Quasi-Local Probabilistic Averaging

• Quasi-Local Probabilistic Averaging is a framework that constructs averages from local components using probabilistic weights, ensuring global coherence.
• It bridges diverse methods such as finite quasi-probability theory, manifold denoising, and covariate-adaptive model aggregation by linking locality to probabilistic consistency.
• The approach underpins practical applications including statistical denoising, Bayesian model averaging, and compact-kernel regularization for robust error control.

Quasi-local probabilistic averaging denotes a family of averaging constructions in which averaging is performed over local or relative components—such as sub-universes, neighborhoods, paths, local statistics, or compactly supported kernels—while the weights, consistency conditions, or guarantees are probabilistic. In current arXiv usage, the expression is applied in several technically distinct settings: finite quasi-probability theory, high-noise manifold denoising, stochastic spin systems, covariate-adaptive model averaging, posterior decomposition over program paths, random average sampling, and cutoff regularization in Euclidean field theory. This suggests that the term functions less as a single standardized formalism than as a recurring structural pattern linking locality to probabilistic coherence, concentration, or weighting (Surace, 12 Feb 2026, Shen et al., 23 Jun 2025, Reddy et al., 2017, Gao et al., 13 May 2026, Ivanov et al., 30 Mar 2026).

1. Conceptual range and recurrent structure

Across the literature, “quasi-local” usually indicates locality relative to a larger ambient object rather than absolute or intrinsic locality. The local domain may be a principal ideal in a Boolean lattice, an extrinsic ball in ambient Euclidean space, a stabilization neighborhood on a graph, a covariate-dependent region in model space, a path-local posterior in a probabilistic program, or the support of a compact averaging kernel. “Probabilistic” then refers, depending on context, to additive valuations, simplex-constrained weights, high-probability concentration bounds, covariance summability, random sampling, or probability-kernel smoothing (Surace, 12 Feb 2026, Shen et al., 23 Jun 2025, Reddy et al., 2017, Gao et al., 13 May 2026, Garg et al., 2024, Ivanov et al., 30 Mar 2026).

Domain	Local object	Probabilistic mechanism
Finite valuation theory	Relative sub-universe $\mathcal L_t$	Quasi-/pre-probabilities and generalized conditionals
Noisy geometry	Extrinsic ball around a landmark	Gaussian concentration and finite-sample guarantees
Random fields and dynamics	Stabilization neighborhood or time window	CLTs, concentration, weighted limit theorems
Model aggregation	Covariate region, path, or model family	Simplex weights, stacking, PAC-Bayes, barycenters
Sampling and regularization	Compact window or kernel support	Random samples or probability-kernel convolution

A common ambiguity is to treat the phrase as if it always meant “local averaging with random noise.” The current literature is broader. In some works the core issue is representational coherence under restriction and embedding; in others it is statistical denoising, asymptotic fluctuation theory, or regularized aggregation over experts or distributions. The shared motif is that averaging is only locally defined in the primitive representation, but is nevertheless made globally coherent.

2. Finite quasi-probabilities, relativisation, and coherent local expectations

In the finite-valuation framework of “Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality,” a universe of discourse is a finite Boolean lattice

$(\mathcal L,\wedge,\vee,\neg,\bot,\top),$

with relative sub-universes realized either as Boolean sublattices or as principal ideals

$$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$

A universal valuation $V:\mathcal L\to\mathbb C$ is required to satisfy Local Deducibility and Universality, meaning that the value of a distinguished statement is deducible from the remaining values by a rule depending only on the number of atoms and the statement’s level, and restrictions of $V$ to sub-universes preserve admissibility (Surace, 12 Feb 2026).

The central representation theorem states that every admissible universal valuation can be reparametrized by a bijection $\phi:\mathbb C\to\mathbb C$ into a finitely additive representative $R=\phi\circ V$, called a pre-probability, satisfying

$$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$

This representative is unique up to composition with an additive automorphism $a$ solving Cauchy’s equation $a(x+y)=a(x)+a(y)$. Under holomorphic regularity, that regraduation freedom collapses to $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$0 with $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$1. In semantic dimension $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$2, canonical normalization fixes the remaining freedom by setting

$(\mathcal L,\wedge,\vee,\neg,\bot,\top),$3

when $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$4; if $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$5, the canonical representative is the invariant valuation $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$6.

Within this framework, classical finite probabilities are exactly the quasi-probabilities that are nonnegative on all statements and stable under relativisation. For $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$7 with $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$8, the relative quasi-probability is

$(\mathcal L,\wedge,\vee,\neg,\bot,\top),$9

and the conditional quasi-probability is $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$0. If $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$1, one must revert to a relative pre-probability. The same formalism yields generalized Bayes formulas in stable, mixed, and invariant cases.

This leads directly to quasi-local probabilistic averaging. For a relative sub-universe $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$2 and an observable $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$3 defined on the atoms of $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$4,

$$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$5

when $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$6. If $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$7 is a disjoint partition of $$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$8, then

$$\mathcal L_t:=\downarrow t=\{\, s\in\mathcal L : s\wedge t = s \,\}.$$9

The framework therefore interprets quasi-local averaging as additive local expectation over sub-universes that remains coherent under embeddings into larger ambient universes. One explicit purpose of the construction is to show that quasi-probabilities need not be treated merely as computational tools; when canonical normalization is available, they become uniquely determined additive representatives of universal valuations.

3. Geometric locality under noise, random sampling, and metric comparability

In high-noise manifold denoising, quasi-local probabilistic averaging refers to local means taken in extrinsic neighborhoods around a noisy anchor rather than intrinsic neighborhoods on the unknown manifold. The data model is

$V:\mathcal L\to\mathbb C$0

with $V:\mathcal L\to\mathbb C$1 a smooth compact $V:\mathcal L\to\mathbb C$2-dimensional manifold. The analyzed estimator is a two-round mini-batch scheme: first average points in an extrinsic ball around an initial anchor $V:\mathcal L\to\mathbb C$3, inject a small Gaussian perturbation, then average again inside a refined extrinsic ball around $V:\mathcal L\to\mathbb C$4. Under the stated inequalities linking $V:\mathcal L\to\mathbb C$5, batch sizes, and radii, the output $V:\mathcal L\to\mathbb C$6 satisfies

$V:\mathcal L\to\mathbb C$7

with probability at least $V:\mathcal L\to\mathbb C$8. The work is explicit that the procedure is quasi-local because the neighborhoods are extrinsic balls in $V:\mathcal L\to\mathbb C$9 tuned to the Gaussian scale, not oracle manifold geodesic neighborhoods (Shen et al., 23 Jun 2025).

A different geometric use appears in random average sampling over local quasi shift-invariant spaces on LCA groups. There the local object is a compact region $V$0, the averaging operation is convolution with a compactly supported window $V$1,

$V$2

and the samples $V$3 are i.i.d. draws in $V$4 with density $V$5. The resulting measurements $V$6 obey high-probability sampling inequalities and reconstruction formulas once the sample size is sufficiently large relative to the finite dimension $V$7 of the local space $V$8. Under the stated compatibility and stability conditions, one obtains

$V$9

on $\phi:\mathbb C\to\mathbb C$0 with very high probability (Garg et al., 2024).

A third measure-theoretic formulation uses local comparability. For a metric measure space $\phi:\mathbb C\to\mathbb C$1, local comparability requires that intersecting balls of the same radius have comparable measures. If the averaging operator is

$\phi:\mathbb C\to\mathbb C$2

then local comparability with constant $\phi:\mathbb C\to\mathbb C$3 implies $\phi:\mathbb C\to\mathbb C$4, and hence $\phi:\mathbb C\to\mathbb C$5 for $\phi:\mathbb C\to\mathbb C$6. In geometrically doubling spaces, local comparability also yields weak-type $\phi:\mathbb C\to\mathbb C$7 bounds for the centered maximal averaging operator. The Gaussian measure is a notable counterexample to any naive equivalence between local comparability and good averaging behavior: local comparability fails for every radius, yet $\phi:\mathbb C\to\mathbb C$8 remains uniformly bounded on $\phi:\mathbb C\to\mathbb C$9, with constants that grow exponentially in the dimension (Aldaz, 2016).

4. Dependent random fields, ergodic windows, and noisy interactive averaging

In spin models on Cayley graphs, quasi-local probabilistic averaging is formulated through exponentially quasi-local score functions. A score $R=\phi\circ V$0 has stabilization radius $R=\phi\circ V$1, and exponential quasi-locality means the tail of that radius decays as

$R=\phi\circ V$2

for some $R=\phi\circ V$3. For the linear statistic

$R=\phi\circ V$4

where $R=\phi\circ V$5, exponential clustering of the spin model and exponential quasi-locality of $R=\phi\circ V$6 imply variance asymptotics

$R=\phi\circ V$7

and, under a variance lower bound, the CLT

$R=\phi\circ V$8

Here quasi-locality is not about geometric averaging of raw values, but about averaging many weakly dependent local or stabilizing functionals over growing balls in the graph (Reddy et al., 2017).

On the discrete torus, the averaging process exhibits a different pair of quasi-local effects: concentration around the mean and fast local smoothness. If $R=\phi\circ V$9 denotes the random mass configuration and $$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$0 the heat-flow mean, then

$$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$1

The paper proves that

$$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$2

a concentration timescale strictly shorter than the diffusive mixing scale $$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$3. It also derives sharp Dirichlet-form bounds showing that local roughness decays at the same rate as the heat flow, up to an $$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$4 correction. This establishes a two-stage picture: early quasi-local smoothing and fluctuation suppression, followed by gradual global relaxation without cutoff (Sau, 2023).

Weighted Birkhoff averages provide a temporal analogue. With a compact-support $$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$5 weight $$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$6, the weighted average

$$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$7

suppresses initial and terminal orbit segments and emphasizes intermediate times. For quasi-periodic, almost periodic, and periodic systems, the paper establishes arbitrary polynomial and exponential convergence under the stated smoothness assumptions, and proves weighted strong laws and a weighted CLT with $$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$8 rate under log-concave unconditional symmetry (Tong et al., 6 May 2025).

Noisy pairwise gossip gives yet another form of quasi-local probabilistic averaging. When only one pair of agents updates in each round and each received value is perturbed by zero-mean Gaussian noise, the potential

$$R\!\left(\bigvee_j s_{i_j}\right)=\sum_j R(s_{i_j}),\qquad R(\bot)=0,\qquad R(\neg s)=R(\top)-R(s).$$9

contracts toward a $a$0 regime in $a$1 rounds, while

$a$2

eventually diverges because the running average performs a random walk. The precise drift identity

$a$3

quantifies the separation between local consensus around the current mean and long-term drift from the initial mean (Mallmann-Trenn et al., 2019).

5. Adaptive aggregation over models, paths, and probability distributions

In localized model averaging for pre-trained models, quasi-local probabilistic averaging means that the weights depend on the covariates. Given fixed candidate predictors $a$4, the ensemble is

$a$5

The paper parameterizes $a$6 through a softmax gating network, trains by empirical risk minimization under a general loss, and proves asymptotic optimality for both in-sample and out-of-sample risks together with consistency of the estimated weights under the stated smoothness, tail, and capacity conditions. A common ambiguity here is to read “probabilistic” as “Bayesian”; in this work it means simplex-valued, covariate-adaptive mixing weights, not posterior model probabilities (Gao et al., 13 May 2026).

Probabilistic programs with stochastic support use a more explicitly Bayesian decomposition. If $a$7 denotes the set of feasible execution paths, then the posterior can be written as

$a$8

where $a$9 is the path-local posterior on the support of path $a(x+y)=a(x)+a(y)$0. Predictive inference under the full posterior is therefore a Bayesian model average over path-local predictives. The paper argues that these default BMA weights can be unstable under misspecification and approximate inference, and replaces them, as a cheap post-processing step, by weights optimized through stacking or a PAC-Bayes objective. In this setting the local objects are the path-local posteriors and predictives, while the averaging remains probabilistic through convex combination over paths (Reichelt et al., 2023).

Transport-based aggregation introduces locality in Wasserstein geometry rather than parameter space. Given distributions $a(x+y)=a(x)+a(y)$1 and weights $a(x+y)=a(x)+a(y)$2, the Wasserstein barycenter minimizes $a(x+y)=a(x)+a(y)$3. In one dimension with $a(x+y)=a(x)+a(y)$4, the barycenter quantile is

$a(x+y)=a(x)+a(y)$5

The paper regularizes the outer weight-selection problem by elastic net penalties, showing sparsity and consistency by a $a(x+y)=a(x)+a(y)$6-convergence argument. The quasi-locality here is geometric: the aggregate remains in the Wasserstein neighborhood or geodesic hull of the input distributions rather than being formed by Euclidean parameter averaging (Androulakis et al., 15 Jul 2025).

A related model-space construction appears in scalable Bayesian model averaging through local information propagation. In LIPS, randomized forward-stepwise trajectories define a latent Markov process over models, local $a(x+y)=a(x)+a(y)$7-step lookahead approximates posterior transitions, and SMC importance weights recover global Bayesian model averaging quantities. This is quasi-local because proposal construction uses bounded-depth neighborhoods around the current partial model, while the averaging is global only after importance reweighting (Ma, 2014).

6. Compact-support kernels, cutoff regularization, and major distinctions

In cutoff regularization for Laplace fundamental solutions, quasi-local probabilistic averaging is realized by a compactly supported probability kernel. The averaging operator is

$a(x+y)=a(x)+a(y)$8

with $a(x+y)=a(x)+a(y)$9, $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$00, and $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$01. Writing $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$02, this is convolution with a probability density supported in $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$03, so the deformation is local up to scale $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$04. For the two-point function, the averaged fundamental solution is

$(\mathcal L,\wedge,\vee,\neg,\bot,\top),$05

The paper derives new spherical-double-average representations, exact formulas for the value at zero, and contact-term identities such as

$(\mathcal L,\wedge,\vee,\neg,\bot,\top),$06

For $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$07, $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$08 has the expected power divergence in $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$09; for $(\mathcal L,\wedge,\vee,\neg,\bot,\top),$10, the divergence is logarithmic. The construction is designed for renormalization in perturbative QFT while preserving manifest locality in position space (Ivanov et al., 30 Mar 2026).

Taken together, these works clarify several recurring misconceptions. Quasi-locality does not imply intrinsic locality: extrinsic Euclidean balls may be the operative neighborhoods in manifold denoising. It does not imply positivity: in finite valuation theory, quasi-local averages may be computed with negative or complex quasi-probabilities. It does not coincide with global regularity conditions such as doubling: local comparability can replace doubling in some metric-measure arguments, fail while useful averaging bounds persist, or be equivalent only under additional connectivity hypotheses. Nor does “probabilistic averaging” always mean posterior model averaging; it may equally denote additive valuation calculus, simplex gating, random sampling, concentration-based error analysis, or compactly supported probability-kernel smoothing (Shen et al., 23 Jun 2025, Surace, 12 Feb 2026, Aldaz, 2016).

The literature therefore supports a broad but technically precise characterization. Quasi-local probabilistic averaging is a mode of constructing averages from local constituents that are not, by themselves, globally canonical. Global validity is recovered by one of several mechanisms: additive coherence under restriction and embedding, concentration and covariance summability, convex weighting on the simplex, transport geometry, or compact-support kernel regularization. The specific mathematics varies sharply by domain, but the recurring objective is stable aggregation under locality constraints without discarding the ambient global structure.