Are these networks based on cosine similarity?

Not in this specialized probabilistic meaning.

Can an observation be omitted locally but included elsewhere?

Yes. Its ability to distinguish hypotheses can depend on the subset being considered.

Can multiple simultaneous faults be modeled?

Extensions exist, but the basic mutually exclusive hypothesis setup does not automatically represent them.

AI Glossary / Graphs and symbolic AI / Advanced

Similarity networks

Definition

In probabilistic AI, similarity networks are collections of local Bayesian networks constructed to distinguish among overlapping subsets of hypotheses. Each local network focuses on variables relevant to its restricted comparison. The approach supports knowledge elicitation and probabilistic reasoning while representing distinctions that may be simpler locally than across the entire hypothesis set.

What it means

Here, “similarity network” has a specialized meaning associated with Heckerman and later work with Geiger. It is not simply a graph joining similar documents or embedding vectors. The central variable represents mutually exclusive and collectively exhaustive hypotheses for the modeled task.
Instead of asking an expert to describe every relationship at once, the method asks narrower questions about selected alternatives. Evidence that distinguishes one pair of hypotheses may not distinguish another. This can expose useful independence relationships. The local models must nevertheless be compatible with the intended probability model; drawing several plausible diagrams is not enough.

How it works

Choose overlapping subsets that cover the hypotheses and connect them through shared members. Construct a Bayesian network for each restricted context, including the variables required by the chosen formal definition of relevance. Assess conditional probabilities and check consistency.
Inference can use the local networks to recover probabilities of the competing hypotheses under appropriate conditions. Different similarity-network types and algorithms have different requirements, including conditions concerning zero probabilities. A variable omitted from one local comparison may still matter elsewhere. The representation can preserve the information needed to distinguish hypotheses without preserving every aspect of the full joint distribution.

Three practical examples

An equipment-diagnosis prototype can compare two easily confused fault types using the observations that separate them.
A document-routing model can organize expert judgments around overlapping sets of mutually exclusive document classes.
A teaching exercise can show why an observation changes the odds between some hypotheses but leaves another pair's odds unchanged.

Think of it this way

Think of preparing several focused comparison cards: one contrasts options A and B, another contrasts B and C. Shared options let the comparisons connect. The analogy is limited because similarity networks require probability distributions and precise independence conditions; ordinary comparison cards can contain inconsistent or merely subjective statements.

Worked example

Consider an original single-fault model with hypotheses A, B, and C. Their prior probabilities are 0.5, 0.3, and 0.2. Let binary observation X have probabilities P(X = yes | A) = 0.8, P(X = yes | B) = 0.8, and P(X = yes | C) = 0.2.
Use overlapping subsets {A, B} and {B, C}. They cover all hypotheses and are connected through B. Within {A, B}, observing X = yes leaves the A:B odds unchanged because both likelihoods equal 0.8. The restricted prior for A is 0.5 / 0.8 = 0.625, and its restricted posterior remains 0.625. In this deliberately simple model, X need not help distinguish A from B.
Within {B, C}, the restricted prior probabilities are 0.6 and 0.4. After X = yes, the unnormalized values are 0.48 and 0.08, giving posterior probabilities 6/7 and 1/7. X does distinguish these alternatives.
Across the full model, the unnormalized posterior values are 0.40, 0.24, and 0.04. Dividing by their sum, 0.68, gives approximately 0.5882, 0.3529, and 0.0588. The A:B ratio remains 5:3 while C loses relative support.
The example illustrates subset-specific relevance using a complete toy probability model. It does not demonstrate that arbitrary pairwise probabilities can always be combined, or that marginal equality alone establishes irrelevance when additional interacting evidence variables are present.

Illustrative case study

In a fictional case, Cedar Diagnostics designs a troubleshooting assistant for a small machine. Engineers initially group every possible fault into one large discussion. They then organize workshops around overlapping groups of similar faults and ask which observations distinguish each group.
One observation is useful for separating electrical from mechanical faults but provides no distinction between two electrical alternatives. The team records that restricted independence and checks the local models together. Before deployment, it also revisits the single-fault assumption because two faults may coexist. This is a fictional knowledge-modeling exercise, not evidence of diagnostic accuracy.

Common misconceptions and mistakes

Local omission does not imply global irrelevance. Nor does a connected cover alone guarantee that inconsistent probability assessments form a coherent model. Relevant formal conditions still matter.
Do not confuse a subset of hypothesis values with a subset of independent binary fault variables. Mutually exclusive hypotheses model one selected alternative; simultaneous faults require an appropriate extension or a different representation. Also avoid claiming that every similarity network encodes the entire joint distribution. Diagnostic inference and recovery of every distributional detail are different requirements.

Business use

Similarity networks can help structure expert knowledge when a classification or diagnostic task has many related alternatives and different observations matter in different comparisons. Their value lies in making the elicitation task more focused, not in eliminating the need for probability assessment.
Maintain a record of hypothesis definitions, subset membership, local independence claims, probability sources, and consistency checks. Validate on representative cases, including observations that challenge the single-fault or class-exclusivity assumptions. Specialists should review the formal model before treating its probabilities as decision-ready.

Check your understanding

Why does X = yes leave the A:B odds unchanged?

Answer: Its likelihood is the same, 0.8, under A and B, so multiplying both prior probabilities by that factor preserves their ratio.

Related terms

Bayesian network
Conditional independence
Probabilistic Graphical Model
hypothesis variable
subset independence
knowledge elicitation
diagnostic inference
connected cover

Learn next

Change P(X = yes | B) from 0.8 to 0.4. The A:B posterior odds become (0.5 × 0.8):(0.3 × 0.4) = 10:3. Explain why X can no longer be omitted on the same independence rationale in that local comparison.

Sources

Sources: Geiger and Heckerman, Inference Algorithms for Similarity Networks, definitions and diagnostic completeness: https://arxiv.org/pdf/1303.1493 . Heckerman, Similarity Networks for the Construction of Multiple-Faults Belief Networks: https://arxiv.org/pdf/1304.1085 .

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