Standard probability theory represents uncertainty through a single number between 0 and 1. But this conflates two very different situations: having strong evidence that an event has probability 0.5 (a fair coin), and having no evidence at all (complete ignorance). Subjective logic, developed by Audun Jøsang, addresses this by representing beliefs as opinions — tuples that separately quantify belief, disbelief, and uncertainty — enabling a richer calculus of trust, evidence combination, and uncertain reasoning.
Opinion Representation
b = belief (evidence-supported confidence in truth)
d = disbelief (evidence-supported confidence in falsehood)
u = uncertainty (uncommitted belief mass due to lack of evidence)
a = base rate (prior probability, used when evidence is absent)
Constraint b + d + u = 1, b, d, u ≥ 0, a ∈ [0, 1]
Projected Probability P(x) = b + a · u
An opinion with u = 0 is a standard probability assignment: full certainty, no residual ignorance. An opinion with u = 1 represents complete ignorance, in which case the projected probability defaults to the base rate a. Between these extremes, opinions capture partial evidence — for instance, after observing 3 heads and 1 tail, the opinion on "heads" might be (0.6, 0.2, 0.2, 0.5), reflecting strong but imperfect evidence favoring heads.
Audun Jøsang introduces the initial framework of subjective logic, combining Dempster-Shafer belief theory with a base rate parameter for decision-making under uncertainty.
Jøsang develops the opinion triangle visualization, where each point in a triangle represents a unique opinion, and proposes operators for combining, discounting, and fusing opinions from multiple sources.
Subjective logic is extended to multinomial and Dirichlet opinions, trust networks, and multi-source fusion. Applications in cybersecurity, reputation systems, and sensor fusion emerge.
Jøsang publishes the comprehensive monograph Subjective Logic: A Formalism for Reasoning Under Uncertainty, consolidating two decades of development.
Connection to Bayesian Inference
Subjective logic has a direct mapping to Bayesian inference with Dirichlet priors. A binomial opinion (b, d, u, a) corresponds to a Beta distribution Beta(α, β) where α = W·b/u + W·a and β = W·d/u + W·(1−a), with W being a weight parameter (typically W = 2). The uncertainty u corresponds to the width of the Beta distribution: high uncertainty maps to a broad, diffuse posterior, while low uncertainty maps to a concentrated one. This equivalence means that subjective logic operations — combination, fusion, discounting — have Bayesian interpretations as operations on Beta or Dirichlet posteriors.
Subjective logic builds on Dempster-Shafer (DS) theory but adds the base rate parameter a and a richer set of operators. In DS theory, combining conflicting evidence can produce counterintuitive results (Zadeh's paradox). Subjective logic addresses this through alternative fusion operators — averaging fusion, cumulative fusion, and constraint fusion — each appropriate for different epistemic situations. The base rate also provides a principled decision rule when uncertainty is high, which DS theory lacks.
Operators
Subjective logic defines a rich algebra of operators on opinions, mirroring the operators of propositional logic:
Conjunction and disjunction: combining opinions on propositions A and B to form opinions on A∧B and A∨B, accounting for dependence or independence assumptions.
Cumulative fusion: combining opinions from multiple independent sources who observe the same proposition, analogous to Bayesian updating with multiple observations.
Averaging fusion: combining opinions from multiple dependent or non-independent sources, giving equal weight to each.
Discounting: adjusting an opinion based on the trustworthiness of the source, reducing belief and increasing uncertainty when the source is less trusted.
Deduction and abduction: chaining opinions through conditional relationships, analogous to forward and backward reasoning in Bayesian networks.
Applications
Subjective logic is widely used in trust and reputation systems, where the explicit representation of uncertainty allows distinguishing between trusted entities (low uncertainty, high belief), distrusted entities (low uncertainty, high disbelief), and unknown entities (high uncertainty). It has been applied to cybersecurity threat assessment, sensor fusion in autonomous systems, fake news detection, and multi-agent decision-making. The opinion triangle provides an intuitive visualization that domain experts find more accessible than raw probability distributions.
"Probability tells you what to believe. Subjective logic also tells you how much you don't know — and that distinction matters enormously when the stakes are high and the evidence is thin." — Audun Jøsang, on the motivation for subjective logic