Common Credible-Set Interpretation Errors

What a Credible Set Does—and Does Not—Say

Fine-mapping results are often summarized as a short list of variants. That list can look deceptively simple: a reader may assume every member is equally plausible, that the reported posterior mass is the probability of a particular variant, or that a highly correlated set proves one of its members is causal. None of those conclusions follows from set membership alone.

The interpretation depends on the quantity being summarized and on the model used to obtain it. In the earlier material on interpreting high posterior inclusion probabilities, each variant’s posterior inclusion probability (PIP) summarizes its marginal posterior probability of being causal. A credible set instead groups variants according to a cumulative posterior-mass rule, often for a particular signal. LD purity describes correlations among the variants in the set, not their posterior probabilities.

The phrase “95% credible set” refers to that set-level posterior criterion, under the specified model and assumptions. It does not mean that each included variant has a 95% probability of being causal. Nor does it mean that 95% of the variants in the set are causal. If there is exactly one causal variant in the modeled signal, the posterior probability that the causal variant is in the set is the sum of the member probabilities. With multiple causal variants, the interpretation requires more care, as shown below.

Set Membership Is Not Equal Probability

A cumulative rule adds posterior probabilities until the target is reached. The final variant may take the cumulative total from below the target to above it, so the total can exceed \(\gamma\). The variants included can have very different posterior probabilities. Being included means that a variant contributes to meeting the set’s construction rule; it does not mean that it is as plausible as every other member.

A further distinction is between posterior credibility and frequentist coverage. A 95% posterior-mass set has at least 95% posterior probability under the fitted model and its priors, conditional on the observed data. It does not automatically guarantee that sets produced in repeated studies contain the true causal variant 95% of the time. Empirical coverage requires assessment across repeated simulations or suitable data, and can be affected by model misspecification.

Posterior Mass and Causal-Variant Count

For a single causal variant, the events \(C_j=1\) are mutually exclusive: at most one candidate is causal. Thus, for any set \(S\), the probability that at least one member is causal is the sum of the member probabilities. The same shortcut does not generally apply when multiple variants may be causal.

With multiple causal variants, the sum of PIPs is the posterior expected number of causal variants in the set. To see this, define \(N_S=\sum_{j\in S}C_j\), the number of causal variants in \(S\). Linearity of expectation gives the following result without requiring the causal indicators to be independent.

But the probability \(P(N_S\geq 1\mid D)\) is a different quantity. It is a union probability, and overlapping events must not be counted twice. In particular, when multiple variants can be causal, summing PIPs can exceed one and cannot then be the probability that at least one member is causal.

A useful justification for the single-signal rule is that mutually exclusive events are additive. If exactly one variant \(J\) is causal, then the event \(J\in S\) is the disjoint union of the events \(J=j\), for \(j\in S\). Therefore \(P(J\in S\mid D)=\sum_{j\in S}P(J=j\mid D)\). When more than one variant can be causal, those events need not be disjoint: configurations with two or more causal members belong to several events at once. Adding the marginal probabilities then counts such configurations more than once. This is why the model’s causal-count assumptions matter to the meaning of “credible mass.”

Purity Measures LD, Not Causality

Credible-set purity is a separate diagnostic commonly used to summarize how strongly correlated the variants in a set are. A typical measure is the minimum pairwise \(r^2\) among the variants, calculated from their genotypes in an appropriate reference or study sample. For a set with at least two variants, it is the smallest pairwise \(r^2\); for a singleton, there is no pair to compare, so minimum pairwise purity is undefined or reported by a stated convention.

Common Interpretation Errors and Checks

• Wrong: “Every variant in the 95% set has a 95% chance of being causal.” Correction: The mass applies to the set-level event under its construction and model. Report individual PIPs separately.
• Wrong: “The set contains five variants, so each has probability 20%.” Correction: Set membership does not imply equal probabilities. Use the variant-specific posterior values.
• Wrong: “The PIPs sum to 1.2, so there is a 120% chance that something is causal.” Correction: With multiple causal variants, that sum is the expected causal count, not a probability. Calculate or report the union probability only if it is available and relevant.
• Wrong: “High purity proves that one of these variants is the cause.” Correction: Purity summarizes pairwise LD. Causality and posterior support come from the association data and the fitted model; purity alone establishes neither.
• Wrong: “A 95% set has 95% repeated-sampling coverage by definition.” Correction: Posterior credibility is conditional on the model and observed data. Repeated-sampling coverage is a separate property that requires evaluation.

When reading a result, check whether the set is signal-specific or region-wide, whether the model permits one or multiple causal variants, how the reported mass was calculated, and how purity was defined. If those details are missing, a precise interpretation may not be possible. These checks do not replace sensitivity or calibration analyses; they prevent the summary statistics from being given meanings they do not support.