Constructing Credible Sets From Posterior Probabilities

From Posterior Probabilities to a Variant Set

Fine-mapping often assigns each candidate variant a posterior inclusion probability (PIP): the posterior probability, given the observed data and specified model, that the variant is causal. A list of PIPs expresses uncertainty variant by variant. A credible set turns that list into a collection of variants whose accumulated posterior probability reaches a chosen target.

The construction is simple when PIPs are distinct: rank variants from largest to smallest, then keep adding them until their cumulative PIP reaches the target. But a tie at the stopping point creates a genuine choice. Including one tied variant may be enough to cross the target, while including every variant with that same PIP gives a larger set. The rule used for ties therefore needs to be stated, not hidden.

Definition: Suppose the candidate variants are \(v_1,\ldots,v_m\), and let \(C_j\) denote the event that variant \(v_j\) is causal. Its posterior inclusion probability is \(\mathrm{PIP}_j=P(C_j\mid D)\), where \(D\) is the observed data. For a chosen target \(q\), with \(0<q\leq 1\), a credible set is a subset \(S\) of the candidate variants whose posterior probability of containing the causal variant is at least \(q\). Under a model in which exactly one candidate variant is causal, this probability is \(\sum_{v_j\in S}\mathrm{PIP}_j\).

This definition uses the single-causal-variant setup: exactly one of the listed variants is causal, and the candidate list exhausts the possibilities under the model. Consequently, the PIPs sum to 1. The target \(q\) is selected before constructing the set; a common choice is \(q=0.95\). A credible set need not have exactly that mass: adding whole variants generally makes its achieved mass equal to or greater than the target.

The Ranking-and-Accumulation Rule

Write the PIPs in non-increasing order as \(\mathrm{PIP}_{(1)}\geq\mathrm{PIP}_{(2)}\geq\cdots\geq\mathrm{PIP}_{(m)}\). Define the cumulative mass after the first \(k\) ranked variants as \(A_k=\sum_{r=1}^{k}\mathrm{PIP}_{(r)}\). The smallest \(k\) for which \(A_k\geq q\) determines the stopping point. If the probabilities are distinct, the first \(k\) ranked variants form the usual smallest-cardinality set obtained by this rule.

k=\min\left\{r:\sum_{j=1}^{r}\mathrm{PIP}_{(j)}\geq q\right\}, \qquad S=\{v_{(1)},\ldots,v_{(k)}\}.

When a PIP tie straddles the stopping point, there are two common reporting choices. A tie-breaking rule selects only as many members of the tied group as are needed to reach the target; the selected members must be identified by an explicit, reproducible rule. A tie-complete rule includes every variant whose PIP equals the PIP at the stopping point. This avoids arbitrarily distinguishing equally ranked variants, but can yield a larger set and more posterior mass than the target requires. Neither choice changes the underlying PIPs; it changes how the set is reported.

Check the candidate space.
Confirm that the PIPs refer to the same single-causal-variant model and sum to 1 over the candidate variants.

Choose the target.
Specify \(q\), such as 0.95, before applying the accumulation rule.

Rank and accumulate.
Order PIPs from largest to smallest and calculate each cumulative sum until it first reaches or exceeds \(q\).

Resolve and report ties.
State whether a tied boundary is resolved by a reproducible tie-break or by including the whole tied group; report the achieved mass.

Why Summing PIPs Gives the Set Probability

The accumulation rule relies on a direct probability identity. Under the single-causal-variant model, the events \(C_1,\ldots,C_m\) are mutually exclusive: two distinct candidate variants cannot both be the one causal variant. The event that the causal variant belongs to \(S\) is the union of the corresponding events \(C_j\), for \(v_j\in S\). Since those events are disjoint, the probability of their union is the sum of their probabilities.

P\left(\bigcup_{v_j\in S}C_j\mid D\right) =\sum_{v_j\in S}P(C_j\mid D) =\sum_{v_j\in S}\mathrm{PIP}_j.

This identity explains why sorting works: among all sets of a fixed number of variants, taking the variants with the largest PIPs gives the greatest sum. To see this, consider a set containing a lower PIP \(a\) while omitting a higher PIP \(b\), where \(b\geq a\). Replacing the variant with PIP \(a\) by the one with PIP \(b\) changes the set's mass by \(b-a\geq 0\). Repeating such replacements produces the top-ranked set without decreasing its mass. Thus the ranking rule reaches the target using as few variants as possible, apart from choices among equal PIPs.

Worked Constructions

Worked Example: Reaching a 0.90 Target With Distinct PIPs

Suppose six hypothetical variants have the following PIPs, already listed in decreasing order: \(0.42, 0.27, 0.16, 0.09, 0.04,\) and \(0.02\). They sum to 1. Construct a set for target \(q=0.90\).

Calculate the cumulative probabilities in order: \(0.42\); \(0.42+0.27=0.69\); \(0.69+0.16=0.85\); and \(0.85+0.09=0.94\). The first three variants have mass \(0.85\), which is below \(0.90\); the first four have mass \(0.94\), which reaches the target.

A_3=0.42+0.27+0.16=0.85<0.90,\qquad A_4=0.85+0.09=0.94\geq0.90.

The credible set therefore contains the four highest-PIP variants and has achieved posterior mass \(0.94\). Its mass exceeds \(0.90\) because variants are included as whole units; the rule does not split a variant's probability to make the sum exactly equal to the target.

Worked Example: A Tie at the Selection Boundary

Consider five hypothetical variants with PIPs \(0.40, 0.20, 0.20, 0.10,\) and \(0.10\), and target \(q=0.55\). After the top variant, cumulative mass is \(0.40\). Either of the two variants tied at \(0.20\) would bring the mass to \(0.60\), which exceeds \(0.55\).

With a tie-breaking rule, one of the \(0.20\) variants is selected along with the \(0.40\) variant, giving a two-variant set with mass \(0.40+0.20=0.60\). Because the tied variants have equal PIP, the model does not favor one over the other. A tie-break should therefore be operational rather than presented as a biological distinction—for example, selecting the variant with the smaller identifier, while recording that this is a reporting convention.

With a tie-complete rule, both variants at the boundary PIP of \(0.20\) are included. The set then contains three variants and has mass \(0.40+0.20+0.20=0.80\). Both sets reach the requested target; the tie-complete set is larger and has greater mass. The report must say which convention was used.

Worked Example: A Target Exactly Reached by One of Two Tied Variants

Suppose five hypothetical variants have PIPs \(0.50, 0.15, 0.15, 0.10,\) and \(0.10\), and the target is \(q=0.65\). The highest-PIP variant contributes \(0.50\). Adding either member of the tied \(0.15\) group gives \(0.50+0.15=0.65\), exactly the target.

A tie-breaking rule can therefore produce a two-variant set with mass exactly \(0.65\). The choice between the two \(0.15\) variants is not determined by their PIPs, so the selected identity depends on the stated tie-break. The tie-complete rule adds both tied variants, producing a three-variant set with mass \(0.50+0.15+0.15=0.80\). The exact equality in the first result does not remove the tie: either tied variant can occupy the second position in a ranked list.

If these displayed probabilities had been rounded, the construction should use the unrounded PIPs when available. Values that appear equal after rounding need not be an actual tie, and rounding before accumulation can also move a cumulative sum across the target.

Reporting and Interpretation

A concise report should state the target, the selected variants, the tie rule if relevant, and the achieved posterior mass. For example: “Using a tie-complete rule and target \(q=0.95\), the set contains the listed variants and has summed PIP 0.97.” Under the specified model, that sum is the posterior probability, conditional on the data, that the single causal variant is among those listed. It is not a claim that each listed variant is causal, nor does it imply that the variants are equally plausible.

The calculation is conditional on the posterior model and its candidate variants. In particular, the sum-to-one check is appropriate only when exactly one causal variant is assumed and the candidate list exhausts the modeled possibilities. If a model assigns posterior probability to a no-causal-variant state or to possibilities outside the list, that probability cannot be silently redistributed among the listed variants. The meaning and construction of a set must then account for those states.

Common Mistake

A frequent error is to stop as soon as one variant has a high PIP and call it a 95% credible set. For instance, a top PIP of \(0.70\) does not by itself reach a target of \(0.95\). The correction is to sum PIPs in ranked order and continue until the cumulative mass is at least \(0.95\). Another error is to include one member of a boundary tie without saying how it was chosen. State the tie rule, or include the whole tied group, and report the resulting set mass.

Key takeaway: Rank the variant PIPs, accumulate them until they reach the prespecified target, and report the achieved mass. At a tied boundary, make the tie rule explicit: including the full tie is transparent but can enlarge the set, while selecting only part of it requires a reproducible convention.

Check Your Understanding

For each question, distinguish the cumulative-mass calculation from the choice of tie rule.

For PIPs \(0.50, 0.30, 0.12,\) and \(0.08\), how many top-ranked variants are needed to reach \(q=0.90\), and what mass do they sum to?
Why can the posterior probability that a set contains the causal variant be calculated by adding its PIPs under the single-causal-variant model?
A target is crossed partway through a group of variants tied at the boundary. What is the difference between a tie-breaking rule and a tie-complete rule?
Why should a report include the achieved posterior mass rather than only naming the target?
If PIPs sum to less than 1 because the model includes a no-causal-variant state, why is it not valid to treat the listed variant PIPs as though they necessarily exhaust all posterior probability?