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Fusion of Probability Density Functions

Günther Koliander, Yousef El-Laham, Petar M. Djurić, Franz Hlawatsch

arXiv:2202.11633v1eess.SPcs.ITmath.ITmath.PR

TL;DR

Pdf fusion has multiple proposed methods, and this paper offers a structured treatment for continuous variables. It organizes axiomatic, optimization-based, and Bayesian approaches and presents new rules, axioms, characterizations, and supra-Bayesian results.

  • Problem

    Pdf fusion has multiple proposed techniques, motivating a structured account of the field’s concepts and methods.

    Source: §I · p. 1

  • Method

    The paper organizes pdf fusion through axiomatic, optimization-based, and Bayesian approaches, combining a structured survey with new results.

    Source: §I · p. 1, §IV · p. 9, D. Supra-Bayesian Fusion for Agents… · p. 24

  • Results

    Contributions include a new pooling function, new axioms and characterization theorems, and finite-dimensional and linear-Gaussian supra-Bayesian results.

    Source: D. Supra-Bayesian Fusion for Agents… · p. 24

  • Takeaways & Limitations

    The work provides a rigorous, coherent treatment of probabilistic opinion pooling for continuous random vectors.

    Source: E. Contributions and Paper Organization · p. 4

    Practical implementations require finite-dimensional pdf representations, which are usually approximations that trade representation complexity against accuracy.

    Source: §X · p. 29

Abstract

from arXiv · show

Fusing probabilistic information is a fundamental task in signal and data processing with relevance to many fields of technology and science. In this work, we investigate the fusion of multiple probability density functions (pdfs) of a continuous random variable or vector. Although the case of continuous random variables and the problem of pdf fusion frequently arise in multisensor signal processing, statistical inference, and machine learning, a universally accepted method for pdf fusion does not exist. The diversity of approaches, perspectives, and solutions related to pdf fusion motivates a unified presentation of the theory and methodology of the field. We discuss three different approaches to fusing pdfs. In the axiomatic approach, the fusion rule is defined indirectly by a set of properties (axioms). In the optimization approach, it is the result of minimizing an objective function that involves an information-theoretic divergence or a distance measure. In the supra-Bayesian approach, the fusion center interprets the pdfs to be fused as random observations. Our work is partly a survey, reviewing in a structured and coherent fashion many of the concepts and methods that have been developed in the literature. In addition, we present new results for each of the three approaches. Our original contributions include new fusion rules, axioms, and axiomatic and optimization-based characterizations; a new formulation of supra-Bayesian fusion in terms of finite-dimensional parametrizations; and a study of supra-Bayesian fusion of posterior pdfs for linear Gaussian models.

I. INTRODUCTION

This section surveys principled probability-density fusion, covering axiomatic, optimization, and supra-Bayesian approaches as well as applications in target tracking, probabilistic machine learning, and forecasting. It also discusses pooling methods, application-specific challenges, and new fusion rules and results.

A. Motivation

B. Probabilistic Opinion Pooling

  • Probabilistic opinion pooling combines agents’ pdfs qk(θ) into an aggregate pdf q(θ), typically through a fusion center.Examples include linear pooling and log-linear pooling; the paper focuses on continuous random variables or vectors.Source: B. Probabilistic Opinion Pooling · p. 2

C. Relevance and Applications

  • Pooling local posterior pdfs requires only those pdfs, unlike calculating the global posterior, which generally requires likelihoods, priors, and dependence information.This makes pooling broadly applicable across differing inference methods, sensors, and local data; the paper generally assumes the fusion center has no additional data, except in its supra-Bayesian setting.Source: C. Relevance and Applications · pp. 2–3

D. Approaches to Probabilistic Opinion Pooling

  • The paper distinguishes axiomatic pooling, defined by required properties; optimization pooling, derived by minimizing or maximizing an objective; and supra-Bayesian pooling, which treats agent pdfs as random observations.Optimization criteria typically seek an aggregate close to all agent pdfs, while the supra-Bayesian approach uses a model for their dependence on θ.Source: D. Approaches to Probabilistic Opinion… · pp. 3–4

E. Contributions and Paper Organization

  • For pdfs, the paper provides a rigorous axiomatic treatment, a generalized multiplicative pooling function, two new axioms, and new axiomatic characterization theorems.The axioms are factorization preservation and generalized Bayesianity; the characterization results adapt some discrete-distribution results and establish others as new.Source: E. Contributions and Paper Organization · p. 4
  • The paper proves that weighted sums of α-divergences, Pearson χ2-divergences, and L2 distances yield weighted H¨older, harmonic, and arithmetic means, respectively.It also derives the solution for a general class of weighted symmetric distance functions.Source: E. Contributions and Paper Organization · p. 4
  • The supra-Bayesian contributions include finite-dimensional fusion of posterior pdfs using local statistics and explicit rules for linear Gaussian models.The framework gives a pooling function for conditionally independent observations and a general procedure for conditionally dependent observations.Source: E. Contributions and Paper Organization · p. 4

A. Target Tracking

  • In target tracking, sensors’ local state-posterior pdfs are typically fused by log-linear pooling or covariance intersection, decoupling fusion from filtering and supporting heterogeneous sensor networks.The approach accommodates different Bayesian filters and sensing modalities across sensor nodes.Source: A. Target Tracking · p. 5
  • In target tracking, log-linear pooling is more sensitive to missed detections, while linear pooling is more sensitive to clutter; both have also been extended to labeled random finite sets.The extensions track target identities as well as states, and some require label association.Source: A. Target Tracking · p. 6

B. Probabilistic Machine Learning

  • In machine-learning ensembles, probabilistic opinion pooling fuses algorithms’ predictive pdfs to combine classification, regression, or clustering results, including deep ensembles, neural network ensembles, and ensemble Gaussian processes.Unlike the multisensor and target-tracking settings discussed earlier, ensemble algorithms may use the same data.Source: B. Probabilistic Machine Learning · p. 6
  • Federated learning combines local models at a fusion center when privacy-sensitive data cannot be shared; agnostic federated learning aggregates sample representations of distributions trained on private data.In some settings, the fusion center also communicates updates back to the nodes.Source: B. Probabilistic Machine Learning · pp. 6–7
  • For big-data Bayesian inference, divide-and-conquer methods combine distributions learned from smaller data subsets; one approach multiplies MCMC-derived subposteriors to approximate the overall posterior.The subposteriors are converted from samples to continuous pdfs using kernel density estimates; the multiplication yields the overall posterior under a suitable conditional-independence assumption.Source: B. Probabilistic Machine Learning · p. 7
  • Probabilistic machine learning remains limited because many popular methods do not provide probabilistic results, though ongoing research is expected to ease this restriction.Source: B. Probabilistic Machine Learning · p. 7

C. Forecasting

  • Density forecasting predicts continuous variables directly with pdfs, unlike earlier probabilistic forecasting that often quantized them into discrete categories; combining density forecasts is a special case of pdf fusion.Forecast combinations can be evaluated as new observations arrive and often concern one-dimensional variables, enabling cdf-based analysis and properties such as calibration.Source: C. Forecasting · p. 7
  • Density-forecast combination research has focused largely on linear pooling, including weight optimization from training data, and Bayesian model averaging, while nonlinear variants include spread-adjusted and Beta-transformed pools,.Source: C. Forecasting · p. 7

III. PROBABILISTIC OPINION POOLING

This section presents probabilistic opinion pooling as a way to combine K agents’ pdfs into an aggregate, surveying pooling rules and axioms for assessing them. It highlights generalized multiplicative pooling and axioms concerning factorization and Bayesianity, including a generalized Bayesianity condition that links fusion with likelihood updating through a fused likelihood independent of the opinion profile.

1) Linear Pooling:

  • Linear pooling forms a weighted arithmetic average and is equivalent to model averaging when agent pdfs are model-conditional pdfs and weights are model probabilities.The presentation uses pdfs on R^dθ for accessibility, while noting that results extend to absolutely continuous probability measures with respect to a σ-finite non-atomic measure.Source: 1) Linear Pooling · p. 8

2) Generalized Linear Pooling:

  • Generalized linear pooling adds an arbitrary pdf q0 to the weighted arithmetic average, with nonnegative weights for validity across opinion profiles.q0 can represent the fusion center’s opinion or serve as regularization.Source: 2) Generalized Linear Pooling · p. 8

3) Log-linear Pooling: / 4) Generalized Log-linear Pooling:

5) H¨older Pooling: / 6) Inverse-linear Pooling:

  • Hölder pooling generalizes linear and log-linear pooling, while inverse-linear pooling is its weighted harmonic-average case.Linear pooling corresponds to α = 1, log-linear pooling to α → 0, and inverse-linear pooling to α = −1; for other cases, the normalizer may require restrictions on the opinion profiles.Source: 5) H¨older Pooling · p. 9, 6) Inverse-linear Pooling · p. 9

7) Multiplicative Pooling:

  • Multiplicative pooling uses a calibrating pdf and is the supra-Bayesian fusion rule for posterior pdfs formed from conditionally independent observations.In that setting, the calibrating pdf is the prior used by the agents; positive opinion profiles and bounded density ratios ensure a well-defined, nonzero normalizer.Source: 7) Multiplicative Pooling · p. 9

8) Generalized Multiplicative Pooling:

  • Generalized multiplicative pooling extends generalized log-linear and multiplicative pooling, allowing arbitrary real agent weights with a positive calibrating pdf.The paper states that this rule is correct for combining posterior pdfs within a supra-Bayesian linear Gaussian model.Source: 8) Generalized Multiplicative Pooling · p. 9

9) Dictatorship Pooling: / 10) Dogmatic Pooling:

  • Dictatorship pooling returns one fixed agent’s opinion, while dogmatic pooling returns a fixed pdf regardless of the opinions; both are generally unsuitable choices.Both are presented as valid pooling functions, though the paper cautions against their use in most applications.Source: 9) Dictatorship Pooling · p. 9, 10) Dogmatic Pooling · p. 9

A. Axioms

  • The axiomatic approach characterizes pooling rules through desirable properties, including likelihood, independence, and Bayesian updating; the paper proposes factorization preservation and generalized Bayesianity.Factorization preservation includes preserving independence between components of the random vector, while generalized Bayesianity extends individualized Bayesianity.Source: §IV · p. 9, A. Axioms · pp. 10–11
  • Generalized Bayesianity requires updating each opinion and then fusing to match fusing first and updating by a profile-independent fused likelihood.The fused likelihood may depend on the individual likelihood functions, but not on the opinion profile.Source: A. Axioms · p. 11
  • The Bayesian-framework axioms do not directly apply to supra-Bayesian pooling, which uses explicit likelihood functions.Therefore, supra-Bayesian pooling need not satisfy axioms concerning arbitrary likelihood functions.Source: A. Axioms · p. 11

B. Relations between Axioms and Pooling Functions

The section maps which axioms pooling functions satisfy and identifies axiom combinations that characterize several pooling rules. It also establishes implications among the axioms for pdf fusion.

Relations between Axioms and Pooling Functions

  • The SSFP characterizes linear pooling; the paper also proves equivalence with the WSFP plus ZPP and with the WSFP plus unanimity preservation for pdfs.The pdf-specific equivalence is proved here; prior results cited in the paper covered pmfs or arbitrary probability measures.Source: B. Relations between Axioms and… · p. 12
  • A pooling function is generalized linear pooling exactly when it satisfies the WSFP, and this rule also satisfies the weak likelihood principle.For pdfs, the generalized linear pooling weights cannot be negative; the paper adapts the proof to this setting.Source: B. Relations between Axioms and… · p. 12
  • Log-linear pooling is equivalent to likelihood principle plus external Bayesianity, or to unanimity preservation, weak likelihood principle, and external Bayesianity.It satisfies the listed axioms, and symmetry holds exactly when all weights are equal.Source: B. Relations between Axioms and… · p. 12
  • Generalized log-linear pooling is characterized by the weak likelihood principle and external Bayesianity, and satisfies factorization preservation and generalized Bayesianity.This characterization depends on fusing pdfs: for pmfs, the corresponding rule may have negative weights.Source: B. Relations between Axioms and… · p. 12
  • Hölder pooling satisfies unanimity preservation and both likelihood principles; inverse-linear pooling inherits these properties as a special case.Hölder pooling is symmetric exactly when its weights are equal.Source: B. Relations between Axioms and… · p. 12
  • Multiplicative pooling is characterized by individualized Bayesianity plus a calibrating pdf preserved when all agents submit it; generalized multiplicative pooling satisfies WSFP and generalized Bayesianity.The first rule also satisfies symmetry, weak likelihood, factorization preservation, and generalized Bayesianity; the generalized rule is symmetric exactly when its weights are equal.Source: B. Relations between Axioms and… · pp. 12–13
  • Dictatorship pooling is characterized by combinations of independence preservation or Bayesianity with strong or weak sure-thing principles; dogmatic pooling is characterized by symmetry, WSFP, and generalized Bayesianity.WSFP plus generalized Bayesianity narrows the possibilities to dogmatic or dictatorship pooling, and symmetry selects dogmatic pooling.Source: B. Relations between Axioms and… · p. 13
  • The implication results include SSFP implying ZPP, unanimity preservation, WSFP, and both likelihood principles; WSFP and likelihood principle each imply weak likelihood principle.The section also establishes independence preservation implies ZPP and individualized Bayesianity implies generalized Bayesianity; Fig. 4 depicts currently known implications, with some apparently nonempty regions possibly empty.Source: B. Relations between Axioms and… · pp. 13–14

V. THE OPTIMIZATION APPROACH

The optimization approach defines a fused pdf by minimizing a weighted average of discrepancies from the agents’ pdfs. Different divergence choices recover established pooling rules, while α-divergences yield a parameterized family of Hölder pooling functions.

V. THE OPTIMIZATION APPROACH

  • The optimization approach treats the aggregate pdf as an average of agent pdfs, chosen to minimize their weighted average discrepancy from it.The paper considers f-divergences and derives associated pooling functions for specific choices.Source: §V · p. 14

A. Kullback-Leibler Divergence

  • Minimizing weighted Kullback-Leibler divergences yields linear pooling, the weighted arithmetic average of the agent pdfs.The paper also provides a proof of this characterization.Source: A. Kullback-Leibler Divergence · pp. 14–15

B. Reverse Kullback-Leibler Divergence

  • Minimizing weighted reverse Kullback-Leibler divergences yields log-linear pooling, the weighted geometric average of the agent pdfs.The reverse KLD reverses the arguments of the KLD.Source: B. Reverse Kullback-Leibler Divergence · p. 15

C. α-Divergences

  • α-divergence minimization yields a family of Hölder pooling functions that includes inverse-linear, log-linear, and linear pooling as special cases.The paper supplies a proof of the result, which it says was not previously available in the literature to the authors’ knowledge.Source: C. α-Divergences · p. 15

D. Reverse α-Divergences

The section extends optimization-based pooling through reverse α-divergences and symmetric discrepancy measures, then describes Gaussian pooling and considerations for choosing pooling parameters. The results connect discrepancy choices to pooling rules and identify practical constraints on Gaussian Hölder pooling.

D. Reverse α-Divergences

  • Reversing the arguments of an α-divergence corresponds to a parameter change α* = 1 − α in the associated Hölder pooling result.For α = 2, the reverse α-divergence gives inverse-linear pooling.Source: D. Reverse α-Divergences · p. 16

E. Symmetric Discrepancy Measures

  • Minimizing weighted squared L2 distances yields linear pooling, while an unconstrained symmetric-distance formulation can recover linear, log-linear, inverse-linear, and Hölder pooling.For arbitrary distance functions, an analytical solution to the constrained problem may not exist because of the requirement that the aggregate be a valid pdf.Source: E. Symmetric Discrepancy Measures · pp. 16–17

A. Linear Pooling

  • Linear pooling of Gaussian agent pdfs produces a Gaussian mixture whose mean and covariance can be computed from the agents’ means and covariances.The mixture is generally non-Gaussian, so those moments do not fully characterize it; it can have heavy tails, multiple modes, or nonzero skewness.Source: A. Linear Pooling · p. 17

B. Log-linear Pooling

  • Log-linear pooling of Gaussian pdfs produces a Gaussian aggregate that is unimodal, symmetric about its mean, and fully characterized by its mean and covariance.The paper connects this case to covariance intersection, whose weights are typically chosen to minimize the fused covariance’s determinant or trace.Source: B. Log-linear Pooling · p. 17

C. Other Pooling Functions

  • For Gaussian inputs, the Hölder pooling normalization factor is generally intractable, making expectations difficult to compute in high dimensions.Numerical integration is needed, and the paper notes that such methods are affected by the curse of dimensionality.Source: C. Other Pooling Functions · p. 18
  • The parameter α controls Hölder pooling’s multimodality and tail behavior: smaller values attenuate modes and produce less heavy tails, while larger values enhance modes and produce heavier tails.The examples use two Gaussian pdfs with different means but equal variances, and with equal means but different variances.Source: C. Other Pooling Functions · p. 18, Figure 5 · p. 19

VII. CHOOSING THE POOLING PARAMETERS

  • Adaptive weights based on agent pdfs depart from the fixed-weight framework of the axiomatic and optimization approaches.With adaptively chosen weights, linear pooling is no longer linear in the agent pdfs.Source: §VII · p. 18
  • Weight choices can reflect model probabilities, predictive performance, or distances between agents, and some methods update weights using observed data.For linear pooling, model-probability weights can be posterior probabilities; predictive-performance weights require data and an assessment mechanism.Source: A. Linear Pooling · p. 18, §VII · p. 18

B. Log-linear Pooling

The section develops pooling choices through KLD-based weight optimization and the effect of the Hölder parameter. It then formulates supra-Bayesian fusion using finite-dimensional local statistics, including results for conditionally independent observations.

B. Log-linear Pooling

  • Minimizing the average KLD from agent pdfs to a log-linear aggregate yields a convex objective, so convex-optimization tools can compute the optimal weights.The minimum-KLD criterion is proposed when there is no basis for determining agent reliability.Source: B. Log-linear Pooling · p. 20

C. H¨older Pooling

  • For Hölder pooling, α controls the aggregate: positive α preserves the union of agent supports, while α = 0 gives a veto property and α < 0 requires shared support.Smaller α attenuates outlying agent contributions; larger α tends to increase uncertainty and promote multimodality or heavy tails.Source: C. H¨older Pooling · pp. 20–21

A. Agents Collecting Conditionally Independent Observations

  • With conditionally independent local observations, multiplicative pooling of local posteriors with a prior correction recovers the oracle posterior.The fusion center can obtain the same result from local likelihoods instead of local posteriors.Source: A. Agents Collecting Conditionally… · p. 21

VIII. THE SUPRA-BAYESIAN FRAMEWORK

  • Supra-Bayesian fusion treats local pdfs as observations and forms a posterior from their likelihood and a prior; finite-dimensional local statistics make this model tractable.Each local pdf is assumed to depend one-to-one on a finite-dimensional statistic, allowing the pdf likelihood to be represented as p(t | θ).Source: B. The Supra-Bayesian Framework and… · p. 22

B. The Supra-Bayesian Framework and Local Statistics

  • In the finite-dimensional model, the supra-Bayesian posterior depends on local pdfs through their uniquely corresponding statistics and the global likelihood.The global likelihood is p(t | θ), and the posterior is proportional to that likelihood times the prior.Source: B. The Supra-Bayesian Framework and… · pp. 22–23

C. Supra-Bayesian Fusion for Conditionally Independent Agents

  • For conditionally independent exponential-family observations, fusion sums the local sufficient statistics and normalization functions; a conjugate prior retains its form with updated hyperparameters.The posterior replaces t0 and a0 with t0 plus the summed local statistics and a0 + K, respectively.Source: C. Supra-Bayesian Fusion for… · pp. 23–24

D. Supra-Bayesian Fusion for Agents Collecting Dependent Observations

For dependent observations, supra-Bayesian fusion operates on the statistics communicated by agents, which may not retain all joint dependence information. In the linear Gaussian setting, the local posterior can be represented by a finite-dimensional statistic, and conditional independence determines whether fusion recovers the oracle posterior.

D. Supra-Bayesian Fusion for Agents Collecting Dependent Observations

  • For dependent observations, the fusion center transforms the joint observation model into a likelihood for communicated statistics, then computes the posterior; explicit pooling rules are derived only for special cases.The procedure identifies statistics representing local posteriors, transforms p(y | θ) into p(t | θ), and applies Bayes' rule.Source: D. Supra-Bayesian Fusion for Agents… · pp. 24–25

A. Local Statistics

  • In the linear Gaussian model, each local posterior corresponds one-to-one with a finite-dimensional statistic recoverable from that posterior given the prior and model matrices.The fusion center can recover the local likelihood from the posterior by dividing out the prior, then obtain the statistic.Source: A. Local Statistics · p. 25, §IX · p. 25
  • When observations are conditionally independent, the linear Gaussian supra-Bayesian result equals the oracle posterior; with dependent observations, that equality no longer generally holds.The communicated global statistic may fail to be sufficient for the full observation vector because it does not capture all dependencies among agents' observations.Source: D. Supra-Bayesian Fusion for Agents… · pp. 24–25, A. Local Statistics · p. 26

B. Global Likelihood Function

The linear Gaussian analysis constructs a global likelihood from local statistics and derives an explicit scalar fusion rule for dependent observations. Its weights can reflect shared information, including negative weights when shared observations would otherwise be counted repeatedly.

B. Global Likelihood Function

  • The global likelihood for the communicated statistics is obtained by transforming the joint Gaussian observation model; the derivation remains valid for positive semidefinite Σ even when a conditional pdf does not exist.The stated conditional pdf requires positive definite Σ, but the subsequent derivations do not require its existence.Source: B. Global Likelihood Function · p. 26

C. Supra-Bayesian Fusion Rule for a Scalar θ

  • For scalar θ, the dependent-observation fusion rule replaces Σ^-1 in the oracle posterior with V⊺(VΣV⊺)^-1V.The resulting generalized multiplicative-pooling weights need not sum to one and may be negative.Source: C. Supra-Bayesian Fusion Rule for a… · pp. 26–27
  • Agents with few private observations can receive negative weights, which can counteract multiple-counting of observations shared across agents.For K = 3, r1 = 1, and r0 = r2 = r3 = 4, the passage gives w1 = −1/7.Source: C. Supra-Bayesian Fusion Rule for a… · pp. 27–28
  • With equal private observations per agent, no shared observations gives multiplicative pooling, while increasing shared observations moves the rule toward symmetric log-linear pooling.If private-observation counts differ, the log-linear connection holds only when all weights are nonnegative.Source: C. Supra-Bayesian Fusion Rule for a… · p. 28

D. Supra-Bayesian Fusion Rule for a Vector θ

For vector-valued linear Gaussian models, the supra-Bayesian posterior remains Gaussian under a Gaussian prior, but differs from the oracle posterior. The outlook identifies finite-dimensional modeling and practical implementation as important boundaries and directions for pdf fusion.

D. Supra-Bayesian Fusion Rule for a Vector θ

  • With a Gaussian prior, vector supra-Bayesian fusion yields a Gaussian posterior, whose information weighting differs from the oracle posterior.The difference replaces Σ^-1 with V⊺(VΣV⊺)^-1V; the fusion rule first preprocesses agent pdfs and then applies generalized multiplicative pooling.Source: D. Supra-Bayesian Fusion Rule for a… · pp. 28–29

X. OUTLOOK

  • Practical use requires finite-dimensional pdf representations and algorithms, balancing approximation accuracy against representation complexity.The outlook also identifies communication cost and decentralized protocols, including avoiding double counting, as practical challenges.Source: §X · p. 29
  • The paper identifies multiobject pdf fusion as an active direction, including testing pooling functions beyond linear and log-linear pooling for multitarget tracking.It also proposes developing supra-Bayesian fusion for finite point processes.Source: §X · p. 29
  • Pdf fusion may combine summaries from partitioned or heterogeneous big-data processing and from decentralized federated-learning devices.In federated learning, devices send summaries rather than raw data, and pooling weights can reflect differing local-data quality.Source: §X · pp. 29–30
  • Explicit supra-Bayesian rules are derived for linear Gaussian models; accuracy for nonlinear or non-Gaussian models with known moments remains unclear.The finite-dimensional framework is also limited to pdfs admitting finite-dimensional parameterizations.Source: §X · p. 30

XI. CONCLUDING REMARKS

  • The axiomatic analysis finds that several desirable axioms are incompatible when imposed together, implying dictatorship pooling.The paper formulates axioms and identifies which considered pooling functions satisfy them.Source: §XI · p. 30
  • Minimizing weighted α-divergences yields Hӧlder mean pooling, which includes linear and log-linear pooling as special cases.Its single parameter determines differing multimodality and tail-decay characteristics.Source: §XI · p. 30
  • The supra-Bayesian framework treats agent pdfs as random observations and derives a global posterior through finite-dimensional local-statistic models.The conclusion also points to implementation, application, decentralized settings, and point processes as future directions.Source: §XI · p. 30

APPENDIX A PROOF OF THEOREM 1

The appendix establishes that linear pooling satisfies the discussed axioms and proves an equivalence characterization: weak sufficient function preservation (WSFP) with unanimity preservation characterizes linear pooling.

A. Axioms Satisfied by the Linear Pooling Function

  • The linear pooling function satisfies the axioms shown in the appendix, including the zero probability property, unanimity preservation, and strong sufficient function preservation.The proof also establishes the weak likelihood principle and symmetry statement for generalized linear pooling.Source: A. Axioms Satisfied by the Linear… · p. 31

B. Equivalence Statement

2) (iii) implies (ii):

  • For nontrivial events, the proof shows that the event-specific functions h_A coincide, yielding one function h that satisfies the required relation for all events.It extends the relation to trivial events using the zero probability property.Source: 2) (iii) implies (ii) · pp. 31–32

3) (ii) implies (i):

These appendix proofs derive linear pooling from strong sufficient function preservation and characterize generalized linear pooling through weak sufficient function preservation. They also establish that generalized linear pooling satisfies the stated axioms.

3) (ii) implies (i):

  • Strong sufficient function preservation makes the event-probability function additive, and its boundedness implies a linear form with weights summing to one.The resulting pooling function is linear.Source: 3) (ii) implies (i) · pp. 32–33

A. Axioms Satisfied by the Generalized Linear Pooling Function

  • The generalized linear pooling function satisfies WSFP and the weak likelihood principle, as well as the stated symmetry property.The proof defines an event-specific function h_A to establish WSFP.Source: A. Axioms Satisfied by the Generalized… · p. 33

B. Converse Statement

  • The converse proof constructs a pdf contribution q0 and weight w0, then subtracts that contribution to obtain a rule satisfying WSFP and the zero probability property.The resulting rule is linear, so the original pooling function is generalized linear pooling.Source: B. Converse Statement · pp. 33–35
  • When w0 = 1, the proof identifies the rule as dogmatic pooling, a special case of generalized linear pooling.For multiplicative pooling, iterative individualized Bayesianity yields the stated form under the bounded-ratio condition.Source: B. Converse Statement · p. 34, Appendix C · p. 35

1) (iii) implies (ii):

The proofs connect independence preservation and external Bayesianity to stronger pooling properties, and show that strong sufficient function preservation with generalized Bayesianity forces dictatorship pooling.

1) (iii) implies (ii):

  • Independence preservation implies the zero probability property: if every agent assigns an event probability zero, the fused probability must also be zero.The proof rules out fused probability one by finding a subset with probability one-half, contradicting independence preservation.Source: 1) (iii) implies (ii) · p. 35

2) (v) implies (iv):

  • WSFP implies the weak likelihood principle, which together with external Bayesianity yields generalized log-linear pooling for positive profiles.The proof combines this with the generalized linear form to derive the SSFP and external Bayesianity conclusion.Source: 2) (v) implies (iv) · pp. 35–36

3) (vi) implies (i):

  • Strong sufficient function preservation with generalized Bayesianity forces every agent weight to be either zero or one, hence dictatorship pooling.The proof obtains the constraint from a quadratic equation in each weight.Source: 3) (vi) implies (i) · pp. 36–37

APPENDIX E PROOF OF LEMMA 9

The proof restricts the weight w0 to 0 or 1, yielding linear or dogmatic pooling; in the linear case, the stated axioms further imply dictatorship pooling.

APPENDIX E PROOF OF LEMMA 9

  • The cubic equation has roots w0 = 0, 1, and 2, but w0 cannot exceed one, leaving only 0 and 1.These cases correspond respectively to linear and dogmatic pooling.Source: Appendix E · pp. 37–38
  • When w0 = 0, g is linear pooling and, because it also satisfies SSFP and generalized Bayesianity, reduces to dictatorship pooling.Source: Appendix E · p. 38
  • When w0 = 1, g is a dogmatic pooling function.Source: Appendix E · pp. 37–38

APPENDIX F PROOF OF THEOREM 11

The proof establishes that individualized Bayesianity implies generalized Bayesianity and verifies the constructed function h against condition (20).

APPENDIX F PROOF OF THEOREM 11

  • Individualized Bayesianity implies generalized Bayesianity, completing implication (v) of the theorem.The proof establishes this by defining a suitable function h.Source: Appendix F · p. 38
  • Iterative application of individualized Bayesianity establishes the relation used to define h.Source: Appendix F · p. 38
  • The h defined in (159) satisfies condition (20).Source: Appendix F · p. 38
  • For f(x) = fα(x), Jensen’s inequality supplies a lower bound on the objective, with equality when the specified function is constant almost everywhere.The proof concludes that the minimizing solution has the form q(θ) = c…, as stated in the passage.Source: Appendix F · p. 39

APPENDIX H CHARACTERIZATION OF THE REVERSE α-DIVERGENCE

The appendices characterize reverse α-divergence and derive minimizers for weighted squared-L2 and general-distance objectives.

APPENDIX H CHARACTERIZATION OF THE REVERSE α-DIVERGENCE

  • Dα(ϕ∥qk) equals Dα*(qk∥ϕ), where α* = 1 − α.The proof uses the dual generator f* and the invariance of an f-divergence to adding c(x − 1).Source: Appendix H · p. 39

THE WEIGHTED AVERAGE OF SQUARED L2 DISTANCES)

OF THE WEIGHTED AVERAGE OF GENERAL DISTANCES)

APPENDIX K PROOFS OF THE FUSION RULE FOR A SCALAR PARAMETER

The appendices derive scalar-parameter fusion rules and Gaussian posterior expressions, including a weight calculation based on matrix inversion.

A. Proof of Theorem 21

  • For the scalar-parameter linear Gaussian model, the local likelihoods yield a global likelihood proportional to λ(θ), from which the posterior fusion rule follows.Source: A. Proof of Theorem 21 · p. 40

B. Calculation of the Weights in Example 3

A. Proof of Theorem 23

B. Proof of Corollary 24

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