Why Source Overlap Matters
A pathway database may list two genes in the same pathway, while an interaction catalog records a link between those genes. These records can appear to be separate reasons to favor a biological relationship. But the interaction record may have been extracted from the same pathway annotation, or both databases may cite the same experiment. Treating every row as an independent piece of evidence can make a knowledge-informed prior far more certain than the underlying evidence warrants.
The first task is to state exactly what claim the sources address. Shared membership in a pathway is not identical to a direct molecular interaction, and neither claim automatically establishes a directed regulatory effect. Combining records is meaningful only after their target claim, molecular entities, and relevant context have been aligned. The earlier material on Filtering Knowledge by Tissue and Experimental Context motivates checking that the records concern the intended biological setting.
Combining Sources Under Conditional Independence
Bayes’ rule gives a direct way to combine evidence. The prior probability \(P(T)\) represents belief in the claim before considering the specified source reports. Prior odds are \(P(T)/P(\neg T)\), where \(P(\neg T)=1-P(T)\). Posterior odds are the corresponding odds after observing the reports. A report is informative for \(T\) when its likelihood is greater under \(T\) than under \(\neg T\).
If the reports are conditionally independent given \(T\), and also conditionally independent given \(\neg T\), then their joint likelihood factors into products. This is a substantive assumption: it says that, within each truth state, learning one report does not change the probability of another. Under that assumption, define \(LR_k=P(D_k\mid T)/P(D_k\mid \neg T)\). Then the joint likelihood ratio is the product of the source likelihood ratios. Convert the resulting odds \(O\) back to a probability using \(O/(1+O)\).
The factorization is justified only when the conditional independence assumptions hold for the evidence being combined. Sources can be dependent even after conditioning on whether \(T\) is true. For example, one catalog may import records from another, or two curators may rely on the same publication. In those cases, the joint likelihood ratio—not the product of marginal likelihood ratios—is the correct multiplier.
Worked Example: Two Distinct, Conditionally Independent Sources
Suppose the target claim is that a specified protein pair interacts in a defined cell context. All numbers in this example are hypothetical. Before consulting two sources, assign \(P(T)=0.10\). A pathway source reports evidence with likelihood ratio \(4\), and a separately curated interaction source reports evidence with likelihood ratio \(3\). Assume the reports refer to the same target claim, have been filtered to the same context, and are conditionally independent both when \(T\) is true and when \(T\) is false.
The prior odds are \(0.10/0.90=1/9\). Under the stated independence assumption, the joint likelihood ratio is \(4\times3=12\). Therefore:
The combined reports raise the probability assigned to this claim from 0.10 to about 0.5714 under the specified model and assumptions. This is not a claim that the interaction is established as fact. If the two sources share evidence, the product of 4 and 3 may overstate what the reports jointly support.
Duplicates and Dependence Are Different Problems
A duplicate record is a repeated representation of the same underlying evidence unit: for example, one study imported into two catalogs under different record identifiers. If the target claim and evidence unit are identical, counting both entries as separate independent reports is not justified. Deduplicating those records is usually the clearest solution.
Dependent sources are not necessarily exact duplicates. They may report different observations or summarize different evidence, but their reports are statistically related. Shared studies, copied annotations, common curation pipelines, or a common source database can induce dependence. Removing exact duplicates alone does not make the remaining reports conditionally independent.
A useful audit records, for each entry, the source, entity identifiers, target claim, context, primary evidence or publication, and any known import or curation relationship. This makes it possible to distinguish repeated rows from genuinely distinct reports and to assess whether conditional independence is plausible. A source’s confidence score is not automatically a likelihood ratio; it needs a calibrated interpretation before it can be used in the odds formula.
Worked Example: Two Catalog Entries From One Study
Suppose a hypothetical prior probability for a specified interaction is \(0.20\). Two catalog entries both trace to the same experiment, and that experiment has a calibrated likelihood ratio of \(5\) for the claim. Treating the two entries as one evidence unit gives:
The evidence changes the assigned probability from \(0.20\) to about \(0.5556\). If the two copies were incorrectly treated as independent, the calculation would use \(5\times5=25\), giving odds \(0.25\times25=6.25\) and probability \(6.25/(1+6.25)=25/29\approx0.8621\). That much larger probability is an artifact of counting the same experiment twice, not a justified increase in evidence.
The assumption check is decisive here: both records refer to the same study, so they are not two independent evidence units. Deduplicating them preserves the experiment’s contribution without multiplying its likelihood ratio by itself.
Use the Joint Likelihood for Dependent Reports
When sources are dependent but their joint behavior can be modeled, combine their reports through the joint likelihood. For two binary reports \(D_1,D_2\), the relevant multiplier for observing both positive reports is \(P(D_1=1,D_2=1\mid T)/P(D_1=1,D_2=1\mid\neg T)\). It need not equal the product of the two marginal likelihood ratios. The model must provide probabilities under both truth states, and those probabilities must be coherent joint distributions.
Worked Example: Two Positively Dependent Reports
Consider two hypothetical catalogs that both report a specified biological relationship as present. Their joint report probabilities, estimated for illustration from a suitable calibration set, are:
| Truth state | \(P(D_1=1,D_2=1)\) | \(P(D_1=1)\) | \(P(D_2=1)\) |
|---|---|---|---|
| \(T\) | 0.60 | 0.80 | 0.75 |
| \(\neg T\) | 0.20 | 0.40 | 0.30 |
These are coherent marginals of possible joint distributions. For example, under \(T\), the probabilities for report pairs \((1,1),(1,0),(0,1),(0,0)\) are \(0.60,0.20,0.15,0.05\), which sum to 1. Under \(\neg T\), they are \(0.20,0.20,0.10,0.50\), also summing to 1. The reports are not conditionally independent: under \(T\), independence would require \(P(D_1=1,D_2=1\mid T)=0.80\times0.75=0.60\), which happens to match, but under \(\neg T\) it would require \(0.40\times0.30=0.12\), not \(0.20\). Thus the pair does not satisfy conditional independence in both truth states.
The joint likelihood ratio for observing both positive reports is \(0.60/0.20=3\). With hypothetical prior probability \(P(T)=0.20\), prior odds are \(0.20/0.80=0.25\). The correct calculation is:
The marginal likelihood ratios are \(0.80/0.40=2\) and \(0.75/0.30=2.5\). Multiplying them would give \(5\), an invalid joint likelihood ratio here. That shortcut would produce posterior odds \(0.25\times5=1.25\) and probability \(5/9\approx0.5556\), rather than the probability \(3/7\) based on the specified joint distribution. This example also shows why checking dependence under both truth states matters: agreement with independence in one state does not establish independence.
Practical Workflow, Diagnostics, and Alternatives
A disciplined workflow makes the assumptions visible before the combined prior is used in a model. The key decisions are not simply how many databases to include, but what each report asserts and how its evidence relates to the other reports.
Specify the entities, relationship type, and context. Do not pool pathway co-membership and direct interaction as though they were identical claims.
Harmonize entity identifiers and record the underlying publication, experiment, import history, and source for each entry. Collapse exact duplicates of the same evidence unit.
Use a product of likelihood ratios only if conditional independence is defensible under both \(T\) and \(\neg T\). Otherwise, use a joint likelihood or a model that explicitly represents the shared evidence.
Multiply prior odds by the joint likelihood ratio, then calculate \(O/(1+O)\). Record the assumptions and evidence units that produced the resulting prior probability.
Warning signs include identical publication identifiers across supposedly separate sources, one catalog documenting that it imports another, sudden certainty increases when adding a near-duplicate database, or a confidence score with no stated probabilistic calibration. The calculation should also be checked against the scope of the source: a likelihood ratio calibrated for one tissue or relationship type may not apply to a different target claim.
There are alternatives to a joint likelihood model, but each answers a different question. A union of source records is useful for forming a candidate set, yet it does not quantify combined evidence. Taking only the strongest source avoids multiplying evidence but discards potentially informative independent reports. A weighted or tempered product may reduce excessive accumulation, but its weights do not automatically correct dependence; they require a stated model or calibration. For multiple sources with known shared provenance, a joint model is the most direct approach, though it can require more data than are available.
No approach can recover evidence that was never documented, and a dependence model estimated from sparse calibration data can itself be uncertain. Keep the resulting prior tied to the precise claim and evidence definition. The probability assigned from knowledge sources is not a substitute for checking whether the knowledge base is relevant to the intended model and data.
Common Pitfall
A tempting argument is: “Two databases report the relationship, so there are two independent confirmations; multiply both likelihood ratios.” The correction is to inspect the evidence provenance and the conditional dependence structure. Two labels in two databases do not imply two independent evidence units. If they repeat one study, deduplicate. If they report distinct but dependent evidence, use the joint likelihood when available. Only multiply marginal likelihood ratios when conditional independence holds given both \(T\) and \(\neg T\).
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A claim has prior probability \(0.25\). Two conditionally independent reports have likelihood ratios 2 and 3. What is the posterior probability?
- A single study is copied into three catalogs. What must be established before treating those entries as three independent pieces of evidence?
- Two positive reports have joint probabilities \(0.50\) under \(T\) and \(0.10\) under \(\neg T\). What is their joint likelihood ratio?
- Why is a pathway co-membership record not automatically interchangeable with a direct interaction record?
Solutions. (1) Prior odds are \(0.25/0.75=1/3\). Conditional independence gives posterior odds \((1/3)\times2\times3=2\), so the posterior probability is \(2/(1+2)=2/3\approx0.6667\). (2) One must establish that the entries refer to distinct underlying evidence and that their reports are conditionally independent under both the claim and its negation; three catalog rows alone establish neither. (3) The joint likelihood ratio is \(0.50/0.10=5\). (4) The records assert different biological relationships, so they do not necessarily provide evidence about the same binary target claim; the claim must be defined and aligned before combining their evidence.