Subjective Logic

A subjective opinion is defined as a triplet (b, d, u) where:

• b (belief): Evidence supporting a proposition
• d (disbelief): Evidence against a proposition
• u (uncertainty): The uncommitted mass representing ignorance

The constraint b + d + u = 1 ensures these masses form a complete opinion. This ternary representation allows reputation systems to distinguish between a lack of evidence (high uncertainty) and conflicting evidence (balanced belief and disbelief), which binary trust models cannot capture.

Every subjective opinion includes a base rate (a) parameter representing the prior probability of a proposition being true in the absence of specific evidence. This anchors subjective beliefs to objective ground truth.

• Default base rate: Typically 0.5 for binary propositions with no prior knowledge
• Domain-specific base rates: Can be calibrated from historical data (e.g., 2% fraud rate in transactions)
• Projected probability: P = b + a·u, combining belief with the base-rate-weighted uncertainty

This mechanism prevents reputation scores from diverging from statistical reality when evidence is sparse.

Opinions are formed by mapping positive evidence (r) and negative evidence (s) onto the belief/disbelief/uncertainty triplet using a Dirichlet distribution:

• b = r / (r + s + W)
• d = s / (r + s + W)
• u = W / (r + s + W)

The non-informative prior weight (W) typically equals 2, representing one piece of imaginary positive and negative evidence. As real evidence accumulates, uncertainty shrinks. This provides a principled way to bootstrap reputation for new entities with zero interaction history.

Subjective logic defines a complete algebra for combining opinions, enabling transitive trust computation across networks:

• Discounting (⊗): When A trusts B's opinion about C, A's trust in B discounts B's opinion about C. Uncertainty compounds through the chain.
• Consensus (⊕): Combines multiple independent opinions about the same proposition, reducing uncertainty as evidence pools.
• Fusion operators: Cumulative and averaging fusion handle dependent and independent evidence sources respectively.

These operators enable trust transitivity across multi-hop reputation graphs without losing the uncertainty signal at each step.

Beyond binary propositions, subjective logic supports multinomial opinions over k mutually exclusive outcomes (e.g., rating levels 1-5 stars). The opinion is a (k+1)-tuple:

• k belief masses: One for each possible outcome
• 1 uncertainty mass: Representing ignorance across all outcomes
• k base rates: Prior probabilities for each outcome

This enables reputation systems to model nuanced trust distributions—such as "likely 4-star, possibly 5-star, with 20% uncertainty"—rather than collapsing to a single scalar score.

Subjective logic explicitly separates two fundamental types of uncertainty:

• Aleatory uncertainty: Irreducible randomness inherent to the system (e.g., coin flip). Represented by the balance of belief and disbelief.
• Epistemic uncertainty: Reducible ignorance due to lack of evidence. Represented by the uncertainty mass u.

This distinction is critical for reputation systems: a new entity with no history has high epistemic uncertainty (u ≈ 1), while an erratic entity with mixed reviews has high aleatory uncertainty (b ≈ d ≈ 0.5). Different risk mitigation strategies apply to each case.

A statistical approach that uses Bayesian inference to update the probability distribution of an entity's trustworthiness. Unlike Subjective Logic's explicit uncertainty mass, Bayesian systems update a beta distribution based on sequential observations of positive and negative outcomes. Both frameworks share a common mathematical heritage in Dirichlet distributions, making them complementary tools for modeling trust under uncertainty.

The logical property enabling trust to flow through a network: if entity A trusts B, and B trusts C, A can derive a measure of trust for C. Subjective Logic formalizes this through discounting operators that propagate uncertainty along the chain. Key mechanisms include:

• Referral trust: B recommends C to A
• Functional trust: A's direct assessment of C's competence
• Uncertainty propagation: Each hop increases the uncertainty mass

A distributed reputation management algorithm for peer-to-peer networks that calculates a global trust value for each peer. It uses transitive trust relationships and converges to a stationary distribution via power iteration, similar to PageRank. Subjective Logic extends this by replacing scalar trust values with opinion tuples (belief, disbelief, uncertainty), enabling nodes to express confidence in their local trust assignments.

The process of assigning initial trust values to new entities lacking historical data—the cold start problem. Subjective Logic addresses this by initializing opinions with high uncertainty mass rather than arbitrary default scores. This explicit representation of ignorance prevents new entities from being unfairly ranked as trustworthy or untrustworthy before evidence accumulates.

Aligning a model's predicted probability of correctness with its actual accuracy. Subjective Logic provides a formal framework for second-order uncertainty—uncertainty about one's own uncertainty estimates. This enables reputation systems to distinguish between:

• Aleatoric uncertainty: Inherent randomness in outcomes
• Epistemic uncertainty: Lack of knowledge that can be reduced with more data

The capability to defend against attacks where an adversary creates multiple pseudonymous identities to gain disproportionate influence. Subjective Logic contributes to Sybil resistance by weighting opinions based on the certainty of the trust relationship. Attackers with newly created identities carry high uncertainty mass, limiting their ability to sway consensus even with numerous fake nodes.