Can ordinary probabilities be represented?

Yes. Assigning all mass to singleton sets produces the corresponding probability measure.

How is complete ignorance represented?

In this formulation, all mass is assigned to the entire frame.

Is Dempster's rule the only combination rule?

No. Other rules exist, with different assumptions and treatments of conflict; their outputs should not be treated as interchangeable.

AI Glossary / Graphs and symbolic AI / Advanced

Dempster-Shafer theory

Definition

Dempster-Shafer theory, also called evidence theory or the theory of belief functions, is a mathematical framework for representing uncertain evidence. It can assign support to sets of possible answers, leaving distinctions within a set unresolved. Belief and plausibility summarize that evidence, while combination rules bring together information under specified assumptions.

What it means

Ordinary probability assignments distribute probability among individual possibilities. A belief-function representation can instead support “either A or B” without deciding how much belongs to A and how much to B. This lets the representation distinguish incomplete knowledge from a precise distribution over the alternatives.
The framework does not manufacture reliable evidence. Someone must define the possibilities, justify the evidence assignments, and explain what the numbers mean. A mathematically valid calculation can still mislead if its sources repeat the same observation or its list of possible answers excludes the actual answer.
Belief and plausibility should not be presented as an automatically calibrated confidence interval for a hidden real-world frequency.

How it works

Start with a frame of discernment: the possible answers under consideration. In the standard finite, normalized formulation, nonnegative masses are assigned to subsets, sum to one, and assign zero to the empty set. A subset with positive mass is called a focal set.
Belief in A sums masses assigned to subsets entirely contained in A. Plausibility sums masses assigned to subsets that intersect A. Equivalently, plausibility of A is one minus belief in its complement.
Dempster's normalized combination rule multiplies masses from two sources and collects products by subset intersection. Products with empty intersections constitute conflict K. The remaining masses are divided by 1 − K. The rule is undefined at total conflict, K = 1, and its use requires suitable source-independence assumptions.

Three practical examples

Equipment diagnosis: An inspection may support “connector or cable fault” without distinguishing the two. Keeping support on that pair preserves ambiguity that would be hidden by an unsupported fifty-fifty split. Additional observations may narrow the possibilities.
Document classification: Evidence may support either an invoice or a receipt while excluding a service agreement. A belief-function representation can retain the two-label possibility instead of forcing an immediate single label. Any final classification still needs a decision rule.
Inventory inspection: Different observations may support competing explanations for a discrepancy. Before combining them, the analyst checks whether they originated from separate checks or the same copied stock record. Two reports are not necessarily two independent sources.

Think of it this way

Imagine placing evidence in labeled boxes. One box is marked “A,” another “B,” and a third “A or B, unresolved.” Evidence in the third box has not been discarded, but it has not been divided between the first two either. A decision to distribute it needs an additional rationale.

Worked example

Consider an invented binary equipment assessment with frame {F, N}, meaning fault present or no fault. Source 1 assigns mass 0.6 to {F} and 0.4 to the whole frame. Thus belief in F is 0.6 and plausibility of F is 1.0. The unresolved 0.4 is not automatically support for N.
Suppose Source 2 assigns mass 0.3 to {N} and 0.7 to the whole frame. Assume the sources satisfy the independence conditions required for this illustrative combination.
The four mass products are 0.6 × 0.3 = 0.18 for the empty intersection, 0.6 × 0.7 = 0.42 for {F}, 0.4 × 0.3 = 0.12 for {N}, and 0.4 × 0.7 = 0.28 for the whole frame. They sum to one.
Conflict K is 0.18, so normalization divides the three nonempty masses by 0.82. The combined masses are approximately 0.5122 for F, 0.1463 for N, and 0.3415 unresolved. Belief in F is approximately 0.5122; plausibility is 0.5122 + 0.3415 = 0.8537.
These values describe the stipulated evidence model. They do not establish a 51.22% observed fault rate. If Source 2 were derived from Source 1, the assumed combination would need reconsideration.

Illustrative case study

Fictional Maple Maintenance introduces an evidence dashboard for equipment checks. Its first design turns every ambiguous inspection into a single fault label. Reviewers find that this conceals whether an inspector identified a specific fault or only narrowed the possibilities.
The revised design retains support for sets of candidate explanations. It also records which observations underpin each report. During a review, the team discovers that two apparently separate reports both used one sensor reading, so it avoids treating them as independent corroboration.
The dashboard helps make uncertainty and evidence provenance visible. It does not automatically determine which repair is best; the team still considers costs, additional tests, and operational consequences. This is an illustrative scenario, not a reported deployment result.

Common misconceptions and mistakes

“Uncommitted mass is evenly divided.” That requires an additional transformation or assumption.
“Belief and plausibility are ordinary confidence limits.” Their meaning depends on the evidence model.
“Different reports are independent evidence.” Reports can share observations or failure mechanisms.
“Normalization resolves disagreement.” It produces a result under a rule; the source conflict still needs interpretation.
“The theory always improves predictions.” Usefulness depends on the application, assignments, and evaluation.

Business use

Document the question, frame, evidence sources, mass assignments, and chosen combination rule. Explain how reliability judgments were obtained and which dependencies were considered. Retain conflict information when presenting combined results.
Test how conclusions change under alternative defensible assignments. Compare the approach with simpler baselines using relevant outcomes. Separate evidence representation from action selection: repair, inspect further, or continue operation can have different consequences even when they use the same belief function.

Check your understanding

What does Source 1's unresolved 0.4 mean?

Answer: It remains on the whole frame rather than favoring either singleton.

Why is the normalization denominator 0.82?

Answer: The conflicting mass is 0.18.

Why inspect evidence provenance?

Answer: Shared evidence can invalidate an independence assumption.

Related terms

Belief function
Uncertainty
Bayesian inference
plausibility
evidence fusion
mass function
frame of discernment
source dependence

Learn next

Construct a three-answer frame and assign support to both singletons and sets. Calculate belief and plausibility for one answer, then explain what further information would reduce the unresolved support.

Sources

Shafer, Belief Functions, introduction: https://www.glennshafer.com/assets/downloads/rur_chapter7.pdf Denoeux, Theory of Belief Functions, Lecture 1, basic concepts and Dempster's rule: https://www.hds.utc.fr/~tdenoeux/dokuwiki/_media/en/bf2023_lecture1.pdf

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