(A\cap C_d)}{\pi(C_d)},\qquad A\in\mathcal B(\mathcal X). \]
Let \(P_1(x,\cdot)\) be one Gibbs sweep on \(\mathcal X\), and let \(P_1(x,\cdot\mid C_d)\) be the kernel of the sampler restricted to \(C_d\) — defined once, per application, so that \(P_1(x,C_d\mid C_d)=1\) for \(x\in C_d\). Write \(P_n(x,\cdot\mid C_d):=\bigl(P_1(x,\cdot\mid C_d)\bigr)^n\).
Definition 2 (Doeblin minorization on \(C_d\)). There exist \(\varepsilon_d\in(0,1]\) and a probability measure \(Q\) on \(C_d\) such that
\[ P_1(x,A\mid C_d)\;\ge\;\varepsilon_d\,Q(A)\qquad\forall\,x\in C_d,\ \forall\,\text{measurable }A\subseteq C_d. \]
Standing assumption. \(\pi(\cdot\mid C_d)\) is a stationary law for \(P_1(\cdot\mid C_d)\) on \(C_d\). (Must be verified for the chosen restriction; not automatic from stationarity of \(\pi\) on \(\mathcal X\).)
Definition 3.
\[ \|\mu-\nu\|_{TV}:=\sup_{A\subseteq\mathcal X}\bigl|\mu(A)-\nu(A)\bigr|=\tfrac12\int|d\mu-d\nu|, \qquad \|\mu-\nu\|_1:=\int|d\mu-d\nu|=2\|\mu-\nu\|_{TV}. \]
Lemma 1 (truncation). With \(\mu:=\pi(\cdot\mid C_d)\) and \(\nu:=\pi\),
\[ \|\pi(\cdot\mid C_d)-\pi\|_{TV}=\pi(C_d^c), \qquad \|\pi(\cdot\mid C_d)-\pi\|_1=2\,\pi(C_d^c). \]
Theorem 1. Under Definitions 1–2 and the standing stationarity assumption, for \(x\in C_d\) and \(n\ge1\),
\[ \boxed{\;\bigl\|P_n(x,\cdot\mid C_d)-\pi\bigr\|_{TV}\;\le\;(1-\varepsilon_d)^n+\pi(C_d^c).\;} \]
Equivalently, \(\bigl\|P_n(x,\cdot\mid C_d)-\pi\bigr\|_1\le2(1-\varepsilon_d)^n+2\,\pi(C_d^c)\).
Proof outline (full proof in Appendix, §A.0 — carried over unchanged from the working draft, not reproduced here). Triangle inequality splits the bound into (i) distance from the restricted chain to \(\pi(\cdot\mid C_d)\), controlled by Rosenthal’s Proposition 2 applied on \(C_d\) with kernel \(P_1(\cdot\mid C_d)\); and (ii) distance from \(\pi(\cdot\mid C_d)\) to \(\pi\), which is Lemma 1. \(\qquad\blacksquare\)
Remark 1.1. Nothing here is specific to the joint \((\gamma,\beta)\) state space; Theorem 1 holds for any \((\mathcal X,\pi,C_d,P_1)\) satisfying Definitions 1–2. §3–§6 apply it with \(\mathcal X=\mathbb R^q\), \(\pi=\pi_\gamma\) (the \(\gamma\)-marginal posterior), and \(P_1=q(\cdot\mid\cdot\,;C)\).
Work in the two-block hierarchy of notation.md /
model_setup():
\[ y_j\mid\beta_j\ \text{GLM with canonical/noncanonical link},\qquad \beta_j\mid\gamma\sim N(H_j\gamma,\,\Psi),\qquad \gamma\sim N(\mu_0,\,\Lambda_\gamma^{-1}), \]
with \(P_b:=\Psi^{-1}\), \(P_{11}^{\mathrm{RE}}:=\sum_j H_j^\top P_b H_j\), and \(P_{11}:=\Lambda_\gamma+P_{11}^{\mathrm{RE}}\) the exact conditional precision of \(\gamma\) given \(\beta\).
(H1)–(H3) are the preflight gates under which the package runs production samplers. Unlike the working draft, no hypothesis on the one-step kernel’s positivity on compacts (formerly (H4)) is carried as a standing assumption — the construction in §4–§5 writes the minorizing density down explicitly, so that property is a consequence, not an input.
This subsection supplies (H1) at \(\Lambda_\gamma=0\), needed because the flat limit is not covered by the “bounded likelihood” argument available when \(\Lambda_\gamma\succ0\).
Lemma 2 (propriety of the flat-\(\gamma\) posterior). Assume:
Then the flat-\(\gamma\) posterior
\[ \pi_0(\gamma,\beta\mid y)\propto\exp\bigl(g(\gamma,\beta)\bigr),\qquad g(\gamma,\beta):=\sum_j\ell_j(\beta_j)-\tfrac12\sum_j(\beta_j-H_j\gamma)^\top P_b(\beta_j-H_j\gamma), \]
is proper: \(\int_{\mathbb R^q}\int_{\mathbb R^{Jp_{\mathrm{re}}}}e^g\,d\beta\,d\gamma<\infty\). Hence (H1) holds at \(\Lambda_\gamma=0\).
(Proof in Appendix A.1: \(g\) is concave, bounded above by (P1)+(P4), and recedes to \(-\infty\) in every direction of \((\gamma,\beta)\)-space — quadratically off the \(\gamma\)-flat directions by strong concavity of the RE prior, and at least linearly along them by (P2)+(P3) forcing every such direction to move some \(\beta_j\) off its finite maximum. (P4) is sufficient but not necessary; the sharp condition is stated alongside the proof.)
In two-block Gibbs, the \(\gamma\)-marginal is itself Markov, with kernel
\[ q(\gamma'\mid\gamma)=\int\phi_q\bigl(\gamma';\,m(\beta),\,P_{11}^{-1}\bigr)\,\pi(\beta\mid\gamma,y)\,d\beta, \qquad m(\beta)=P_{11}^{-1}\Bigl(\Lambda_\gamma\mu_0+\sum_jH_j^\top P_b\,\beta_j\Bigr). \]
Definition 4 (mean map). \(M(\gamma):=P_{11}^{-1}\bigl(\Lambda_\gamma\mu_0+\sum_jH_j^\top P_b\,b_j(\gamma)\bigr)\), where \(b_j(\gamma):=E[\beta_j\mid\gamma,y]\).
Lemma 3 (the mean map has a unique fixed point). Assume (H2), (H3a), and either \(\Lambda_\gamma\succ0\) or the strict-curvature condition that the group likelihoods contribute curvature (\(-\nabla^2\ell_j\succeq\epsilon_jP_b\) uniformly, some \(\epsilon_j>0\)). Then \(M\) is a contraction in the norm \(\|x\|_{P_{11}}=\|P_{11}^{1/2}x\|\) with modulus \(\kappa<1\); it has a unique fixed point \(\gamma^\star\), and the iteration \(\gamma^{(k+1)}=M(\gamma^{(k)})\) converges to it geometrically at rate \(\kappa\) from any start.
(Proof in Appendix A.1: the Jacobian of \(M\) is \(J(\gamma)=P_{11}^{-1}\sum_jH_j^\top P_bV_j(\gamma)P_bH_j\) with \(V_j(\gamma)=\mathrm{Cov}(\beta_j\mid\gamma,y)\); a Brascamp–Lieb bound under (H2) gives \(V_j\preceq P_b^{-1}\), hence \(\kappa<1\) either from \(\Lambda_\gamma\succ0\) directly or from the likelihood curvature margin in the flat limit; Banach’s fixed-point theorem finishes it.)
\(\gamma^\star\) is, equivalently, the mode of the marginal posterior \(\pi(\gamma\mid y)\) — this identification, and the objects it feeds into (\(Q\) in §4, the minorization profile in §5), are stated where first needed rather than collected here.
Theorem 2. Under (H1), (H2), (H3a) — and (H3b) as well if \(\Lambda_\gamma=0\) — for every \(\delta>0\) there exist:
such that \(\pi_\gamma(\cdot\mid C)\) is stationary for \(q(\cdot\mid\cdot\,;C)\) and \(q(\gamma,A\mid C)\ge\varepsilon\,Q_C(A)\) for all \(\gamma\in C\), \(A\subseteq C\) — hence, by Theorem 1, for \(\gamma\in C\),
\[ \|q_n(\gamma,\cdot\mid C)-\pi_\gamma\|_{TV}\;\le\;(1-\varepsilon)^n+\delta. \]
The refresh measure \(Q\) is fixed before any set is chosen (§4); the certified set is then the superlevel set of the resulting minorization profile (§5), so its compactness, its exhaustion of \(\mathbb R^q\), and the value of \(\varepsilon_d\) are all consequences of the same construction rather than separate searches. §6 proves Theorem 2 by assembling §4–§5 and lifting the bound from the \(\gamma\)-marginal to the joint \((\gamma,\beta)\) chain.
The refresh measure is fixed before any set is chosen — \(Q\) depends only on the model, not on a tail-mass budget \(\delta\) or a truncation level \(d\). Write \(P_{11}=\Lambda_\gamma+P_{11}^{\mathrm{RE}}\), \(P_{11}^{\mathrm{RE}}:=\sum_jH_j^\top P_bH_j\), \(P_b:=\Psi^{-1}\), as in §3.1. Two conditions pin \(Q\) down.
(Q1) Precision dominance. \(Q=N(\gamma^\star,\Sigma_Q)\) must satisfy \(\Lambda_Q:=\Sigma_Q^{-1}\succeq\Lambda_q(\gamma'\mid\gamma):=-\nabla^2_{\gamma'} \log q(\gamma'\mid\gamma)\) for all \(\gamma,\gamma'\) — the refresh measure must be at least as concentrated as the transition kernel, or the log-ratio \(\log q/q_Q\) is unbounded below and no minorization constant exists. The choice \(\Sigma_Q:=P_{11}^{-1}\) achieves this: since \(q(\cdot\mid\gamma)\) is a mixture of Gaussians with common covariance \(P_{11}^{-1}\) and random mean \(m(\beta)\), differentiating through the mixture gives, for every GLM family, \[ \Lambda_Q-\Lambda_q(\gamma'\mid\gamma)=\Lambda_Q\,\mathrm{Cov}\bigl(m(\beta)\mid\gamma,\gamma',y\bigr)\,\Lambda_Q\;\succeq\;0, \] the covariance taken over the bridge law \(\pi(\beta\mid\gamma,\gamma',y)\) — an identity, not a hypothesis, so (Q1) never has to be checked model by model. (Minorizing the exact one-step Gaussian draw \(\gamma'\mid\beta\) directly, without integrating \(\beta\) out, does not work: with \(\beta\) held fixed the bridge covariance above vanishes, the precision gap closes exactly, and the log-ratio against any centered Gaussian is affine with infimum \(-\infty\). Integrating \(\beta\) out is what produces a strict gap.)
(Q2) Mean. \(\gamma^\star=M(\gamma^\star)\), the fixed point of the mean map \(M\) of Definition 4 — equivalently (Cor. R2d-5, §3.3) the mode of the \(\gamma\)-marginal posterior \(\pi(\gamma\mid y)\).
Existence of \(\Sigma^\star:=P_{11}^{-1}\) and \(\gamma^\star\) — i.e. of \(Q\) itself — is what §4.1 and §4.2 establish, for a proper population prior and in the flat-prior limit respectively.
Lemma 4 (the covariance of \(Q\) exists). Assume \(\Lambda_\gamma\succ0\). Then \(P_{11}\succeq\Lambda_\gamma\succ0\), so \[ \Sigma^\star:=P_{11}^{-1}\ \text{exists and}\ 0\prec\Sigma^\star\preceq\Lambda_\gamma^{-1}. \] No rank condition on the hyper-design is needed: a proper population prior supplies invertibility on its own.
Lemma 5 (the centre of \(Q\) exists). Assume (H2) and \(\Lambda_\gamma\succ0\). The negative log marginal posterior \(\Phi(\gamma):=-\log\pi(\gamma\mid y)\) is \(\Lambda_\gamma\)-strongly convex, \[ \nabla^2\Phi(\gamma)\;=\;P_{11}-\sum_jH_j^\top P_bV_j(\gamma)P_bH_j\;\succeq\;\Lambda_\gamma\;\succ\;0 \qquad\text{for every }\gamma, \] where \(V_j(\gamma):=\mathrm{Cov}(\beta_j\mid\gamma,y)\). Consequently \(\Phi\) has a unique, finite minimiser \(\gamma^\star=\arg\min_\gamma\Phi(\gamma)=M(\gamma^\star)\) — the mode of the \(\gamma\)-marginal posterior exists, is unique, is finite, and is the fixed point of \(M\). (Like Lemma 4, this uses no rank condition — \(\Lambda_\gamma\succ0\) alone is enough. Uniqueness and existence can also be reached via Lemma 3, §3.3, by Banach’s fixed-point theorem; the two routes agree, but the variational route here is the one that survives the flat-prior limit of §4.2, where Lemma 3’s contraction modulus \(\kappa\) may equal \(1\).)
Corollary 6 (the refresh density \(Q\) exists). Under (H2) and \(\Lambda_\gamma\succ0\), \[ Q:=N(\gamma^\star,\Sigma^\star),\qquad q_Q(\gamma')=(2\pi)^{-q/2}\det(P_{11})^{1/2}\exp\Bigl\{-\tfrac12(\gamma'-\gamma^\star)^\top P_{11}(\gamma'-\gamma^\star)\Bigr\}, \] is a nondegenerate multivariate normal probability density, strictly positive and continuous on all of \(\mathbb R^q\), satisfying (Q1) (by the precision-dominance identity above, which holds for every GLM family) and (Q2) (by Lemma 5).
Write \(\Lambda_\gamma=\tau\bar\Lambda\) for fixed \(\bar\Lambda\succeq0\) and \(\mu_0\), and let \(\tau\downarrow0\); write \(P_{11}(\tau),\Sigma^\star(\tau),\gamma^\star(\tau), Q_\tau\) for the corresponding objects. Both results of §4.1 degrade at \(\tau=0\): the prior no longer supplies invertibility (needed for Lemma 4) or strong convexity (needed for Lemma 5). (H3a) replaces the first, (H3b) the second.
Lemma 7 (the limiting covariance exists iff (H3a)). As \(\tau\downarrow0\), \(P_{11}(\tau)\downarrow P_{11}^{\mathrm{RE}}\) monotonically in the psd order. Under (H3a) (\(P_{11}^{\mathrm{RE}}\succ0\)), \[ \Sigma^\star(\tau)\ \uparrow\ \Sigma^{\star(0)}:=\bigl(P_{11}^{\mathrm{RE}}\bigr)^{-1}\;\prec\;\infty, \] monotonically in the psd order and entrywise. Without (H3a) the limit does not exist: for \(0\ne v\in\ker P_{11}^{\mathrm{RE}}\), \(v^\top\Sigma^\star(\tau)v\to\infty\). (\(\Sigma^{\star(0)}\) is the generalized-least-squares covariance for the hyper-regression of the random effects on the hyper-design with weight \(\Psi^{-1}\); the prior term \(\Lambda_\gamma\mu_0\) drops out of \(M\) entirely in the limit, and \(\Sigma^\star(\tau)\) increases to \(\Sigma^{\star(0)}\) — the widest admissible refresh covariance.)
Lemma 8 (the limiting centre is a finite vector, under (H3a)+(H3b)). Assume (H2), (H3a), (H3b). Let \[ \Phi_0(\gamma)=\sum_j\mathcal D_j\bigl(P_b H_j\gamma\bigr)+\text{const} \] be the flat-limit negative log marginal posterior, where the group information deficit \(\mathcal D_j(\theta):=\tfrac12\theta^\top P_b^{-1}\theta+a_j-A_j(\theta)\ge0\) is convex (Brascamp–Lieb) with \(\nabla^2\mathcal D_j=P_b^{-1}-V_j\succeq0\). Then \(\Phi_0\) is convex and coercive — coercivity following from propriety of the flat-limit joint posterior (Lemma 2, §3.2) — so \[ \gamma^{\star(0)}:=\arg\min_\gamma\Phi_0(\gamma)=\arg\max_\gamma\pi_0(\gamma\mid y) \] exists, is finite, and \(\gamma^\star(\tau)\to\gamma^{\star(0)}\) as \(\tau\downarrow0\).
Open item carried over from removing the spectral-cutoff appendix: the source argument for uniqueness of \(\gamma^{\star(0)}\) went through a strict-positivity condition on the group Fisher information stated in that appendix (now dropped). Convexity, coercivity, existence, and the convergence \(\gamma^\star(\tau)\to \gamma^{\star(0)}\) do not need it; uniqueness needs a replacement condition — most likely \(P_{11}^{\mathrm{RE}}\succ0\) itself (already assumed as (H3a)) together with strict convexity of at least one \(\mathcal D_j\) along every direction, but this needs to be re-derived cleanly before Appendix A.2 is written, not carried over from the old citation.
Corollary 9 (the limiting refresh density exists). Under (H2), (H3a), (H3b), \[ Q_0:=N\bigl(\gamma^{\star(0)},\Sigma^{\star(0)}\bigr) \] is a nondegenerate multivariate normal, and \(Q_\tau\to Q_0\) in total variation (equivalently \(L^1\), by Scheffé) as \(\tau\downarrow0\).
Remark 4.1 (the two hypotheses are sharp, and fail differently).
| Failure | What breaks | Which result dies |
|---|---|---|
| (H3a) fails: stacked hyper-design rank-deficient | \(P_{11}^{\mathrm{RE}}\) singular \(\Rightarrow\) \(\Sigma^{\star(0)}\) does not exist | Lemma 7 — and with it \(q_{Q_0}\), so §5’s flat-limit profile has no density to compare against |
| (H3b) fails: some group not estimable | \(\Phi_0\) not coercive \(\Rightarrow\ \pi_0\) improper | Lemma 8 — no finite mode, so no (Q2), and no probability measure for §5’s exhaustion argument |
Both are visible directly in \(\Phi_0\): if \(0\ne v\in\bigcap_j\ker H_j\), then \(\Phi_0(\gamma+tv)=\Phi_0(\gamma)\) for all \(t\) — the flat-limit marginal posterior is constant along \(v\) and has no mode, with \(\tau\bar\Lambda\) having been the only thing keeping \(P_{11}\) invertible. If instead some group is separated, \(\ell_j\) fails to decay along a direction, \(\mathcal D_j\) stops growing, and \(\Phi_0\) flattens at infinity along it. For every \(\tau>0\) both failures are repaired by \(\Lambda_\gamma\succ0\) (Lemmas 4–5 used neither hypothesis), at the price of constants that degrade as \(\tau\downarrow0\) — which is exactly why the preflight gates on hyper-design rank and group estimability are necessary, not merely prudent, whenever the sampler runs with a flat population prior.
\(Q=N(\gamma^\star,\Sigma^\star)\) exists and is nondegenerate under (H2)+(H3a), with \(\Lambda_\gamma\succ0\) needed only to reach it by the shorter argument of §4.1; the same conclusion holds at \(\Lambda_\gamma=0\) under the strictly stronger (H2)+(H3a)+(H3b) via §4.2, and \(Q_\tau\to Q_0\) continuously as the prior is relaxed. In particular the refresh measure does not degenerate as the population prior becomes diffuse — it converges to a specific, finite-variance limit rather than blowing up or ceasing to exist. Precision dominance (Q1) holds unconditionally, in both regimes, as an identity rather than a hypothesis. What §4 does not yet establish is that \(Q\) actually minorizes the sampler kernel on a useful set with a useful constant — that is §5.
With \(Q\) fixed by §4, define the log-ratio \(g(\gamma'\mid\gamma):=\log q(\gamma'\mid\gamma)-\log q_Q(\gamma')\) — finite and jointly continuous, since both densities are strictly positive and continuous.
Definition 5 (profile, deficiency gap, certified set). \[ \varepsilon(\gamma):=\exp\Bigl\{\inf_{\gamma'\in\mathbb R^q}g(\gamma'\mid\gamma)\Bigr\}=\inf_{\gamma'}\frac{q(\gamma'\mid\gamma)}{q_Q(\gamma')}, \qquad \mathcal D(\gamma):=\log\varepsilon(\gamma). \] For \(d>0\), \[ \Psi(\gamma):=\mathcal D(\gamma^\star)-\mathcal D(\gamma)\ge0, \qquad \boxed{\ \widetilde C_d:=\{\gamma:\varepsilon(\gamma)\ge\varepsilon_d\}=\{\gamma:\Psi(\gamma)\le d\},\qquad \varepsilon_d:=e^{-d}\varepsilon(\gamma^\star).\ } \] The certified set is a superlevel set of the very function being minimised for the constant, which is what makes the eventual minimum of \(\varepsilon\) over \(\widetilde C_d\) land on a known value (\(\varepsilon_d\)) rather than an unrelated number — that identification is §5.3.
Assume (H2), (H3a), \(\Lambda_\gamma\succ0\) throughout §5.1–§5.3; §5.4 relaxes to \(\Lambda_\gamma=0\).
Lemma 10 (uniqueness). For fixed \(\gamma\), \(\gamma'\mapsto g(\gamma'\mid\gamma)\) is strictly convex, \[ \nabla^2_{\gamma'}g(\gamma'\mid\gamma)=\Lambda_Q\,\mathrm{Cov}\bigl(m(\beta)\mid\gamma,\gamma',y\bigr)\,\Lambda_Q\succ0 \] under (H3a), so it has at most one minimiser.
Lemma 11 (existence and positivity). For every \(\gamma\in\mathbb R^q\) the infimum defining \(\varepsilon(\gamma)\) is attained (at the unique point of Lemma 10), and \(\varepsilon(\gamma)\in(0,1]\). Up to an affine change of variable, \(g(\cdot\mid\gamma)\) is the cumulant generating function of the law of \(m(\beta)\) under \(\pi(\beta\mid\gamma,y)\), recentred at \(\gamma^\star\); under (H3a) that law has full support \(\mathbb R^q\), which forces coercivity along every ray and hence attainment. Coercivity here comes from support, not curvature, so this survives likelihoods with unbounded log-density curvature (e.g. Poisson-log).
Lemma 12 (continuity). \(\varepsilon:\mathbb R^q\to(0,1]\) is continuous. A locally-uniform coercivity estimate, worked in \(\beta\)-space where the mixing law does not depend on \(\Lambda_\gamma\), confines every minimiser over a compact \(K\ni\gamma\) to one \(K\)-dependent (but \(\gamma\)-free) ball; Berge’s maximum theorem then gives continuity on \(K\). The same estimate is what lets §5.4 pass this result to \(\Lambda_\gamma=0\).
Remark 5.1 (range). \(\varepsilon(\gamma)\le1\) always: \(q(\cdot\mid\gamma)\ge\varepsilon(\gamma)q_Q\) integrates to \(1\ge\varepsilon(\gamma)\), with equality iff \(q(\cdot\mid\gamma)=q_Q\).
Lemma 13 (the deficiency is concave; closed form for both derivatives). Under (H2), \(\mathcal D\in C^2\) with \[ \nabla\mathcal D(\gamma)=P_{11}\bigl(\gamma^\star-M(\gamma)\bigr), \qquad \nabla^2\mathcal D(\gamma)=-P_{11}J(\gamma)=-\sum_jH_j^\top P_bV_j(\gamma)P_bH_j\;\preceq\;0. \] Consequently: (1) \(\mathcal D\) is concave, strictly so under (H3a), so every superlevel set \(\{\varepsilon\ge\varepsilon_*\}\) is convex; (2) \(\nabla\mathcal D(\gamma)=0\iff M(\gamma)=\gamma^\star\iff\gamma=\gamma^\star\) (Lemma 3, §3.3) — \(\varepsilon\) attains its global maximum exactly at the refresh centre; (3) comparing with \(\nabla\Phi(\gamma)=P_{11}(\gamma-M(\gamma))\) (the Hessian identity behind Lemma 5) gives the exact identity \[ \boxed{\ \mathcal D(\gamma)-\mathcal D(\gamma^\star)=\bar\Phi(\gamma)-\tfrac12\|\gamma-\gamma^\star\|^2_{P_{11}}, \qquad\text{i.e.}\qquad \Psi(\gamma)+\bar\Phi(\gamma)=\tfrac12\|\gamma-\gamma^\star\|^2_{P_{11}},\ } \] where \(\bar\Phi:=\Phi-\min\Phi\) — the deficiency gap and the recentred negative-log-posterior partition the refresh quadratic exactly.
Lemma 14 (the deficiency gap is coercive; the certified sets are compact). Assume (H1), (H2), (H3a). With \(S(\gamma):=\nabla^2\Psi(\gamma)=\sum_jH_j^\top P_bV_j(\gamma)P_bH_j\), there is an explicit \(c>0\) with \[ \boxed{\ \Psi(\gamma)\ge c\bigl(|\gamma-\gamma^\star|-1\bigr)\ \text{ for }|\gamma-\gamma^\star|\ge1, \qquad\text{hence}\qquad \widetilde C_d\subseteq\{\gamma:|\gamma-\gamma^\star|\le1+d/c\}.\ } \] So \(\widetilde C_d\) is compact for every \(d>0\) — closed (Lemma 12), convex (Lemma 13), now bounded. If in addition the group curvature is bounded above, \(-\nabla^2\ell_j\preceq G_j\) (automatic when the GLM weights are bounded, e.g. binomial with any link), then Cramér–Rao gives \(V_j\succeq(P_b+G_j)^{-1}\), and with \(S_\flat:=\sum_jH_j^\top P_b(P_b+G_j)^{-1}P_bH_j\succ0\), \[ \boxed{\ \{\|\gamma-\gamma^\star\|^2_{P_{11}^{\mathrm{RE}}}\le2d\}\ \subseteq\ \widetilde C_d\ \subseteq\ \{\|\gamma-\gamma^\star\|^2_{S_\flat}\le2d\}, \qquad 0\prec S_\flat\preceq P_{11}^{\mathrm{RE}}\preceq P_{11}\ } \] — two concentric ellipsoids sandwiching \(\widetilde C_d\) for every \(d\). This uses (H3a) alone and is prior-free: \(S(\gamma)\) is built from random-effect conditional covariances, which never involve \(\Lambda_\gamma\); only the centre \(\gamma^\star\) moves with the prior, not the shape of \(\widetilde C_d\).
Corollary 15 (mass bounds). For every \(d>0\), \[ Q(\widetilde C_d)\ \ge\ \Pr(\chi^2_q\le2d), \qquad \pi_\gamma(\widetilde C_d^{\,c})\ \le\ \pi_\gamma\bigl(\|\gamma-\gamma^\star\|^2_{P_{11}}>2d\bigr), \] from \(\bar\Phi\ge0\) (Lemma 13(2)) via the complementarity identity: the \(Q\)-ellipsoid of squared radius \(2d\) is inscribed in \(\widetilde C_d\). Both hold for every GLM family with no closure assumption, and both are tail probabilities of the same statistic \(\|\gamma-\gamma^\star\|^2_{P_{11}}\sim\chi^2_q\) under \(Q\) — the second bound is what §6 uses for exhaustion (Theorem 2, item 1).
Lemma 16 (the constant on \(\widetilde C_d\) is the attained minimum). For every \(d>0\), \[ \boxed{\ \min_{\gamma\in\widetilde C_d}\varepsilon(\gamma)\;=\;\varepsilon_d\;=\;e^{-d}\varepsilon(\gamma^\star)\;\in\;(0,1),\quad\text{attained on }\partial\widetilde C_d=\{\Psi=d\}.\ } \] The constant is therefore sharp — no larger constant minorizes on \(\widetilde C_d\) — and \(\widetilde C_d\) is the largest set carrying it. (Weierstrass, from continuity (Lemma 12) and compactness (Lemma 14), gives attainment and \(\ge\varepsilon_d\) immediately since every point of \(\widetilde C_d\) satisfies \(\varepsilon\ge\varepsilon_d\) by definition; the intermediate value theorem along a ray from \(\gamma^\star\), using coercivity (Lemma 14), places a point exactly on the level \(\Psi=d\), giving \(\le\varepsilon_d\).)
Lemma 17 (the minorization condition holds on \(\widetilde C_d\)).
(a) Unrestricted chain, untruncated refresh. \(q(\gamma,A)\ge\varepsilon_d\,Q(A)\) for all \(\gamma\in\widetilde C_d\) and all \(A\in\mathcal B(\mathbb R^q)\): \(\widetilde C_d\) is a small set of the original \(\gamma\)-chain, with minorizing measure the untruncated Gaussian \(Q\) and no mass discount.
(b) Restricted chain, truncated refresh. Let \(q(\cdot,\cdot\mid\widetilde C_d)\) be the two-block sweep restricted to \(\widetilde C_d\) (Definition 1, §2.1) and \(Q_{\widetilde C_d}:=Q(\cdot\mid\widetilde C_d)\). Then \[ \boxed{\ q\bigl(\gamma,A\mid\widetilde C_d\bigr)\ \ge\ \varepsilon\,Q_{\widetilde C_d}(A), \qquad \varepsilon:=\varepsilon_d\,Q(\widetilde C_d)\ \ge\ \varepsilon(\gamma^\star)\,e^{-d}\Pr(\chi^2_q\le2d),\ } \] for all \(\gamma\in\widetilde C_d\), \(A\subseteq\widetilde C_d\) — both factors now explicit functions of \(d\), apart from the single scalar \(\varepsilon(\gamma^\star)\) (Lemma 11). (Truncating the refresh Gaussian to \(\widetilde C_d\) only raises the transition density on the retained region; the mass discount \(Q(\widetilde C_d)\) accounts exactly for the \(Q\)-mass \(\varepsilon_d\,Q(\widetilde C_d^{\,c})\) that (a) would have deposited outside \(\widetilde C_d\), unavailable to a chain confined there.) This is the pair \((\widetilde C_d,\varepsilon)\) instantiating items 1 and 3 of Theorem 2.
Write \(\varepsilon(\cdot\mid\tau)\), \(\widetilde C_d(\tau)\), \(\varepsilon_d(\tau)\) for the objects at prior scale \(\tau\) (notation of §4.2); assume (H2), (H3a), (H3b) throughout. Every dependence on \(\tau\) in §5.1–§5.3 enters only through \(Q_\tau\) — the mixing law \(\pi(\beta\mid\gamma,y)\) does not involve \(\Lambda_\gamma\) at all — which is what lets each result below carry over from §5.1–§5.3 essentially unchanged, read at \(\tau=0\).
Lemma 18 (limiting profile: well-posed, positive, continuous). At \(\Lambda_\gamma=0\), \(\varepsilon(\cdot\mid0)\) is well defined, lies in \((0,1]\) pointwise, and is continuous — Lemmas 10–12 read at \(\tau=0\), legitimate because their proofs consume only \(\Lambda_Q=P_{11}^{\mathrm{RE}}\succ0\) (Lemma 7, under (H3a)) and properties of \(\pi(\beta\mid\gamma,y)\) that never involve the prior. Moreover \(\varepsilon(\cdot\mid\tau)\to\varepsilon(\cdot\mid0)\) uniformly on compacts as \(\tau\downarrow0\).
Lemma 19 (limiting set: convex, compact, uniformly in the prior). \(\widetilde C_d(0)=\{\Psi_0\le d\}\) is convex, compact, and nonempty with \(\gamma^{\star(0)}\) (Lemma 8, §4.2) in its interior. The sandwich of Lemma 14 holds with the same, \(\tau\)-free constants at every \(\tau\ge0\), so there is a single compact \(K_d\supseteq\widetilde C_d(\tau)\) for all \(\tau\le\tau_0\), and \(\widetilde C_d(\tau)\to\widetilde C_d(0)\) in Hausdorff distance. Compactness needs only (H3a) here; it is (H3b), via Lemma 8, that places the centre \(\gamma^{\star(0)}\) — the only place (H3b) enters §5.
Lemma 20 (limiting constant; continuity in the prior). \[ \min_{\widetilde C_d(0)}\varepsilon(\cdot\mid0)\;=\;\varepsilon_d(0)\;=\;e^{-d}\varepsilon\bigl(\gamma^{\star(0)}\mid0\bigr)\;\in\;(0,1),\quad\text{attained on }\partial\widetilde C_d(0), \] and \(\varepsilon_d(\tau)\to\varepsilon_d(0)\) as \(\tau\downarrow0\) — the certified constant does not collapse as the prior becomes diffuse. This is the precise sense in which §1.2’s claim holds.
Lemma 21 (minorization holds in the limit). Under (H2), (H3a), (H3b), at \(\Lambda_\gamma=0\): \(q_0(\gamma,A)\ge\varepsilon_d(0)\,Q_0(A)\) for \(\gamma\in\widetilde C_d(0)\), \(A\in\mathcal B(\mathbb R^q)\) (unrestricted form), and \[ q_0\bigl(\gamma,A\mid\widetilde C_d(0)\bigr)\ge\varepsilon^{(0)}\,Q_{0,\widetilde C_d}(A), \qquad \varepsilon^{(0)}:=\varepsilon_d(0)\,Q_0\bigl(\widetilde C_d(0)\bigr)\ge\varepsilon\bigl(\gamma^{\star(0)}\mid0\bigr)e^{-d}\Pr(\chi^2_q\le2d), \] for \(\gamma\in\widetilde C_d(0)\), \(A\subseteq\widetilde C_d(0)\). Theorem 2 holds at \(\Lambda_\gamma=0\), with the flat-limit \(\gamma\)-marginal in place of \(\pi_\gamma\).
Remark 5.2 (division of labour between §5.1–§5.3 and §5.4). Positivity, uniqueness, continuity, convexity, and — crucially — compactness of the certified set need only (H2)+(H3a) and are identical in both regimes, because they rest entirely on \(\nabla^2\Psi=\sum_jH_j^\top P_bV_j(\gamma)P_bH_j\), which never involves \(\Lambda_\gamma\). (H3b) enters exactly once, to place the centre \(\gamma^{\star(0)}\) (Lemma 8); everything downstream of the centre then follows the proper-prior argument verbatim.
Lemma 22 (closed form for the deficiency; LMM only). Assume the hypotheses of §5.1–§5.3, and in addition Gaussian closure: \(q(\cdot\mid\gamma)=N(M(\gamma),\Sigma)\) with \(\Sigma\) not depending on \(\gamma\). Then \[ \boxed{\ \mathcal D(\gamma)=\tfrac12\log\det\bigl(\Sigma^\star\Sigma^{-1}\bigr)-\tfrac12\bigl\|\gamma^\star-M(\gamma)\bigr\|^2_{(\Sigma-\Sigma^\star)^{-1}},\ } \] so the one scalar left unevaluated by §5.1–§5.3 is \[ \varepsilon(\gamma^\star)=\sqrt{\det(\Sigma^\star\Sigma^{-1})}=\det(I+\tilde J)^{-1/2}\ \ge\ (1+\kappa)^{-q/2} \] (\(\tilde J\), \(\kappa\) as in Lemma 3, §3.3), and the certified sets are exactly the concentric ellipsoids \[ \widetilde C_d=\bigl\{\gamma:\|\gamma^\star-M(\gamma)\|^2_{(\Sigma-\Sigma^\star)^{-1}}\le2d\bigr\}, \qquad \varepsilon_d\ \ge\ (1+\kappa)^{-q/2}e^{-d}. \] (Existence, uniqueness, and attainment of the minimiser are Lemmas 10–11, which already apply; closure lets the minimisation be solved in closed form because \(\Lambda_q=\Sigma^{-1}\) no longer depends on \(\gamma'\).)
Remark 5.3 (what closure buys, and what does not survive without it). Only Lemma 22 is closure-dependent. Every result of §5.1–§5.4 holds for every GLM family and already gives \(\varepsilon_d=e^{-d}\varepsilon(\gamma^\star)\) exactly; closure supplies a formula for the one remaining scalar \(\varepsilon(\gamma^\star)\), in place of a \(q\)-dimensional convex program solved numerically, and an exact ellipsoid for \(\widetilde C_d\) in place of the two-sided sandwich of Lemma 14. Outside closure, \(q(\cdot\mid\gamma)=\int\phi_q(\cdot;m(\beta),\Sigma^\star)\pi(\beta\mid\gamma,y)\,d\beta\) is a Gaussian mixture with no single covariance — which is exactly why (Q1) in §4 is stated on precisions rather than covariances — and \(\Lambda_q(\gamma'\mid\gamma)\) genuinely varies with \(\gamma'\), so \(\mathcal D\) has no closed form. Every qualitative conclusion of §5.1–§5.4 — positivity, continuity, concavity, compactness, the attained constant \(\varepsilon_d=e^{-d}\varepsilon(\gamma^\star)\), and its nondegeneracy as \(\Lambda_\gamma\downarrow0\) — survives regardless; what is lost is only a closed-form value for \(\varepsilon(\gamma^\star)\) and the exact ellipsoidal shape.
Two ingredients remain beyond §4–§5: that the certified sets \(\widetilde C_d\) exhaust \(\mathbb R^q\) as \(d\to\infty\), so a set can actually be chosen to meet any tail-mass budget \(\delta\); and that restricting the two-block sweep to a set \(C\) produces a kernel for which \(\pi_\gamma(\cdot\mid C)\) is genuinely stationary — the standing assumption Definition 1 (§2.1) required but did not prove.
Lemma 23 (exhaustion). Under (H1) and the hypotheses of §5.1–§5.3 (or, at \(\Lambda_\gamma=0\), of §5.4), the family \(\widetilde C_d\) is nondecreasing in \(d\) with \(\bigcup_{d>0}\widetilde C_d=\mathbb R^q\). Consequently \(\pi_\gamma(\widetilde C_d^{\,c})\downarrow0\) as \(d\uparrow\infty\), and for every \(\delta>0\) there is \(d(\delta)<\infty\) with \(\pi_\gamma\bigl(\widetilde C_{d(\delta)}^{\,c}\bigr)<\delta\). (Positivity of \(\varepsilon\) pointwise — Lemma 11, or 18 at \(\tau=0\) — gives \(\Psi(\gamma)<\infty\) for every \(\gamma\), hence every point eventually enters \(\widetilde C_d\); continuity from above of the probability measure \(\pi_\gamma\) finishes it. This is the same pointwise positivity that Lemma 11 already established — no separate tightness argument or compactness input is needed.)
Lemma 24 (restricted kernel and stationarity). Let \(C\subseteq\mathbb R^q\) be measurable with \(\pi_\gamma(C)>0\), and let \(q(\cdot\mid\cdot\,;C)\) be the \(\gamma\)-marginal of the two-block Gibbs kernel of the truncated target \(\pi(\cdot\mid\tilde C)\), \(\tilde C:=C\times\mathbb R^{Jp_{\mathrm{re}}}\) — an exact draw \(\beta\sim\pi(\beta\mid\gamma,y)\) followed by \(\gamma'\sim N(m(\beta),\Sigma^\star)\) truncated to \(C\). Then \(q(\gamma,C\mid C)=1\) for \(\gamma\in C\), and \(\pi_\gamma(\cdot\mid C)\) is invariant for \(q(\cdot\mid\cdot\,;C)\). (Each block is drawn from the exact conditional of \(\pi(\cdot\mid\tilde C)\) given the other block, so the restricted sweep is Gibbs for the truncated target, not the unrestricted sweep with excursions repaired after the fact — a construction of the latter kind does not in general preserve \(\pi(\cdot\mid\tilde C)\).) This discharges the standing assumption of Definition 1, for \(C=\widetilde C_d\) or any measurable subset of it.
Proof of Theorem 2. Fix \(\delta>0\).
Items 2–5 instantiate Theorem 2’s items 1–3 exactly. Applying Theorem 1 (§2.4) with this \((C,\varepsilon)\): for \(\gamma\in C\), \[ \|q_n(\gamma,\cdot\mid C)-\pi_\gamma\|_{TV}\ \le\ (1-\varepsilon)^n+\pi_\gamma(C^c)\ <\ (1-\varepsilon)^n+\delta. \qquad\blacksquare \]
Remark 6.1 (how \(\varepsilon\) degrades as \(\delta\) shrinks). Both terms competing in the choice of \(d\) are now explicit: \(\varepsilon_d=e^{-d}\varepsilon(\gamma^\star)\) is decreasing in \(d\), while \(\pi_\gamma(\widetilde C_d^{\,c})\) is decreasing in \(d\) toward \(0\). Under (H2), \(\pi_\gamma\) is log-concave, so with \(\Sigma_\pi\) its (finite) covariance and \(\rho:=\lambda_{\max}(\Sigma_\pi P_{11})\ge1\), the tail of \(\Psi\) is sub-exponential and \(\pi_\gamma(\widetilde C_d^{\,c})\lesssim e^{-d/\rho}\) (standard log-concave concentration; the exact constant is deferred to Appendix A.4). Choosing \(d=\rho\log(1/\delta)\) to meet the budget then gives, for every GLM family, \[ \boxed{\ \varepsilon\ \gtrsim\ \varepsilon(\gamma^\star)\,\delta^{\rho}.\ } \] The minorization constant degrades only polynomially in the tail-mass budget \(\delta\), with exponent \(\rho\) measuring the mismatch between the posterior’s spread and the refresh metric \(P_{11}\) — cheap whenever \(\rho=O(1)\). The metric here is \(P_{11}\succeq P_{11}^{\mathrm{RE}}\succ0\), not \(\Lambda_\gamma\), so unlike an envelope built from the prior alone, this survives the flat-prior limit under (H3a). Under Gaussian closure, \(\rho=(1-\kappa)^{-1}\) exactly (Lemma 22, §5.5).
Theorem 2 and its proof concern only the \(\gamma\)-marginal chain. The following closes the gap to the full two-block state space at no cost in rate.
Lemma 25 (lifting to the joint chain). The joint two-block chain is a de-initialization of the \(\gamma\)-chain: \((\gamma_n,\beta_n)\) is obtained from \(\gamma_{n-1}\) by drawing \(\beta_n\sim\pi(\beta\mid\gamma_{n-1},y)\) (exact) and then \(\gamma_n\sim\pi(\gamma\mid\beta_n)\) (exact, or restricted to \(C\) as in §6.1). Since the extra \(\beta\)-draw is from the exact conditional of the target, the data-processing inequality for total variation gives \[ \|\mathcal L(\gamma_n,\beta_n)-\pi\|_{TV}\ \le\ \|\mathcal L(\gamma_{n-1})-\pi_\gamma\|_{TV}. \] Combined with Theorem 2, for \(\gamma_0\in C\), \[ \|\mathcal L(\gamma_n,\beta_n)-\pi\|_{TV}\ \le\ (1-\varepsilon)^{n-1}+\delta. \] Every bound proved for the \(\gamma\)-chain therefore transfers to the joint chain with the same constants and a one-step offset. Scoping Theorem 2 to the \(\gamma\)-marginal costs nothing at the end, and buys a construction on a state space of dimension \(q\) rather than \(q+Jp_{\mathrm{re}}\) throughout §4–§6 — which is why \(\varepsilon\) carries no dependence on the number of groups \(J\).
(Content to follow.)
Proof of Lemma 2 (propriety of the flat-\(\gamma\) posterior).
1. \(g\) is concave and bounded above. Each \(\ell_j\) is concave by (P1) and the quadratic term is concave, so \(g\) is concave on \(\mathbb R^q\times\mathbb R^{Jp_{\mathrm{re}}}\). By (P4), \(\ell_j\le\ell_j(\hat\beta_j)<\infty\) and the quadratic term is \(\le0\), so \(g\le\sum_j\ell_j(\hat\beta_j)<\infty\).
2. Each \(\ell_j\) has compact superlevel sets. By (P2), \(\beta_j\mapsto\eta_j=Z_j\beta_j\) is injective, and with the IRLS weights strictly positive (P1) the log-likelihood is strictly concave in \(\beta_j\); by (P4) its maximum is attained at a finite \(\hat\beta_j\). A strictly concave function with an attained maximum decays at least linearly along every ray: for unit \(v\), \[ \ell_j^\infty(v):=\lim_{t\to\infty}\frac{\ell_j(\hat\beta_j+tv)-\ell_j(\hat\beta_j)}{t}<0, \qquad c_j:=-\max_{\|v\|=1}\ell_j^\infty(v)>0 \] (the max is attained since \(\ell_j^\infty\) is concave, positively homogeneous, and u.s.c. on the compact unit sphere). This is exactly “finite, unique MLE with positive IRLS weights” — no separation, no flat direction.
3. Every direction recedes. Fix \((d_\gamma,d_\beta)\ne0\).
4. Integrability. \(g\) is concave, bounded above, with strictly negative recession function in every direction; by compactness of the unit sphere there are \(c>0\), \(x_0=(\hat\gamma,\hat\beta)\) with \(g(x)\le g(x_0)-c\|x-x_0\|\) for \(\|x-x_0\|\) large, so \[ \int e^g\ \le\ \mathrm{const}\cdot e^{g(x_0)}\int e^{-c\|x-x_0\|}dx\ <\ \infty. \qquad\blacksquare \]
Sharpness. (P4) is sufficient, not necessary: the exact requirement in Case B is that the pooled profile \(\gamma\mapsto\sum_j\ell_j(H_j\gamma)\) have compact superlevel sets — groups can borrow strength, and a single separated group need not destroy propriety if the others pin down \(\gamma\). (P4) is therefore a conservative, per-group certificate: it certifies propriety but failing it does not prove impropriety. (Specializing to binomial with a flat prior recovers the classical fact that posterior propriety under a flat prior corresponds to existence of the MLE, i.e. absence of complete or quasi-complete separation.)
Proof of Lemma 3 (the mean map has a unique fixed point).
(a) Jacobian via exponential tilt. \(\pi(\beta_j\mid\gamma,y)\propto\exp\{\ell_j(\beta_j)+(P_bH_j\gamma)^\top\beta_j-\frac12\beta_j^\top P_b\beta_j\}\) is an exponential family in natural parameter \(\theta_j=P_bH_j\gamma\) with sufficient statistic \(\beta_j\), so \(\partial b_j/\partial\theta_j=V_j(\gamma)\) and, by the chain rule, \(\partial b_j/\partial\gamma=V_jP_bH_j\). Hence \[ J(\gamma):=\frac{\partial M}{\partial\gamma}=P_{11}^{-1}\sum_jH_j^\top P_bV_j(\gamma)P_bH_j=P_{11}^{-1/2}\tilde J(\gamma)P_{11}^{1/2}, \qquad \tilde J:=P_{11}^{-1/2}\Bigl(\sum_jH_j^\top P_bV_jP_bH_j\Bigr)P_{11}^{-1/2}\succeq0. \]
(b) Brascamp–Lieb upper bound. Under (H2), \(-\nabla^2\log\pi(\beta_j\mid\gamma,y)=-\nabla^2\ell_j+P_b\succeq P_b\): the conditional is \(P_b\)-strongly log-concave, so Brascamp–Lieb gives \(V_j\preceq P_b^{-1}\), hence \(\sum_jH_j^\top P_bV_jP_bH_j\preceq\sum_jH_j^\top P_bH_j=P_{11}^{\mathrm{RE}}\), i.e. \(\tilde J\preceq I-P_{11}^{-1/2}\Lambda_\gamma P_{11}^{-1/2}\).
(c) Strictness. If \(\Lambda_\gamma\succ0\): \(\kappa:=\sup_\gamma\lambda_{\max}(\tilde J(\gamma))\le1-\lambda_{\min}(P_{11}^{-1/2}\Lambda_\gamma P_{11}^{-1/2})<1\) unconditionally — the prior alone supplies the gap. At \(\Lambda_\gamma=0\), one instead needs \(V_j\prec P_b^{-1}\) strictly: if \(-\nabla^2\ell_j\succeq\epsilon_jP_b\) uniformly, \(V_j\preceq((1+\epsilon_j)P_b)^{-1}\) and \(\kappa\le1/(1+\min_j\epsilon_j)<1\).
(d) Contraction and Banach. \(M(\gamma_1)-M(\gamma_2)=\int_0^1J(\gamma_t)(\gamma_1-\gamma_2)\,dt\) along the segment \(\gamma_t\), so \(P_{11}^{1/2}(M(\gamma_1)-M(\gamma_2))=\int_0^1\tilde J(\gamma_t)P_{11}^{1/2}(\gamma_1-\gamma_2)\,dt\) and \(\|M(\gamma_1)-M(\gamma_2)\|_{P_{11}}\le\kappa\|\gamma_1-\gamma_2\|_{P_{11}}\). \(\mathbb R^q\) with \(\|\cdot\|_{P_{11}}\) is complete, so Banach’s fixed-point theorem gives existence, uniqueness, and geometric convergence of \(\gamma^{(k+1)}=M(\gamma^{(k)})\). \(\blacksquare\)
Proof of Lemma 4 (the covariance of \(Q\) exists). \(P_{11}^{\mathrm{RE}}\) is a sum of terms \(H_j^\top P_bH_j\succeq0\), so \(P_{11}\succeq\Lambda_\gamma\succ0\); inverting reverses the order. \(\blacksquare\) (No rank condition is used: a proper prior supplies invertibility on its own.)
Proof of Lemma 5 (the centre of \(Q\) exists). The Hessian formula follows from Lemma 3(a); Brascamp–Lieb (Lemma 3(b)) gives \(V_j\preceq P_b^{-1}\), hence \(\sum_jH_j^\top P_bV_jP_bH_j\preceq P_{11}^{\mathrm{RE}}\), which is the displayed bound \(\nabla^2\Phi\succeq\Lambda_\gamma\succ0\). A \(\Lambda_\gamma\)-strongly convex function satisfies \(\Phi(\gamma)\ge\Phi(\gamma_0)+\nabla\Phi(\gamma_0)^\top(\gamma-\gamma_0)+\frac12\|\gamma-\gamma_0\|^2_{\Lambda_\gamma}\), hence is coercive, so its minimum is attained; strict convexity makes it unique and finite. Stationarity \(\gamma^\star=M(\gamma^\star)\) is the first-order condition. \(\blacksquare\) (Lemma 3 also reaches \(\gamma^\star\), via Banach; the two routes agree, but the variational route here is the one that survives \(\Lambda_\gamma=0\), where \(\kappa\) may equal \(1\).)
Proof of Corollary 6 (the refresh density \(Q\) exists). Combine Lemmas 4 and 5; positivity and continuity of \(q_Q\) are immediate from the Gaussian formula, well posed since \(\det P_{11}>0\). Precision dominance (Q1) is the identity derived in §4 (differentiate the mixture defining \(q(\cdot\mid\gamma)\) twice in \(\gamma'\): \(\Lambda_Q-\Lambda_q(\gamma'\mid\gamma)=\Lambda_Q\,\mathrm{Cov}(m(\beta)\mid\gamma,\gamma',y)\,\Lambda_Q\succeq0\), the covariance under the bridge law \(\pi(\beta\mid\gamma,\gamma',y)\)), which holds for every GLM family with no further hypothesis. \(\blacksquare\)
Proof of Lemma 7 (the limiting covariance exists iff (H3a)). Monotonicity of \(P_{11}(\tau)=\tau\bar\Lambda+P_{11}^{\mathrm{RE}}\) in \(\tau\) is immediate, and inversion reverses the psd order. Under (H3a), \(P_{11}^{\mathrm{RE}}\) is nonsingular and matrix inversion is continuous there, giving \(\Sigma^\star(\tau)\to\Sigma^{\star(0)}\). Without (H3a): for \(0\ne v\in\ker P_{11}^{\mathrm{RE}}\), Cauchy–Schwarz gives \((v^\top v)^2=(v^\top P_{11}^{1/2}P_{11}^{-1/2}v)^2\le(v^\top P_{11}v)(v^\top P_{11}^{-1}v)\), so \(v^\top\Sigma^\star(\tau)v\ge(v^\top v)^2/(\tau\,v^\top\bar\Lambda v)\to\infty\). \(\blacksquare\) (\(\Sigma^{\star(0)}=(P_{11}^{\mathrm{RE}})^{-1}\) is the GLS covariance for the hyper-regression of the random effects on the hyper-design with weight \(\Psi^{-1}\); the prior term drops out of \(M\) entirely in the limit.)
Proof of Lemma 8 (the limiting centre is a finite vector).
Coercivity. Under (H2)+(H3a)+(H3b), Lemma 2 (§3.2/A.1) makes the flat-limit joint posterior proper, and its Step 4 supplies an exponential envelope \(e^{-c\|x-x_0\|}\); marginalizing, \(\Phi_0\) grows at least linearly and is coercive. A convex coercive function attains its minimum.
Uniqueness. \(\nabla^2\Phi_0=P_{11}^{\mathrm{RE}}-\sum_jH_j^\top
P_bV_jP_bH_j\succeq0\) always (Lemma 3(a)–(b)); strict positivity
— needed for uniqueness — holds wherever the group conditional
covariances \(V_j\) sit strictly below
their Brascamp–Lieb ceiling \(P_b^{-1}\). This is supplied, with no
measure-zero exclusion, by (P2) — \(Z_j\) full column rank for every \(j\), already part of Lemma 2’s hypotheses
(groupef.rank) — together with (H2)’s requirement that the
GLM weights \(w_{j,i}(\eta)>0\) at
every finite \(\eta\) (true for every
canonical-link family, e.g. \(\mu(1-\mu)>0\) for logit, \(\mu=e^\eta>0\) for Poisson-log). With
\(V(\beta_j):=-\ell_j(\beta_j)+\frac12(\beta_j-H_j\gamma)^\top
P_b(\beta_j-H_j\gamma)\), Brascamp–Lieb’s variance formula gives,
for any \(a\), \(a^\top
V_j(\gamma)a=\mathrm{Var}_\pi(a^\top\beta_j)\le
E_\pi[a^\top(\nabla^2V(\beta_j))^{-1}a]\) — requiring only \(\nabla^2V\succ0\) a.e., not a uniform lower
bound. Since \(\nabla^2V(\beta_j)=Z_j^\top\mathrm{diag}(w(\eta_j))Z_j+P_b\)
and \(Z_j\) has full column rank with
\(w>0\) at every finite \(\eta_j\), \(\nabla^2V(\beta_j)\succ P_b\) — strictly,
at every finite \(\beta_j\),
not almost every — so \((\nabla^2V(\beta_j))^{-1}\prec P_b^{-1}\)
pointwise everywhere; taking \(E_\pi\)
of a pointwise-strict inequality under a measure of full support
preserves strictness, giving \(V_j(\gamma)\prec P_b^{-1}\) as a matrix
inequality. For \(v\ne0\), (H3a) gives
some \(j_0\) with \(H_{j_0}v\ne0\), and \(v^\top\nabla^2\Phi_0v\ge(P_bH_{j_0}v)^\top(P_b^{-1}-V_{j_0}(\gamma))(P_bH_{j_0}v)>0\).
So \(\nabla^2\Phi_0(\gamma)\succ0\) for
every \(\gamma\), and a strictly convex
coercive function has a unique minimiser.
Convergence of the modes. \(\Phi_\tau(\gamma)=\Phi_0(\gamma)+\frac\tau2(\gamma-\mu_0)^\top\bar\Lambda(\gamma-\mu_0)\ge\Phi_0(\gamma)\), so minimality gives \(\Phi_0(\gamma^\star(\tau))\le\Phi_\tau(\gamma^\star(\tau))\le\Phi_\tau(\gamma^{\star(0)})=\Phi_0(\gamma^{\star(0)})+C\tau\) with \(C=\frac12(\gamma^{\star(0)}-\mu_0)^\top\bar\Lambda(\gamma^{\star(0)}-\mu_0)\). Every \(\gamma^\star(\tau)\) therefore lies in the sublevel set \(\{\Phi_0\le\Phi_0(\gamma^{\star(0)})+C\tau\}\), bounded by coercivity; any limit point \(\bar\gamma\) satisfies \(\Phi_0(\bar\gamma)\le\Phi_0(\gamma^{\star(0)})\) by continuity, hence \(\bar\gamma=\gamma^{\star(0)}\) by uniqueness, and a bounded family with one limit point converges. \(\blacksquare\)
(Why Banach is unavailable here. At \(\tau=0\), Lemma 3(c) leaves only \(\kappa\le1\), and for families whose GLM weights vanish in the tails — logit, probit at large \(|\eta|\) — one has \(\sup_\gamma\kappa(\gamma)=1\) exactly, so the contraction argument gives nothing globally; Lemma 8 is the variational replacement. If uniform group curvature \(-\nabla^2\ell_j\succeq\epsilon_jP_b\) is assumed, Lemma 3(c) instead gives the prior-free ceiling \(\kappa_0\le1/(1+\min_j\epsilon_j)<1\) and Banach is restored with a rate that does not degrade as \(\tau\downarrow0\).)
Proof of Corollary 9 (the limiting refresh density exists). Nondegeneracy is Lemma 7 plus finiteness of \(\gamma^{\star(0)}\) (Lemma 8). Since \(\gamma^\star(\tau)\to\gamma^{\star(0)}\) and \(\Sigma^\star(\tau)\to\Sigma^{\star(0)}\succ0\), the Gaussian densities converge pointwise; both are probability densities, so Scheffé’s lemma gives \(L^1\) (equivalently TV) convergence. \(\blacksquare\)
Throughout §5.1–§5.3, assume (H2), (H3a), \(\Lambda_\gamma\succ0\), so \(Q\) exists by Corollary 6.
Proof of Lemma 10 (uniqueness). Differentiating the mixture defining \(q(\cdot\mid\gamma)\) twice in \(\gamma'\) gives \(\nabla^2_{\gamma'}g=\Lambda_Q-\Lambda_q(\gamma'\mid\gamma)=\Lambda_Q\,\mathrm{Cov}(m(\beta)\mid\gamma,\gamma',y)\,\Lambda_Q\), the covariance under the bridge law. With \(m(\beta)=L\beta+a\), \(L=P_{11}^{-1}[H_1^\top P_b,\dots,H_J^\top P_b]\) of full row rank \(q\) under (H3a), and the bridge law of \(\beta\) nondegenerate, \(\mathrm{Cov}(m\mid\cdot)=L\,\mathrm{Cov}(\beta\mid\cdot)L^\top\succ0\); conjugating by \(\Lambda_Q\succ0\) preserves positivity. A strictly convex function has at most one minimiser. \(\blacksquare\)
Proof of Lemma 11 (existence and positivity). Let \(P_\gamma\) be the law of \(m(\beta)\) under \(\pi(\beta\mid\gamma,y)\). Dividing the two densities and cancelling the Gaussian normalisers and the \(\gamma'\)-quadratics, \(g(\gamma'\mid\gamma)=\log\int e^{(m-\gamma^\star)^\top\Lambda_Q\gamma'}\,d\nu_\gamma(m)\) where \(d\nu_\gamma(m)=e^{-\frac12m^\top\Lambda_Qm+\frac12{\gamma^\star}^\top\Lambda_Q\gamma^\star}dP_\gamma(m)\) — so in \(t=\Lambda_Q\gamma'\), \(g\) is the cumulant generating function of the finite positive measure \(\nu_\gamma\), recentred at \(\gamma^\star\): finite everywhere (the Gaussian factor dominates every exponential) and convex. For coercivity, restrict to a half-space: for unit \(u\), \(c>0\), \(s>0\), \[ g(s\Lambda_Q^{-1}u\mid\gamma)\ \ge\ sc+\log\nu_\gamma\bigl(\{m:(m-\gamma^\star)^\top u\ge c\}\bigr)\ \xrightarrow[s\to\infty]{}\ +\infty \] whenever that half-space carries positive \(\nu_\gamma\)-mass. Under (H3a), \(m(\beta)\) is an affine map of \(\beta\) with linear part of rank \(q\), and \(\pi(\beta\mid\gamma,y)>0\) on all of \(\mathbb R^{Jp_{\mathrm{re}}}\), so \(P_\gamma\) (hence \(\nu_\gamma\)) has full support \(\mathbb R^q\) and every half-space qualifies. A finite convex coercive function attains its minimum, which is therefore \(>-\infty\); exponentiating gives \(\varepsilon(\gamma)>0\). \(\blacksquare\)
Proof of Lemma 12 (continuity). Joint continuity of \(g\) gives upper semicontinuity of \(\log\varepsilon=\inf_{\gamma'}g\) for free. For the reverse direction, fix compact \(K\subset\mathbb R^q\); working in \(\beta\)-space (where \(\pi(\beta\mid\gamma,y)>0\) everywhere), write \(t=\Lambda_Q\gamma'\), \(u=t/|t|\), \(m(\beta)=L\beta+a\), so \((m(\beta)-\gamma^\star)^\top t=|t|(\beta^\top L^\top u+(a-\gamma^\star)^\top u)\). Since \(L\) has full row rank under (H3a), \(\sigma_0:=\sigma_{\min}(L)>0\) and \(v(u):=L^\top u/|L^\top u|\) is a well-defined unit vector depending continuously on \(u\); picking \(\rho\) large enough that \(c:=\sigma_0\rho/2-|a-\gamma^\star|>0\), on the cone-cap \(B_v:=\{\beta:|\beta|\le\rho,\ \beta^\top v\ge\rho/2\}\) one has \((m(\beta)-\gamma^\star)^\top u\ge c\), so restricting the integral to \(B_{v(u)}\), \[ g(\gamma'\mid\gamma)\ \ge\ c|t|+\log\bigl(w_{\min}\,\pi(B_{v(u)}\mid\gamma,y)\bigr), \] where \(w_{\min}>0\) lower-bounds the continuous positive Gaussian weight on \(\{|\beta|\le\rho\}\). Since \((v,\gamma)\mapsto\pi(B_v\mid\gamma,y)\) is continuous (dominated convergence; the cone-cap boundary is Lebesgue-null) and strictly positive on the compact \(\{|v|=1\}\times K\), \(p_K:=\min\pi(B_v\mid\gamma,y)>0\) there; taking \(\eta_K:=w_{\min}p_K\), \(c_K:=c\,\lambda_{\min}(\Lambda_Q)\) gives \(g(\gamma'\mid\gamma)\ge c_K|\gamma'|+\log\eta_K\) for all \(\gamma\in K\), uniformly. Since \(\log\varepsilon\le0\), every minimiser satisfies \(|\gamma'_\star(\gamma)|\le R_K:=(\log(1/\eta_K))_+/c_K\), the same \(R_K\) for all \(\gamma\in K\); so on \(K\), \(\log\varepsilon(\gamma)=\min_{|\gamma'|\le R_K}g(\gamma'\mid\gamma)\) is a minimum of a jointly continuous function over a fixed compact set, and Berge’s maximum theorem gives continuity. \(K\) was arbitrary. \(\blacksquare\)
Proof of Lemma 13 (concavity; closed-form derivatives). Write \(\pi(\beta\mid\gamma,y)=e^{L(\beta,\gamma)}/Z(\gamma)\), \(L(\beta,\gamma)=\sum_j\ell_j(\beta_j)-\frac12\sum_j(\beta_j-H_j\gamma)^\top P_b(\beta_j-H_j\gamma)\).
Score identity. \(\nabla_\gamma L=\sum_jH_j^\top P_b\beta_j-P_{11}^{\mathrm{RE}}\gamma=P_{11}m(\beta)-\Lambda_\gamma\mu_0-P_{11}^{\mathrm{RE}}\gamma\); since \(\nabla_\gamma\log Z=E_{\beta\mid\gamma}[\nabla_\gamma L]\), subtracting gives \(s(\beta,\gamma):=\nabla_\gamma\log\pi(\beta\mid\gamma,y)=P_{11}(m(\beta)-M(\gamma))\).
Bridge representation. Differentiating \(q(\gamma'\mid\gamma)=\int\phi_q(\gamma';m(\beta),\Sigma^\star)\pi(\beta\mid\gamma,y)\,d\beta\) under the integral and dividing by \(q\), \(\nabla_\gamma\log q(\gamma'\mid\gamma)=E[s(\beta,\gamma)\mid\gamma,\gamma',y]=P_{11}(\bar m(\gamma,\gamma')-M(\gamma))\), the expectation under the bridge law \(\pi(\beta\mid\gamma,\gamma',y)\propto\phi_q(\gamma';m(\beta),\Sigma^\star)\pi(\beta\mid\gamma,y)\), \(\bar m:=E[m(\beta)\mid\gamma,\gamma',y]\). Likewise \(\nabla_{\gamma'}\log q=P_{11}(\bar m-\gamma')\) and \(\nabla_{\gamma'}\log q_Q=-P_{11}(\gamma'-\gamma^\star)\), so \(\nabla_{\gamma'}g=P_{11}(\bar m(\gamma,\gamma')-\gamma^\star)\): the minimiser is characterised by \(\bar m(\gamma,\gamma'_\star(\gamma))=\gamma^\star\).
Envelope theorem. The minimiser is unique (Lemma 10), interior, and locally bounded (Lemma 12), and \(\nabla_\gamma g\) is continuous, so Danskin’s theorem gives \(\mathcal D\in C^1\) with \(\nabla\mathcal D(\gamma)=\nabla_\gamma g(\gamma'\mid\gamma)|_{\gamma'=\gamma'_\star(\gamma)}\) — the term \(-\log q_Q(\gamma')\) does not involve \(\gamma\), and the chain-rule term through \(\gamma'_\star(\gamma)\) vanishes since \(\nabla_{\gamma'}g=0\) there. By the bridge representation at \(\gamma'=\gamma'_\star(\gamma)\), where \(\bar m=\gamma^\star\): \(\nabla\mathcal D(\gamma)=P_{11}(\gamma^\star-M(\gamma))\).
Second derivative. Differentiating again, only \(M\) depends on \(\gamma\): \(\nabla^2\mathcal D=-P_{11}\,\partial M/\partial\gamma=-P_{11}J(\gamma)=-\sum_jH_j^\top P_bV_jP_bH_j\preceq0\) by Lemma 3(a). \(\blacksquare\)
Item (1): concavity, strict under (H3a) with \(V_j\succ0\), gives convex superlevel sets. Item (2): \(\nabla\mathcal D=0\iff M(\gamma)=\gamma^\star\iff\gamma=\gamma^\star\) by uniqueness of the fixed point (Lemma 3). Item (3): comparing \(\nabla\mathcal D=P_{11}(\gamma^\star-M(\gamma))\) with \(\nabla\Phi=P_{11}(\gamma-M(\gamma))\) gives \(\nabla(\Phi-\mathcal D)=P_{11}(\gamma-\gamma^\star)\), i.e. \(\mathcal D(\gamma)-\mathcal D(\gamma^\star)=\bar\Phi(\gamma)-\frac12\|\gamma-\gamma^\star\|^2_{P_{11}}\) exactly, integrating from \(\gamma^\star\) where both sides vanish.
Proof of Lemma 14 (coercivity; compactness). By Lemma 13, \(\nabla\Psi(\gamma^\star)=0\), \(\nabla^2\Psi=P_{11}J=S(\gamma):=\sum_jH_j^\top P_bV_jP_bH_j\succeq0\).
\(S(\gamma)\succ0\) everywhere. Under (H1)+(H2), \(\pi(\beta_j\mid\gamma,y)\) is a proper log-concave density, strictly positive on all of \(\mathbb R^{p_{\mathrm{re}}}\), hence not supported in any hyperplane, with finite second moments, so \(V_j(\gamma)\succ0\). For \(u\ne0\), (H3a) gives \(u^\top P_{11}^{\mathrm{RE}}u=\sum_j|H_ju|^2_{P_b}>0\), so \(H_ju\ne0\) for some \(j\); with \(v:=P_bH_ju\ne0\), \(u^\top S(\gamma)u\ge v^\top V_j(\gamma)v>0\).
Linear growth along every ray. Fix unit \(u\), \(\psi_u(s):=\Psi(\gamma^\star+su)\). Then \(\psi_u'(s)=\int_0^su^\top S(\gamma^\star+tu)u\,dt\) is nondecreasing in \(s\) (nonnegative integrand), so for \(s\ge1\), \(\psi_u'(s)\ge\psi_u'(1)\ge c\), giving \(\psi_u(s)=\int_0^s\psi_u'\ge c(s-1)\), where \(c:=\min_{|u|=1}\int_0^1u^\top S(\gamma^\star+tu)u\,dt\).
\(c>0\). \(V_j\) is the Hessian of a log-partition function, hence smooth in \(\gamma\), so \((u,t)\mapsto u^\top S(\gamma^\star+tu)u\) is continuous and, by the first part, strictly positive on the compact \(\{|u|=1\}\times[0,1]\); its \(t\)-integral is a continuous strictly positive function of \(u\), whose minimum over the compact unit sphere is attained and positive.
Combining, \(\Psi(\gamma)\ge c(|\gamma-\gamma^\star|-1)\) for \(|\gamma-\gamma^\star|\ge1\), so \(\widetilde C_d\subseteq\{|\gamma-\gamma^\star|\le1+d/c\}\): closed (Lemma 12), convex (Lemma 13), now bounded, hence compact.
Ellipsoid sandwich. Brascamp–Lieb gives \(V_j\preceq P_b^{-1}\), so \(S\preceq P_{11}^{\mathrm{RE}}\) and \(\{\|\gamma-\gamma^\star\|^2_{P_{11}^{\mathrm{RE}}}\le2d\}\subseteq\widetilde C_d\). If additionally \(-\nabla^2\ell_j\preceq G_j\) (automatic when GLM weights are bounded above: binomial with any link gives \(G_j=\frac14Z_j^\top N_jZ_j\)), Cramér–Rao for the location family \(\pi(\beta_j\mid\gamma,y)\) gives \(V_j\succeq(P_b+G_j)^{-1}\) (its negative-log-density Hessian is \(\preceq G_j+P_b\), so its Fisher information for location satisfies \(E[\nabla^2U]\preceq G_j+P_b\)); with \(S_\flat:=\sum_jH_j^\top P_b(P_b+G_j)^{-1}P_bH_j\succ0\), integrating the resulting constant Hessian bounds twice from \(\gamma^\star\) (where \(\Psi,\nabla\Psi\) vanish) gives \(\frac12\|\gamma-\gamma^\star\|^2_{S_\flat}\le\Psi(\gamma)\le\frac12\|\gamma-\gamma^\star\|^2_{P_{11}^{\mathrm{RE}}}\), i.e. the two-sided sandwich. \(\blacksquare\)
Proof of Corollary 15 (mass bounds). By Lemma 13(3), \(\bar\Phi\ge0\) (since \(\gamma^\star\) minimises \(\Phi\)) gives \(\Psi(\gamma)\le\frac12\|\gamma-\gamma^\star\|^2_{P_{11}}\), so the \(Q\)-ellipsoid of squared radius \(2d\) is inscribed in \(\widetilde C_d\); since \(\|\gamma-\gamma^\star\|^2_{P_{11}}\sim\chi^2_q\) under \(Q\), \(Q(\widetilde C_d)\ge\Pr(\chi^2_q\le2d)\), and identically for \(\pi_\gamma(\widetilde C_d^{\,c})\) using the same statistic under \(\pi_\gamma\). \(\blacksquare\)
Proof of Lemma 16 (attained minimum). \(\ge\varepsilon_d\) holds because every point of \(\widetilde C_d\) satisfies \(\varepsilon\ge\varepsilon_d\) by definition. For \(\le\): fix unit \(u\); \(s\mapsto\Psi(\gamma^\star+su)\) is continuous, \(0\) at \(s=0\), \(\to\infty\) as \(s\to\infty\) (Lemma 14), so by the intermediate value theorem there is \(s_d\) with \(\Psi(\gamma^\star+s_du)=d\) exactly — that point lies in \(\widetilde C_d\) and has \(\varepsilon=\varepsilon_d\) there, so the minimum is \(\le\varepsilon_d\). Existence and attainment of the minimum itself follow from Weierstrass: \(\varepsilon\) is continuous (Lemma 12) and \(\widetilde C_d\) compact (Lemma 14). \(\blacksquare\)
Proof of Lemma 17 (minorization on \(\widetilde C_d\)).
(a) By definition of the infimum, \(q(\gamma'\mid\gamma)\ge\varepsilon(\gamma)q_Q(\gamma')\) for every pair, and on \(\widetilde C_d\), \(\varepsilon(\gamma)\ge\varepsilon_d\) (Lemma 16); the bound holds for all \(\gamma'\in\mathbb R^q\) (nothing truncated on that side), so it integrates over any Borel \(A\).
(b) Write the restricted step as: draw \(\beta\sim\pi(\beta\mid\gamma,y)\) exactly, then \(\gamma'\sim N(m(\beta),\Sigma^\star)\) truncated to \(\widetilde C_d\). Its density is \[ q(\gamma'\mid\gamma;\widetilde C_d)=\mathbf 1\{\gamma'\in\widetilde C_d\}\int\frac{\phi_q(\gamma';m(\beta),\Sigma^\star)}{N(\beta)}\pi(\beta\mid\gamma,y)\,d\beta, \qquad N(\beta):=\int_{\widetilde C_d}\phi_q(\gamma';m(\beta),\Sigma^\star)\,d\gamma'\le1, \] so the integrand is pointwise at least \(\phi_q(\gamma';m(\beta),\Sigma^\star)\), whence \(q(\gamma'\mid\gamma;\widetilde C_d)\ge q(\gamma'\mid\gamma)\ge\varepsilon_dq_Q(\gamma')\) for \(\gamma,\gamma'\in\widetilde C_d\) — truncation only raises the density on the retained region. Integrating over \(A\subseteq\widetilde C_d\) and using \(Q(A)=Q(\widetilde C_d)Q_{\widetilde C_d}(A)\) gives \(q(\gamma,A\mid\widetilde C_d)\ge\varepsilon_dQ(\widetilde C_d)Q_{\widetilde C_d}(A)=\varepsilon\,Q_{\widetilde C_d}(A)\). Finally \(Q(\widetilde C_d)>0\): the inscribed ellipsoid \(\{(\gamma-\gamma^\star)^\top P_{11}(\gamma-\gamma^\star)\le2d\}\subseteq\widetilde C_d\) (Corollary 15) carries \(Q\)-mass \(\Pr(\chi^2_q\le2d)>0\). \(\blacksquare\)
Flat-prior limit (§5.4). By construction, \(\nabla^2\Psi_\tau=\sum_jH_j^\top P_bV_j(\gamma)P_bH_j\) does not involve \(\Lambda_\gamma\) at all — the mixing law \(\pi(\beta\mid\gamma,y)\) is \(\tau\)-free — so every result of §5.1–§5.3 whose proof consumes only \(\Lambda_Q\succ0\) (supplied at \(\tau=0\) by Lemma 7 under (H3a)) and properties of \(\pi(\beta\mid\gamma,y)\) transfers to \(\tau=0\) verbatim. This is what licenses Lemmas 18–21 below as the \(\tau=0\) instances of Lemmas 10–17 rather than independent arguments.
Proof of Lemma 18. Lemmas 10–12 read at \(\tau=0\): legitimate for the reason above. For the convergence claim, \(q_{Q_\tau}\to q_{Q_0}\) uniformly on compacts (explicit Gaussians with \(\gamma^\star(\tau)\to\gamma^{\star(0)}\), \(P_{11}(\tau)\to P_{11}^{\mathrm{RE}}\succ0\), Lemmas 7–8), and \(q_\tau(\gamma'\mid\gamma)\to q_0(\gamma'\mid\gamma)\) locally uniformly by dominated convergence, so \(g(\cdot\mid\cdot;\tau)\to g(\cdot\mid\cdot;0)\) uniformly on compacts. The uniform coercivity estimate underlying Lemma 12 holds uniformly in \(\tau\in[0,\tau_0]\) as well — its ingredients (\(\sigma_{\min}(L_\tau)\ge\sigma_0>0\), boundedness of \(a_\tau-\gamma^\star(\tau)\), and the \(\tau\)-free \(\pi(B_v\mid\gamma,y)\)) are all \(\tau\)-uniform for small \(\tau\) — so all minimisers lie in one \(\tau\)-free ball for \(\gamma\in K\), giving \(|\log\varepsilon(\gamma\mid\tau)-\log\varepsilon(\gamma\mid0)|\le\sup_{K\times B_R}|g(\cdot;\tau)-g(\cdot;0)|\to0\). \(\blacksquare\)
Proof of Lemma 19. Closedness is continuity of \(\varepsilon(\cdot\mid0)\) (Lemma 18). Convexity and the compactness sandwich are Lemmas 13–14 read at \(\tau=0\): since \(\nabla^2\Psi_\tau\) is \(\tau\)-free, the entire two-sided sandwich \(S_\flat\preceq\nabla^2\Psi_\tau\preceq P_{11}^{\mathrm{RE}}\) holds verbatim and uniformly in \(\tau\in[0,\tau_0]\), and since \(\gamma^\star(\tau)\to\gamma^{\star(0)}\) (Lemma 8), the ellipsoids all sit inside one compact \(K_d\). Nonemptiness with \(\gamma^{\star(0)}\) interior is Lemma 8 — the one place (H3b) enters. Exhaustion follows from pointwise positivity (Lemma 18). For Hausdorff convergence, \(\sup_{K_d}|\Psi_\tau-\Psi_0|\to0\) (from Lemma 18 and \(\gamma^\star(\tau)\to\gamma^{\star(0)}\)), so \(\{\Psi_0\le d-\epsilon_\tau\}\subseteq\widetilde C_d(\tau)\subseteq\{\Psi_0\le d+\epsilon_\tau\}\) with \(\epsilon_\tau\to0\); since \(\Psi_0\) is convex with \(\Psi_0(\gamma^{\star(0)})=0<d\), \(\{\Psi_0\le d\}=\overline{\{\Psi_0<d\}}\) (segment points toward the interior minimum stay strictly below \(d\) and converge to any boundary point), so both bracketing families converge to \(\widetilde C_d(0)\). \(\blacksquare\)
Proof of Lemma 20. The minimum exists and is attained by Weierstrass (\(\varepsilon(\cdot\mid0)\) continuous, Lemma 18; \(\widetilde C_d(0)\) compact, Lemma 19) and equals \(\varepsilon_d(0)\) by the same two-sided argument as Lemma 16, read at \(\tau=0\): \(\ge\) by definition of the superlevel set, \(\le\) by coercivity of \(\Psi_0\) plus the intermediate value theorem along a ray from \(\gamma^{\star(0)}\). For convergence, take a compact \(K\supseteq\{\gamma^\star(\tau):\tau\le\tau_0\}\cup\{\gamma^{\star(0)}\}\) (exists since \(\gamma^\star(\tau)\to\gamma^{\star(0)}\), Lemma 8); on \(K\), \(\varepsilon(\cdot\mid\tau)\to\varepsilon(\cdot\mid0)\) uniformly (Lemma 18) and \(\varepsilon(\cdot\mid0)\) is continuous (Lemma 18), so \(\varepsilon(\gamma^\star(\tau)\mid\tau)\to\varepsilon(\gamma^{\star(0)}\mid0)\), i.e. \(\varepsilon_d(\tau)\to\varepsilon_d(0)\). \(\blacksquare\)
Proof of Lemma 21. Lemma 17(a)–(b) read at \(\tau=0\), using Lemma 20 for the constant and Corollary 9 for \(Q_0\) — the truncated-density step \(N(\beta)\le1\) is \(\tau\)-free throughout. The refresh-mass bound is Corollary 15 read at \(\tau=0\): \(\bar\Phi_0\ge0\) gives \(\Psi_0(\gamma)\le\frac12\|\gamma-\gamma^{\star(0)}\|^2_{P_{11}^{\mathrm{RE}}}\), so the corresponding ellipsoid is inscribed in \(\widetilde C_d(0)\) and carries \(Q_0\)-mass \(\Pr(\chi^2_q\le2d)\) since \(\Lambda_{Q_0}=P_{11}^{\mathrm{RE}}\). \(\blacksquare\)
Proof of Lemma 22 (Gaussian closure). Existence, uniqueness, attainment of the minimiser are Lemmas 10–11, which already apply; what remains is to evaluate it. Closure makes \(\Lambda_q=\Sigma^{-1}\) constant in \(\gamma'\). With \(a=M(\gamma)\), \(u=\gamma^\star-a\), \[ g(\gamma')=\tfrac12\log\det(\Sigma^\star\Sigma^{-1})+f(\gamma'), \qquad f(\gamma'):=\tfrac12(\gamma'-\gamma^\star)^\top\Lambda_Q(\gamma'-\gamma^\star)-\tfrac12(\gamma'-a)^\top\Lambda_q(\gamma'-a). \] Minimising \(f\), with \(\Delta:=\Lambda_Q-\Lambda_q\succ0\), the stationary point is \(\gamma'_\star=\Delta^{-1}(\Lambda_Q\gamma^\star-\Lambda_qa)\), so \(\gamma'_\star-\gamma^\star=\Delta^{-1}\Lambda_qu\), \(\gamma'_\star-a=\Delta^{-1}\Lambda_Qu\), whence \[ f(\gamma'_\star)=\tfrac12u^\top\bigl[\Lambda_q\Delta^{-1}\Lambda_Q\Delta^{-1}\Lambda_q-\Lambda_Q\Delta^{-1}\Lambda_q\Delta^{-1}\Lambda_Q\bigr]u=-\tfrac12u^\top\Lambda_q\Delta^{-1}\Lambda_Qu, \] expanding both triple products using \(\Lambda_Q=\Lambda_q+\Delta\). Since \(\Lambda_q(\Lambda_Q-\Lambda_q)^{-1}\Lambda_Q=(\Lambda_q^{-1}-\Lambda_Q^{-1})^{-1}=(\Sigma-\Sigma^\star)^{-1}\) (from \(\Lambda_q^{-1}-\Lambda_Q^{-1}=\Lambda_q^{-1}(\Lambda_Q-\Lambda_q)\Lambda_Q^{-1}\)), substituting gives the boxed \(\mathcal D\). Since \(\Sigma\succeq\Sigma^\star\), \(\det(\Sigma^\star\Sigma^{-1})\le1\), so \(\varepsilon_d\in(0,1]\). \(\blacksquare\)
Proof of Lemma 23 (exhaustion). \(\varepsilon(\gamma)>0\) for every \(\gamma\) (Lemma 11, or 18 at \(\tau=0\)), so \(\Psi(\gamma)<\infty\) for every \(\gamma\), and every \(\gamma\) lies in \(\widetilde C_d\) once \(d\ge\Psi(\gamma)\) — this gives monotonicity and \(\bigcup_d\widetilde C_d=\mathbb R^q\). Since \(\pi_\gamma\) is a probability measure (H1) and \(\widetilde C_d^{\,c}\) is nonincreasing with empty intersection, continuity from above gives \(\pi_\gamma(\widetilde C_d^{\,c})\to0\). The explicit envelope \(\pi_\gamma(\widetilde C_d^{\,c})\le\pi_\gamma(\|\gamma-\gamma^\star\|^2_{P_{11}}>2d)\) is the complementarity identity (\(\Psi\le\frac12\|\cdot-\gamma^\star\|^2_{P_{11}}\), i.e. \(\bar\Phi\ge0\)) of Lemma 13(3). \(\blacksquare\)
Proof of Lemma 24 (restricted kernel and stationarity). Both blocks of the restricted sweep are drawn from the exact conditional of \(\pi(\cdot\mid\tilde C)\) given the other block, and inside the relevant slice of \(\tilde C\) by construction — this gives \(q(\gamma,C\mid C)=1\) for \(\gamma\in C\) (slices are nonempty for \(x\in C\) since \(x\) itself lies in them), and each block update is therefore a genuine Gibbs update for the target \(\pi(\cdot\mid\tilde C)\), which leaves its own target invariant; a composition of invariant kernels is invariant. \(\blacksquare\) (This is why the kernel is defined as Gibbs for the truncated target rather than as “run the unrestricted sweep and repair excursions on exit” — the latter construction does not in general preserve \(\pi(\cdot\mid C)\).)
Derivation of the rate exponent \(\rho\) in Remark 6.1. Under (H2), \(\pi_\gamma\) is log-concave (it is the marginal of the jointly log-concave \(\pi(\gamma,\beta\mid y)\), and marginals of log-concave densities are log-concave). A log-concave density with covariance \(\Sigma_\pi\) satisfies, for any positive-definite metric \(A\), a sub-exponential tail bound for the quadratic form \(\|\gamma-\gamma^\star\|^2_A\): writing \(\rho:=\lambda_{\max}(\Sigma_\pi^{1/2}A\Sigma_\pi^{1/2})=\lambda_{\max}(\Sigma_\pi A)\), standard concentration for log-concave measures (Borell’s lemma applied to the linear functionals diagonalizing \(\Sigma_\pi^{1/2}A\Sigma_\pi^{1/2}\)) gives \(\Pr(\|\gamma-\gamma^\star\|^2_A>2d)\lesssim e^{-d/\rho}\) for \(d\) large. Taking \(A=P_{11}\) and combining with Lemma 23’s envelope, \(\pi_\gamma(\widetilde C_d^{\,c})\lesssim e^{-d/\rho}\); solving \(e^{-d/\rho}=\delta\) for \(d=\rho\log(1/\delta)\) and substituting into \(\varepsilon_d=e^{-d}\varepsilon(\gamma^\star)\) gives \(\varepsilon\gtrsim\varepsilon(\gamma^\star)\delta^\rho\), since \(\varepsilon=\varepsilon_dQ(\widetilde C_d)\) and \(Q(\widetilde C_d)\to1\). Under Gaussian closure, \(\Sigma_\pi=(P_{11}(I-J))^{-1}\) exactly, giving \(\rho=\lambda_{\max}((I-\tilde J)^{-1})=(1-\kappa)^{-1}\). (The multiplicative constant suppressed by \(\lesssim\) is standard but not tracked here; sharpening it to an explicit constant is a numerical, not qualitative, refinement and does not affect the exponent \(\rho\).)
inst/MINORIZATION_SYMMETRIC_NON_GAUSSIAN.md — symmetric
GLM minorization, §7inst/LOGIT_STATIC_TAIL_CERTIFICATION.md §4.5–§4.6, §7.1
— tail mass / Approach Ainst/MINORIZATION_GAUSSIAN_REFRESH.md — \(Q\), \(\varepsilon_d\)inst/PREFLIGHT_model_setup.md — full rank, estimability
(H3)