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An Information Aggregation Operator

Heyang Gong

TL;DR

This paper introduces InfoAgg, a stochastic aggregation operator that fuses probability distributions by forming $p(x) = \frac{1}{C_{norm}} p_1(x) p_2(x)$, enabling direct, prior-free information fusion. It proves that InfoAgg forms an Abelian Group over the space of population distributions, with commutativity, associativity, a uniform identity, and inverses, providing a robust and reversible framework for combining probabilistic information. Grounded in a personalized incentive scenario on a large-scale platform, the authors use abduction to obtain posterior $P(u|e)$ and combine prior information $P^c$ with $P(\cdot|U=u)$ to optimize incentive allocation, illustrating the approach's practical utility. The work further extends InfoAgg to handle sets and evidences, introduces prior-adjusted aggregation variants, and outlines directions for extending beyond finite populations and bridging discrete-continuous variable aggregation, positioning InfoAgg as a versatile toolkit for probabilistic reasoning in decision-making under uncertainty.

Abstract

This study explores a new mathematical operator, symbolized as $\cupplus$, for information aggregation, aimed at enhancing traditional methods by directly amalgamating probability distributions. This operator facilitates the combination of probability densities, contributing a nuanced approach to probabilistic analysis. We apply this operator to a personalized incentive scenario, illustrating its potential in a practical context. The paper's primary contribution lies in introducing this operator and elucidating its elegant mathematical properties. This exploratory work marks a step forward in the field of information fusion and probabilistic reasoning.

An Information Aggregation Operator

TL;DR

This paper introduces InfoAgg, a stochastic aggregation operator that fuses probability distributions by forming , enabling direct, prior-free information fusion. It proves that InfoAgg forms an Abelian Group over the space of population distributions, with commutativity, associativity, a uniform identity, and inverses, providing a robust and reversible framework for combining probabilistic information. Grounded in a personalized incentive scenario on a large-scale platform, the authors use abduction to obtain posterior and combine prior information with to optimize incentive allocation, illustrating the approach's practical utility. The work further extends InfoAgg to handle sets and evidences, introduces prior-adjusted aggregation variants, and outlines directions for extending beyond finite populations and bridging discrete-continuous variable aggregation, positioning InfoAgg as a versatile toolkit for probabilistic reasoning in decision-making under uncertainty.

Abstract

This study explores a new mathematical operator, symbolized as , for information aggregation, aimed at enhancing traditional methods by directly amalgamating probability distributions. This operator facilitates the combination of probability densities, contributing a nuanced approach to probabilistic analysis. We apply this operator to a personalized incentive scenario, illustrating its potential in a practical context. The paper's primary contribution lies in introducing this operator and elucidating its elegant mathematical properties. This exploratory work marks a step forward in the field of information fusion and probabilistic reasoning.
Paper Structure (10 sections, 3 theorems, 31 equations, 1 figure)

This paper contains 10 sections, 3 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Concise Introduction on Stochastic Aggregation
  3. Information Aggregation
  4. The Stochastic Aggregation Operation
  5. Tailored to a Personalized Decision-making Scenario
  6. The InfoAgg Abelian Group
  7. Extensions of InfoAgg
  8. Conclusion and Discussion
  9. Preliminaries on DiscoSCM
  10. Derivations in DPO

Key Result

Theorem 1.1

Let $H$ be a joint distribution function of random variables with marginal distribution functions $F_1, F_2, \ldots, F_n$. Then there exists a copula $C$ such that for all $u_1, u_2, \ldots, u_n$, If $H$ has a density $h$, and the marginals $F_i$ have densities $f_i$, then where $c$ is the density of $C$. If all marginals $F_i$ are continuous, then $C$ is unique.

Figures (1)

  • Figure 1: Causal Model for Personalized Incentives: This diagram illustrates the causal relationships among group assignment $S$, incentive treatment $T$, pre-treatment features $\mathbf{X}$, and the outcome variable $Y$. The model integrates a unit representation $U$, capturing all relevant endogenous information (excluding $T$) that determines the Layer valuations regarding to $(T, Y)$.

Theorems & Definitions (19)

  • Theorem 1.1: Sklar's Theorem renyi1959measures
  • Definition 2.1: Stochastic Aggregation for Distributions
  • Example 2.2
  • Definition 2.3: Stochastic Aggregation for Random Variables
  • Example 2.4
  • Definition 2.5: Information Aggregation (InfoAgg)
  • proof
  • Definition 3.1: Set Aggregation
  • Definition 3.2: Evidence Aggregation
  • Example 3.3
  • ...and 9 more