Overlapping Credible Sets Across Signals

Why Overlap Needs Careful Interpretation

In multi-signal fine-mapping, each inferred signal has its own posterior distribution over candidate variants and its own credible set. Two such sets can overlap because the data do not clearly distinguish which nearby variant belongs to which signal. That overlap is useful information about uncertainty, but it is not, by itself, evidence that the signals are the same effect—or that they are independent effects.

The distinction matters because a credible set is a summary of marginal posterior uncertainty for one signal. Whether assignments across signals are dependent is a joint-probability question. A shared variant may be a candidate for either signal, while the posterior may make the two assignments strongly dependent, nearly independent, or subject to model constraints. Those possibilities cannot generally be distinguished by looking at the sets alone.

This definition presumes a specified multi-signal model, a common set of candidate variants, and a posterior assignment distribution for each signal. The target level describes posterior mass under that model and its assumptions; it is not a guarantee that every set will contain a causal variant in repeated studies. If posterior probabilities are obtained from an approximation, the credible-set interpretation inherits that approximation.

Write \(S_k\) for the credible set of signal \(k\). The sets overlap when \(S_k\cap S_\ell\) is nonempty. Their intersection identifies variants included as candidates for both signals. It does not tell us how probable it is that both signals are assigned to the same variant: that requires the joint distribution \(P(C_k,C_\ell\mid D)\).

Reading the Overlap

A variant in an intersection has posterior support in the marginal assignment distribution for each of the relevant signals. It is therefore a shared candidate, not automatically a shared causal effect. Conversely, two sets with no shared variants do not establish that the signals are statistically independent: their assignments could still be dependent through the joint posterior.

A simple measure of set overlap is the Jaccard similarity:

This describes the proportion of the combined set members that appear in both sets. It is a descriptive comparison, not a probability of shared causality or a measure of posterior dependence. In particular, it treats a variant barely included at the credible-set threshold the same as one with a large assignment probability.

Overlap can arise for several reasons. Nearby variants may be difficult to distinguish because their genotypes are correlated through linkage disequilibrium. A signal may be weak, leaving its assignment probabilities spread over many variants. Multiple signals may also compete to explain related evidence. These explanations are not interchangeable: a broad set can reflect limited resolution for one signal, while cross-signal dependence concerns how assignments vary together.

The earlier material on posterior uncertainty within a credible set and credible-set purity and LD helps distinguish uncertainty among variants in one set from the relationship between sets. A wide intersection can be a clue to shared ambiguity, but inspecting the variants’ assignment probabilities and the model’s joint behavior is necessary before drawing stronger conclusions.

Marginal Probabilities Do Not Determine Dependence

To assess dependence between two signal assignments, define the joint assignment probabilities \(q_{k\ell}(j,m)=P(C_k=j,C_\ell=m\mid D)\). They must agree with the marginal probabilities when summed over the other signal:

The marginals constrain the joint distribution but do not usually determine it. The following short derivation makes that limitation explicit. If the two assignments were independent, then \(q_{k\ell}(j,m)=\alpha_{kj}\alpha_{\ell m}\). But other joint tables can have the same row and column sums while assigning different probabilities to matching or differing variants. Therefore, neither marginal probabilities nor credible-set membership alone identify dependence.

Assessing Dependence and Model Interpretation

When a joint posterior over assignments is available, one direct summary is the probability of matching assignments, \(\sum_j P(C_k=j,C_\ell=j\mid D)\). This is meaningful only when the signal definitions and variant universe make a match interpretable. It is not, on its own, proof that two modeled signals are biologically one effect: distinct effects can map to the same measured variant under some model formulations, and the model’s assumptions determine which configurations are permitted.

Another useful question is conditional: how does the assignment distribution for one signal change given the assignment of another? For example, compare \(P(C_\ell=m\mid C_k=j,D)\) with the marginal \(P(C_\ell=m\mid D)\). A difference indicates dependence for those assignments, provided the conditioning event has positive posterior probability. Such comparisons can reveal competition or coordination that a single overall matching probability might conceal.

In some computational approaches, a posterior approximation factorizes across signal components. Under that approximation, assignments may be treated as independent in the approximation even when the exact posterior would not factorize. Dependence conclusions should therefore be attributed to the joint distribution actually represented by the fitted method—not inferred from an algorithm’s separate marginal summaries. If only per-signal assignment probabilities are reported, the joint dependence is not identified from those summaries alone.

Diagnostics, Alternatives, and Limitations

Before interpreting overlap, check the assignment probabilities and the model output that produced them. Confirm that probabilities for each signal are normalized over the stated candidate universe, that both sets use the same variants and target level, and that any ties or thresholding choices are handled consistently. If a signal has diffuse probabilities or is weak, small changes in its ranking can change set membership even when the underlying uncertainty has changed very little.

Also inspect whether the same variant appears in many signals, whether assignment probabilities are concentrated or diffuse, and whether the fitted model’s joint output permits assessment of dependence. A pattern of repeated or very similar signal sets can flag weak resolution or unstable component allocation; it is a diagnostic prompt, not a stand-alone proof that the model has duplicated a signal. In particular, labels for fitted components should not be treated as distinct biological entities solely because the computation reports separate components.

Several summaries answer different questions:

A common mistake is to see one variant in two 95% credible sets and conclude that the model has found two independent causal effects at that variant. The reasoning is wrong because membership records that the variant contributes to each signal’s marginal credible set; it does not show that the two assignment events occur jointly, nor that they are independent. The correction is to report shared membership as shared candidacy, then use joint posterior output and the model’s assumptions to assess assignment dependence and interpret the signal structure.

The main limitation is that all these interpretations are conditional on the fitted model, candidate variants, data, and any approximation used to compute posterior probabilities. Overlap can describe unresolved localization without resolving the biological number of effects. A model-based joint probability adds information beyond marginal sets, but it does not remove uncertainty caused by limited data or model misspecification.