Trans-Ancestry Credible Sets and Posterior Evidence

Why Credible Sets Need a Precise Target

A fine-mapping analysis can assign posterior support across many variants, but a ranked list of probabilities is not yet a concise account of uncertainty. A credible set summarizes which variants carry enough posterior support to reach a chosen threshold. In a trans-ancestry analysis, that summary is often compared with sets from analyses of individual ancestries. Such comparisons are useful only when the target, model, and meaning of “coverage” are clear.

The key distinction is between a posterior probability for each variant and a posterior probability for an event involving a set of variants. For a single causal signal, the variants are mutually exclusive explanations, so their posterior probabilities add directly to the probability that the causal variant is in a set. With multiple causal variants, marginal inclusion probabilities need not add up to the probability that a set contains at least one causal variant. A set-construction rule must match the posterior quantity being reported.

Definition and Single-Signal Construction

Definition: Let \(C_j\) indicate whether variant \(j\) is causal for a specified signal, and let \(D\) denote the data and model being used. The posterior inclusion probability (PIP) for variant \(j\) is \(q_j=P(C_j=1\mid D)\). A credible set is a selected set of variants whose posterior probability of containing the causal variant, or another explicitly defined causal event, reaches a prespecified threshold such as 0.95.

For a model with exactly one causal variant among the candidates, the indicators \(C_j\) are mutually exclusive and exactly one equals 1. Therefore, \(\sum_j q_j=1\). If \(S\) is any set of candidates, the event that the causal variant is in \(S\) is the disjoint union of the events \(C_j=1\) for \(j\in S\). Its posterior probability is thus

P\left(\sum_{j\in S}C_j=1\mid D\right) =\sum_{j\in S}P(C_j=1\mid D) =\sum_{j\in S}q_j.

To make a compact set, rank variants from largest to smallest PIP and take the shortest prefix whose cumulative PIP reaches the target. This gives a set with at least the target posterior mass; it can exceed that threshold because variants are added in whole units. Ranking is size-efficient: among all sets of a fixed number of variants, the set containing the largest PIPs has the greatest summed posterior probability. If several variants tie at the boundary, more than one set of the same size may attain the same mass.

Worked Example: A 95% Set for One Shared Signal

Suppose a hypothetical trans-ancestry model assigns posterior inclusion probabilities \(0.42, 0.31, 0.23,\) and \(0.04\) to variants A, B, C, and D, respectively. Assume the model represents exactly one shared causal variant among these candidates.

The PIPs sum to \(0.42+0.31+0.23+0.04=1.00\), as required by this single-causal-variant setup. Rank the variants in the listed order. The first two accumulate \(0.42+0.31=0.73\), below 0.95. Adding C gives

$$ 0.42+0.31+0.23=0.96. $$

The 95% credible set is therefore \(\{\mathrm{A},\mathrm{B},\mathrm{C}\}\), with 0.96 posterior probability that the causal variant is among its members under this model and data. The set exceeds 0.95; it does not have exactly 0.95 posterior mass. Adding D is unnecessary for the stated threshold.

What Changes With Multiple Causal Variants?

In a multi-causal model, a PIP remains a marginal probability: it describes the posterior probability that one variant is included, averaging over possible configurations of all the other variants. PIPs can sum to more than 1 because the posterior can support several causal variants at once. Their sum over a set is the posterior expected number of causal variants in that set, not generally the probability that the set contains at least one causal variant.

Important distinction: For a set \(S\) in a multi-causal model, posterior coverage of at least one causal variant is \(P(\sum_{j\in S}C_j\geq 1\mid D)\). By contrast, \(\sum_{j\in S}q_j=E(\sum_{j\in S}C_j\mid D)\). These are different quantities unless additional conditions make them equal.

When the target is coverage of at least one causal variant, the joint posterior over causal configurations is needed. A PIP-ranked list by itself does not encode how often variants are included together. In practice, a credible set may instead be defined for a particular signal, using posterior probabilities that assign that signal to candidate variants. The definition should state which signal or event is being covered. Do not use a set-building rule for single-signal probabilities as though it automatically gave coverage for an arbitrary multi-causal event.

Worked Example: Why Marginal PIPs Do Not Determine Set Coverage

Consider a hypothetical posterior over four variants A, B, C, and D. With probability 0.50, the causal configuration is \(\{\mathrm{A},\mathrm{B}\}\); with probability 0.50, it is \(\{\mathrm{C},\mathrm{D}\}\). No other configuration has posterior probability.

Each variant has PIP 0.50. For example, A is included in the first configuration, which has probability 0.50, and is absent in the second. Ranking by PIP allows either pair to be chosen first because of the tie. The top two marginal PIPs sum to \(0.50+0.50=1.00\), but the set \(\{\mathrm{A},\mathrm{B}\}\) contains at least one causal variant only in the first configuration:

P(C_{\mathrm{A}}+C_{\mathrm{B}}\geq 1\mid D)=0.50.

Thus, the summed marginal PIP of 1.00 is not 1.00 posterior coverage. It is the expected number of causal variants in that pair: the pair contains two in half the configurations and none in the other half, so \(2(0.50)+0(0.50)=1.00\). This example shows why a multi-causal credible set for a union event must be evaluated from joint configurations, not inferred by summing PIPs.

Comparing Trans-Ancestry and Ancestry-Specific Sets

A trans-ancestry set is built from posterior probabilities under the joint model and combined data. An ancestry-specific set is built from the posterior under an analysis restricted to that ancestry. Even when all analyses consider the same harmonized variants and use the same nominal threshold, the PIPs can differ because the evidence and posterior distributions differ. A joint model may draw on association patterns across ancestries while representing ancestry-specific linkage disequilibrium and effects according to its assumptions. The resulting set is not obtained by mechanically averaging the ancestry-specific PIPs.

For a fair comparison, identify the same target signal, use the same candidate variant definition where possible, and report both set membership and size. “Coverage” here refers to posterior coverage under each analysis’s own model and data, not a guarantee that the set contains the true causal variant in a particular study. A smaller joint set at the same nominal threshold indicates a more concentrated posterior under that joint analysis; it does not by itself establish that the model is better calibrated or that the causal variant has been proven.

Worked Example: Comparing Set Size Across Analyses

Suppose three hypothetical analyses consider the same five variants, V1–V5, for a single shared signal. Their PIPs, in descending order within each analysis, are:

Analysis	V1	V2	V3	V4	V5
Trans-ancestry joint model	0.55	0.32	0.09	0.03	0.01
Ancestry A only	0.72	0.18	0.06	0.03	0.01
Ancestry B only	0.25	0.22	0.21	0.18	0.14

Each row sums to 1.00, consistent with the stated single-signal setup. In the joint analysis, the first two variants accumulate \(0.55+0.32=0.87\); adding V3 gives \(0.96\). Its 95% set is \(\{\mathrm{V1},\mathrm{V2},\mathrm{V3}\}\), with posterior mass 0.96. In ancestry A, the first two accumulate \(0.72+0.18=0.90\); adding V3 gives \(0.96\), so its set also has three variants. In ancestry B, the first four accumulate \(0.25+0.22+0.21+0.18=0.86\); adding V5 gives 1.00, so all five variants are required.

Under these hypothetical posterior distributions, the joint set is narrower than the ancestry B set at the same threshold, while ancestry A’s set has the same size as the joint set. This describes posterior concentration, not a direct comparison of frequentist coverage. The differing set sizes are not contradictions: each analysis conditions on its own data and model.

Interpreting Changes in Variant Support

A variant’s PIP is posterior support for inclusion under a specified model and data, not a permanent attribute of the variant. It can rise or fall between an ancestry-specific analysis and a joint analysis, or between model specifications, because the likelihood, prior, candidate set, or dependence among variants has changed. In trans-ancestry fine-mapping, ancestry-specific linkage disequilibrium can alter how well the observed association patterns distinguish correlated candidates. The PIP change should therefore be reported with the analyses being compared, not described as the variant becoming causal or ceasing to be causal.

Posterior odds can help express the size of a change for a particular inclusion event. For a PIP \(q_j\), the posterior odds of inclusion versus exclusion are \(q_j/(1-q_j)\). Comparing these odds across analyses is descriptive. It does not isolate the contribution of new data unless the event, prior odds, and model are held appropriately comparable. Nor does a larger PIP by itself establish causality; posterior probabilities remain conditional on the model and its assumptions.

Worked Example: A Variant’s Support Shifts Across Analyses

In a hypothetical single-causal-variant analysis, variant X has PIP 0.58 in ancestry A and PIP 0.21 in a trans-ancestry joint analysis. The posterior odds of inclusion in ancestry A are

\frac{0.58}{1-0.58}=\frac{0.58}{0.42}=1.381.

In the joint analysis they are

\frac{0.21}{1-0.21}=\frac{0.21}{0.79}=0.2658.

The posterior odds are lower in the joint analysis, and X may no longer appear in the joint 95% set if other variants accumulate the threshold first. The correct interpretation is that the joint data and model assign less posterior support to X relative to its alternatives than the ancestry A analysis does. This change does not mean X’s causal status changed between populations. It may reflect the added evidence, differences in LD patterns, or other model features; identifying which explanation applies requires examining those features rather than reading causality directly from the PIP shift.

Why the Construction Works, and a Common Pitfall

For a single causal signal, the cumulative-PIP construction has a simple justification. Sort the PIPs so that \(q_{(1)}\geq q_{(2)}\geq\cdots\). For any fixed set size \(k\), the sum of the first \(k\) values is at least as large as the sum for any other \(k\) variants: replacing a selected lower-ranked value with an omitted higher-ranked value cannot decrease the sum. Since the posterior probability that a set contains the causal variant equals its PIP sum in this setup, the top-ranked prefix reaches a given threshold with the fewest variants.

This justification depends on the single-causal target. In the multi-causal case, marginal PIPs do not determine the probability of a union event, so the same argument cannot establish posterior coverage for at least one causal variant. The joint posterior configurations must be used for that event, or the target must be defined for a specific signal with probabilities that sum over its mutually exclusive candidate assignments.

Mistake: “The 95% set has exactly 95% posterior mass.” Correction: The shortest ranked set has at least 95%; discrete PIPs can make its mass larger.
Mistake: “The PIPs in a multi-causal set sum to its coverage.” Correction: That sum is the posterior expected number of included causal variants; compute the joint event probability for coverage.
Mistake: “A variant with a higher PIP in one ancestry is more causal there.” Correction: PIP is conditional on the analysis; a difference is a change in posterior support, not evidence that causal status itself varies.

Key takeaway: State the causal event your credible set is intended to cover. For a single shared causal signal, rank its mutually exclusive posterior probabilities and stop at the first cumulative total at least as large as the target. For a multi-causal union event, evaluate coverage from the joint posterior, and interpret cross-analysis PIP changes as model-conditional evidence.

Check Your Understanding

Use the stated posterior target and causal model in each answer.

For a single causal variant with ranked PIPs 0.50, 0.30, 0.15, and 0.05, what is the 95% credible set and its posterior mass?
Why can PIPs sum to more than 1 in a multi-causal model?
In a multi-causal model, what does the sum of PIPs over a selected set represent, and what quantity is needed to assess the probability that at least one member is causal?
What does it mean if a joint trans-ancestry set is smaller than an ancestry-specific set at the same nominal threshold?
A variant’s PIP decreases between two analyses. Give one appropriate interpretation and one conclusion that the PIP change alone does not justify.