Grouped Geometric Pooled Posterior Framework

• The framework groups inferential objects and applies explicit geometric operators—such as prediction metrics and Hamming distances—to reallocate posterior mass meaningfully.
• It yields pooled posterior laws via diverse methods like pushforward maps, logarithmic pools, and kernel mixtures, improving model selection and predictive accuracy.
• It demonstrates robust applications in high-dimensional regression, experimental design, and Bayesian model averaging with strong theoretical guarantees.

“Grouped Geometric Pooled Posterior Framework” is best understood as an Editor’s term for a family of posterior constructions in which inferential objects are first organized into groups, then related through an explicit geometry, and finally combined into a pooled posterior law. In the literature supplied here, the grouped object may be coefficient blocks in high-dimensional regression, regions of support space in Bayesian model averaging, outer samples in Bayesian experimental design, studies in grouped GMM, or non-exchangeable data groups linked by a DAG. The geometry may be the prediction metric induced by a design matrix, Hamming or group-Hamming distance on discrete model space, the Fisher–Rao geometry of probability densities under the square-root map, or a logarithmic pool in likelihood space. The resulting pooled law may be a pushforward posterior, a mixture of region-restricted components, a geometric pool, a linear opinion pool, or a quasi-posterior obtained by exponentiating a summed objective (Pal et al., 2024, Yang et al., 20 Apr 2026, Li et al., 19 Jun 2026, Fairall et al., 30 Jun 2026).

1. Definitional scope

As an Editor’s term, the framework does not denote a single standardized model. It denotes a recurring structural pattern: begin with a collection of posterior or quasi-posterior objects indexed by groups, define a geometry or deterministic map that expresses similarity, sparsity, or comparability across those groups, and then produce a pooled inferential law that is simpler, more structured, or better aligned with the downstream task.

A useful way to distinguish the main variants is by the object being grouped and by the pooling operator:

Setting	Grouped object	Pooled posterior form
High-dimensional grouped regression	Coefficient groups	$\Pi_T = \Pi \circ T^{-1}$
Bayesian model averaging under redundancy	Support-space regions or kernels	$$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$
Bayesian experimental design	Outer-sample groups	$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$
Grouped GMM	Studies	$$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$

This family resemblance is substantive rather than merely terminological. In each case, the pooled object is not formed by naive averaging of point estimates. It is formed at the distributional level, with the geometry encoded either in an optimization problem, a distance on model space, a manifold structure on densities, or a hierarchical objective. This suggests that the common core of the framework is not the specific algebraic form of pooling, but the use of geometry to reorganize posterior mass into grouped structures that are inferentially meaningful.

2. Geometric operators and pooled-posterior forms

One canonical form is the pushforward posterior. In grouped regression, a dense Gaussian posterior on $\beta\in\mathbb R^p$ is transformed sample-wise by a deterministic projection map

$$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$

and the pooled law is the image measure $\Pi_T=\Pi\circ T^{-1}$. The geometry is induced by the prediction metric $\frac1n\|X\beta-Xu\|^2$, while the grouping is induced by block penalties such as Group LASSO, Group SCAD, or Adaptive Group LASSO (Pal et al., 2024).

A second canonical form is the density-ratio pooled reporting law on discrete support space. For a reference BMA posterior $\pi_0(\gamma\mid D)$, hard regions or soft kernels define restricted components

$$\pi_m(\gamma)=\frac{\pi_0(\gamma\mid D)w_m(\gamma)}{\alpha_m}, \qquad \alpha_m=E_{\pi_0}\{w_m(\Gamma)\},$$

and a pooled report is

$$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$0

Here the geometry lives directly on support space through Hamming balls, group-Hamming balls, interval kernels, or posterior-cluster regions, while the density ratio gives exact total-variation and Kullback–Leibler distortion formulas (Li et al., 19 Jun 2026).

A third form is the logarithmic or geometric pool used as a shared proposal in Bayesian experimental design. Given outer observations $$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$1 and weights $$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$2, $$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$3, the geometric pooled posterior is

$$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$4

equivalently a normalized logarithmic pool of the individual posteriors $$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$5. Grouping partitions the outer samples into subsets $$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$6, producing group-specific pooled proposals

$$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$7

This is explicitly the grouped geometric pooled posterior of the BED paper (Yang et al., 20 Apr 2026).

A fourth form is the linear opinion pool. In likelihood-free inference, separate LFI posteriors built from different summaries, discrepancies, or algorithms are combined by

$$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$8

with a recentered variant designed to avoid variance inflation. This is grouped pooling, but not geometric pooling in the logarithmic sense. The distinction is explicit: the pooled density is a convex sum rather than a product of powered component densities (Frazier et al., 2022).

A fifth geometric substrate is provided by the Fisher–Rao manifold of densities. Under the square-root density representation $$\bar\pi_q^{\,W} = \sum_{m=0}^M q_m \pi_m = \pi_0 h_q^W$$9, the manifold of strictly positive densities becomes the positive orthant of the unit sphere in $$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$0, and the Fisher–Rao distance becomes

$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$1

This paper develops geodesics, exponential and inverse-exponential maps, and variational optimization on that sphere. It does not itself define grouped pooling, but it supplies a natural geometry in which grouped posteriors could be pooled by Riemannian barycenters. This suggests a density-manifold realization of the grouped geometric pooled posterior idea (Saha et al., 2017).

3. Sparse projection-posteriors for grouped regression

The most direct instantiation of the framework in continuous parameter space is the sparse projection-posterior for grouped regression. The underlying model is

$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$2

with the design matrix partitioned into disjoint groups $$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$3 and coefficients grouped as $$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$4. Sparsity is assumed at the group level, but the prior does not encode it. Instead, the base prior is the Gaussian “super-parameter” prior

$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$5

which yields the dense posterior

$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$6

The grouped geometric step is then to immerse each dense draw into a structured sparse space by solving a penalized prediction-matching problem (Pal et al., 2024).

Three projection maps are developed. The Group LASSO Projection Posterior uses

$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$7

The Group SCAD Projection Posterior applies the SCAD penalty to each group norm $$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$8, with nonconvex behavior and oracle-style shrinkage for large groups. The Adaptive Group LASSO Projection Posterior introduces weights

$$p_{\mathrm{GPP},k}(\theta\mid Y_k,d)\propto p(\theta)\prod_{i\in\mathcal I_k}p(y_i\mid\theta,d)^{\nu_i'}$$9

computed from a preliminary frequentist Group LASSO estimator, producing a convex but reweighted penalty

$$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$0

In all three cases, the induced posterior is the law of $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$1 under the dense base posterior.

The theory is unusually complete. For all three projection posteriors, the paper derives posterior contraction for groupwise $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$2-type error, $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$3 estimation error, and prediction error, with rates

$$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$4

In the ungrouped case $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$5 and $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$6, this recovers $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$7 and prediction error $$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$8. Model selection consistency is established for all three maps, with different assumptions: Group LASSO requires beta-min and irrepresentability, Group SCAD achieves an oracle property under its own beta-min regime, and Adaptive Group LASSO avoids irrepresentability.

A further geometric correction addresses post-selection bias. The Debiased Group LASSO Projection Map is

$$\Pi^{\mathrm{full}}_{n,\lambda}\propto \exp\{q_{2,n}+\lambda q_1\}\pi$$9

where $\beta\in\mathbb R^p$0 is built by groupwise nodewise regressions. Under the stated design and sparsity assumptions, the paper proves a Bernstein–von Mises type result for $\beta\in\mathbb R^p$1 and derives marginal credible intervals with asymptotically exact frequentist coverage.

The same grouped projection logic extends to nonparametric additive models by replacing $\beta\in\mathbb R^p$2 with a B-spline design matrix $\beta\in\mathbb R^p$3. With $\beta\in\mathbb R^p$4 and $\beta\in\mathbb R^p$5, the projected additive estimator satisfies

$\beta\in\mathbb R^p$6

and model selection consistency holds for the active additive components.

The computational design is also part of the framework. Base posterior sampling is direct because the unrestricted posterior is Gaussian. Each draw then requires only a standard group-penalized regression solve. The projection step is therefore sample-wise and embarrassingly parallel. Empirically, the projection posteriors mirror the behavior of their frequentist analogs in grouped linear and additive simulations, while the debiased projection achieves near-nominal $\beta\in\mathbb R^p$7 coverage. In the ADNI brain MRI application, GS and GS-P achieve the lowest test prediction errors, approximately $\beta\in\mathbb R^p$8–$\beta\in\mathbb R^p$9, whereas BGL is reported near $$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$0, and the selected anatomical regions include CA2, CA3, SUB, ERC, BA35, BA36, and PHC.

4. Support-space pooling and density-ratio compression

In discrete model space, the framework takes a different form. A support is a binary vector $$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$1, and Bayesian model averaging induces a posterior $$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$2 on individual supports. Under predictor redundancy, posterior mass can spread across many nearly interchangeable supports, making exact-support summaries unstable even when prediction is stable. The response of the density-ratio posterior compression paper is not to change the Bayesian target, but to report it through grouped regions of support space (Li et al., 19 Jun 2026).

For a hard region $$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$3, the retained mass is

$$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$4

and the restricted posterior is

$$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$5

For kernels $$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$6, one analogously defines kernel-restricted components and then forms a pooled law

$$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$7

This identity yields exact distortion formulas: $$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$8

$$\beta^*(\beta)=\arg\min_u\left\{\frac1n\|X\beta-Xu\|_2^2+\sum_{k=1}^K\mathcal P_{\lambda_n}(\|u_k\|)\right\},$$9

and, when $\Pi_T=\Pi\circ T^{-1}$0 almost surely,

$\Pi_T=\Pi\circ T^{-1}$1

For a single hard region $\Pi_T=\Pi\circ T^{-1}$2, the simplification is exact: $\Pi_T=\Pi\circ T^{-1}$3

The geometry on support space is explicit. The paper uses fixed hard regions, Hamming balls

$\Pi_T=\Pi\circ T^{-1}$4

group-Hamming balls based on group-activation maps $\Pi_T=\Pi\circ T^{-1}$5,

$\Pi_T=\Pi\circ T^{-1}$6

posterior-cluster regions, pooled-pruned region dictionaries, capacity kernels, and interval kernels for ordered predictors. The grouped geometric pooled posterior here is therefore a pool over geometrically defined regions in a discrete space, rather than over continuous densities.

The paper’s redundancy theorem makes the rationale precise. If posterior mass lies on a combinatorially large redundancy class

$\Pi_T=\Pi\circ T^{-1}$7

then exact-support truncations may require a number of atoms proportional to $\Pi_T=\Pi\circ T^{-1}$8 to achieve small total-variation error, whereas a single region report can have retained mass $\Pi_T=\Pi\circ T^{-1}$9 and zero distortion. Under bounded-nonuniform redundancy, one hard region plus a small fallback weight already yields explicit TV and KL bounds.

A defining feature of this variant is diagnostic transparency. Retained mass $\frac1n\|X\beta-Xu\|^2$0, fallback weight $\frac1n\|X\beta-Xu\|^2$1, density-ratio distortions, and mass-floor discrepancy are all part of the formal object being reported. This also marks a conceptual boundary: the compressed law is a reporting distribution, not a new Bayesian posterior in the modeling sense.

5. Hierarchical, modular, and graph-structured variants

A grouped geometric pooled posterior can also arise from explicit hierarchy. In grouped GMM, the Quasi-Bayesian Hierarchical Model places a pooling term on economically comparable parameters while leaving each group-specific objective function intact. If group $\frac1n\|X\beta-Xu\|^2$2 contributes the GMM criterion

$\frac1n\|X\beta-Xu\|^2$3

and the hierarchy contributes $\frac1n\|X\beta-Xu\|^2$4, then the full quasi-posterior is

$\frac1n\|X\beta-Xu\|^2$5

This is a grouped product-of-experts construction: the lower-level quasi-likelihood factors and the upper-level pooling factor are combined multiplicatively. When the number of studies is fixed, the quasi-posterior mean has the same asymptotic distribution as GMM for strongly identified study parameters. For weakly identified studies, the paper develops a weak-GMM limit experiment in which the upper-level pooling relation induces a family of priors over weak values, and the weak-limit QBHM rule is a Bayes rule under squared loss for the hierarchy-induced weak-limit prior. Pooling can reduce pointwise asymptotic mean squared error when the bias–variance tradeoff is favorable, and the paper works this out for Gaussian likelihood, nonlinear weak-GMM, and weak-IV examples (Fairall et al., 30 Jun 2026).

For non-exchangeable grouped data, the Graphical Dirichlet Process supplies a different hierarchy. If $\frac1n\|X\beta-Xu\|^2$6 is a DAG over groups and $\frac1n\|X\beta-Xu\|^2$7 is the parent set of node $\frac1n\|X\beta-Xu\|^2$8, then a child group-specific random measure is

$\frac1n\|X\beta-Xu\|^2$9

The child base measure is therefore a weighted average of parent measures, with parent weights given a Dirichlet prior and concentrations following a gamma-DAG. The resulting joint stochastic process respects the Markov property of the DAG. Hypergraph, stick-breaking, restaurant-type, and finite-mixture-limit representations all encode the same principle: group-specific clustering distributions borrow strength through a graph-structured pooling of parent random measures. In the colorectal cancer grouped scRNA-seq example, GDP achieves CHI $\pi_0(\gamma\mid D)$0, DBI $\pi_0(\gamma\mid D)$1, and SI $\pi_0(\gamma\mid D)$2, whereas HDP yields CHI $\pi_0(\gamma\mid D)$3, DBI $\pi_0(\gamma\mid D)$4, and SI $\pi_0(\gamma\mid D)$5 (Chakrabarti et al., 2023).

Likelihood-free inference introduces a modular rather than hierarchical variant. Different summary sets, discrepancies, or algorithms define distinct module-level posteriors, and the pooled posterior is a linear opinion pool. The recentered version preserves the pooled posterior mean while yielding covariance

$\pi_0(\gamma\mid D)$6

thereby avoiding part of the variance inflation of the naive mixture. The asymptotic theory is about the pooled posterior mean: under compatibility and posterior normality assumptions, the optimal pooling weight minimizes asymptotic frequentist risk, and the pooled mean is asymptotically at least as good as the better of the two component LFI posteriors. If one summary set is incompatible and the other is compatible, the optimal weight tends to zero on the incompatible component (Frazier et al., 2022).

These hierarchical and modular cases clarify that “grouped geometric pooled posterior” does not require a single universal operator. The grouped structure may be encoded through an upper-level quasi-posterior, a DAG of dependent random measures, or a module combination rule. What remains common is that cross-group information is introduced at the distributional level rather than through post hoc averaging of estimators.

6. Theoretical regimes, computation, applications, and limitations

Across the supplied literature, the main theoretical questions are contraction, model selection, distortion control, risk, and robustness to heterogeneity. In grouped regression, projection posteriors achieve optimal contraction rates, model selection consistency, and—after debiasing—asymptotically exact marginal credible-set coverage. In support-space compression, density-ratio identities yield exact TV and KL formulas, bounded functional error control, and a validation-split theorem for fitted region reports. In grouped GMM, strong components retain standard root-$\pi_0(\gamma\mid D)$7 behavior, while weak components converge to Bayes rules in a weak-limit experiment. In LFI, the pooled posterior mean enjoys asymptotic risk dominance under explicit covariance conditions. In grouped BED, the main claim is variance reduction via improved proposal–target overlap rather than exact posterior optimality (Pal et al., 2024, Li et al., 19 Jun 2026, Fairall et al., 30 Jun 2026, Frazier et al., 2022, Yang et al., 20 Apr 2026).

Computation is similarly heterogeneous but structurally aligned with the grouped-pooling idea. Sparse projection-posteriors require direct Gaussian sampling followed by one standard penalized regression solve per draw, so the map is sample-wise and embarrassingly parallel. Density-ratio compression constructs a kernel dictionary, estimates retained masses, fits mixture weights under a distortion–cost criterion, and prunes low-weight components. GDP uses a Metropolis-within-blocked-Gibbs sampler, with SALTSampler for simplex-constrained latent weights. Grouped geometric pooling in BED is computationally distinctive: ensemble Kalman inversion generates samples for multiple group-specific pooled proposals without extra forward-model evaluation cost, because different pooled observation means can be processed with the same forecast ensemble and Kalman gain (Pal et al., 2024, Li et al., 19 Jun 2026, Chakrabarti et al., 2023, Yang et al., 20 Apr 2026).

The applications show why grouped pooling is attractive. In grouped additive regression with ADNI MRI data, projection posteriors identify brain regions associated with Alzheimer’s progression and remain competitive in prediction. In support-indexed BMA, region reports on group-Hamming balls preserve predictive behavior while drastically shortening summaries under redundancy. In grouped scRNA-seq, GDP captures biologically interpretable similarities across treatment groups that a globally exchangeable HDP does not. In Bayesian experimental design for PDE-governed discrepancy calibration, grouping problematic outer samples reduces gradient-estimation variance substantially: in one reported experiment, moving from no grouping to grouping at fixed large inner sample size reduces gradient standard deviation from about $\pi_0(\gamma\mid D)$8 or $\pi_0(\gamma\mid D)$9 to about $$\pi_m(\gamma)=\frac{\pi_0(\gamma\mid D)w_m(\gamma)}{\alpha_m}, \qquad \alpha_m=E_{\pi_0}\{w_m(\Gamma)\},$$0, while total PDE-solve counts remain far below per-outer-sample or diffusion-based baselines (Pal et al., 2024, Li et al., 19 Jun 2026, Chakrabarti et al., 2023, Yang et al., 20 Apr 2026).

Several misconceptions recur and are addressed explicitly by the papers. First, not every pooled law is a literal posterior under a generative model: the support-space compressed law is a reporting distribution, and QBHM is quasi-Bayesian because it exponentiates a GMM objective rather than a likelihood. Second, “geometry” is context-dependent: it may refer to the $$\pi_m(\gamma)=\frac{\pi_0(\gamma\mid D)w_m(\gamma)}{\alpha_m}, \qquad \alpha_m=E_{\pi_0}\{w_m(\Gamma)\},$$1-prediction metric, discrete support-space distances, the Fisher–Rao sphere, or logarithmic pooling in likelihood space. Third, pooling is not uniformly beneficial. Group SCAD requires nonconvex optimization; beta-min conditions are nontrivial in sparse regression; support compression assumes a reliable reference posterior; GDP assumes a known DAG; grouped BED relies on an ESS diagnostic and one-step EKI approximations; and LFI pooling theory is developed rigorously only for two groups and mainly for the posterior mean (Pal et al., 2024, Li et al., 19 Jun 2026, Chakrabarti et al., 2023, Yang et al., 20 Apr 2026, Frazier et al., 2022).

Taken together, these works support a general description. A grouped geometric pooled posterior framework is a distributional mechanism for reallocating posterior mass across grouped structures by means of an explicit geometry. Depending on the application, the pooling operator may be a pushforward map, a logarithmic pool, a kernel mixture, a linear opinion pool, or a hierarchical quasi-posterior. The unifying principle is that grouping and geometry are not auxiliary summaries added after inference; they are built into the posterior object itself.