Constructing Bayesian Credible Sets

From Variant Probabilities to a Credible Set

Fine-mapping often leaves uncertainty about which variant in a region is causal. A posterior inclusion probability (PIP) summarizes that uncertainty for one variant: it is the posterior probability that the variant is causal under the specified model and data. A credible set collects variants whose combined posterior support reaches a chosen target, such as 0.95. The aim here is not to declare one variant causal, but to find a compact set that retains at least the target posterior mass.

The construction below applies to the single-causal-variant setting established in the earlier material on the single-causal-variant posterior and PIP calculation. Its key condition is that exactly one causal variant is assumed among the \(m\) candidate variants, so the causal-variant events are mutually exclusive and exhaustive. The set’s posterior mass is then the sum of the PIPs it contains.

Definition: Let \(Z_j=1\) mean that candidate variant \(j\) is the causal variant, and let \(\mathrm{PIP}_j=P(Z_j=1\mid D)\), where \(D\) denotes the data and the specified model. Under a model with exactly one causal variant among the candidates, \(\sum_{j=1}^{m}\mathrm{PIP}_j=1\). For a prespecified target \(q\), where \(0<q\leq 1\), a credible set \(S\) satisfies \(\sum_{j\in S}\mathrm{PIP}_j\geq q\). The smallest credible set is a set satisfying this condition with the fewest variants.

The threshold \(q\) is a chosen posterior-mass target, not a data-derived significance level. A 95% credible set, for example, is constructed with \(q=0.95\). Its interpretation depends on the posterior model: conditional on the data and that model, the probability that the causal variant is in the set is at least 0.95. This is not a claim that the procedure has 95% frequentist coverage across repeated studies.

The Ranking and Accumulation Rule

Order the variants so their PIPs are nonincreasing: \(\mathrm{PIP}_{(1)}\geq \mathrm{PIP}_{(2)}\geq\cdots\geq \mathrm{PIP}_{(m)}\). The parentheses indicate rank, not the original variant label. Add probabilities in that order until the cumulative sum first reaches or exceeds \(q\). If that occurs at rank \(k\), the selected set contains the \(k\) variants ranked first through \(k\).

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

The inequality is “at least” rather than “equal to” because posterior probabilities usually do not add to the target exactly. The cumulative sum just before the final variant is below \(q\); adding the next-ranked variant crosses the threshold with the fewest variants in this ranked construction.

Worked Example: Reaching a 90% Target

Suppose a hypothetical region has five candidate variants and the fitted single-causal-variant model gives PIPs 0.42, 0.27, 0.16, 0.09, and 0.06. These sum to \(1.00\), consistent with the model’s assumption that one of the five candidates is causal. Construct a set with target \(q=0.90\).

The PIPs are already in descending order. Accumulating them gives \(0.42\), then \(0.42+0.27=0.69\), then \(0.69+0.16=0.85\), and then \(0.85+0.09=0.94\). The first three variants fall short of the target, while the first four reach it. Thus the credible set is the four variants with PIPs 0.42, 0.27, 0.16, and 0.09, with total posterior mass \(0.94\).

It is a smallest set: the three largest PIPs sum to 0.85, and no other set of three variants can have a larger total. Under this model and conditional on the data, the posterior probability that the causal variant is among these four is 0.94. The set is larger than necessary for a 90% target only in the sense that discrete variant probabilities make its mass exceed 0.90.

Why Ranking Gives a Minimum-Size Set

The ranking rule is not merely a convenient convention. For any fixed number \(r\) of variants, the \(r\) largest PIPs have at least as much total probability as any other \(r\)-variant set. Therefore, if those top \(r\) probabilities do not reach \(q\), no set of \(r\) variants can reach it.

Justification: Consider any set \(A\) of \(r\) variants. Write the ranks of its members as \(i_1<i_2<\cdots<i_r\). Since \(i_t\geq t\) and the ranked PIPs are nonincreasing, \(\mathrm{PIP}_{(i_t)}\leq \mathrm{PIP}_{(t)}\) for each \(t\). Summing these inequalities gives \(\sum_{j\in A}\mathrm{PIP}_j\leq\sum_{t=1}^{r}\mathrm{PIP}_{(t)}\). Consequently, if the top \(r\) PIPs sum to less than \(q\), no \(r\)-variant set can be credible at target \(q\). The first rank \(k\) whose cumulative sum reaches \(q\) therefore gives the minimum possible cardinality.

This argument establishes minimum size, not uniqueness. When several variants tie at the boundary, different sets of the same minimum size can have the same posterior mass. A tie-breaking rule may be needed to produce one reported set, but a variant’s biological annotation or genomic position should not be treated as extra posterior evidence unless it was part of the model.

Worked Example: A Tie at the Cutoff

For a hypothetical set of five variants, suppose the PIPs are 0.30, 0.25, 0.20, 0.20, and 0.05. They sum to 1.00. With target \(q=0.75\), accumulation gives \(0.30+0.25=0.55\), followed by \(0.55+0.20=0.75\). Three variants suffice, and two do not.

The two variants with PIP 0.20 are tied. Either one can be included with the first two variants, giving a three-variant set with posterior mass 0.75. Both choices have minimum size. Including both tied variants instead produces a four-variant set with mass \(0.30+0.25+0.20+0.20=0.95\); that is a valid set, but it is not the smallest set for the 0.75 target.

A report should make any boundary tie clear. One option is to state that either tied variant can occupy the final position under the chosen tie rule. Including every variant tied at the boundary avoids arbitrarily choosing between them, but increases the set size and should not be described as minimum-cardinality when a smaller tied choice reaches the target.

Check the Model and the Calculation

Before interpreting a set, verify what the probabilities refer to. The sum-to-one property used above holds when the model assigns exactly one causal variant among the listed candidates. The candidate set, likelihood, prior probabilities, and conditioning assumptions determine the PIPs; changing them can change both the ranking and the resulting set. A set cannot recover a causal variant omitted from the candidate list.

The procedure also requires care when the model allows more than one causal variant. In that setting, a PIP is the posterior probability that a variant is included among the causal variants, and the PIPs need not sum to one. Their sum can represent the posterior expected number of included variants. Adding PIPs within a set then does not, in general, give the posterior probability that the set contains at least one causal variant. The single-causal-variant accumulation rule must not be given that interpretation outside its assumptions.

Worked Example: Why Multiple PIPs Need Not Be Set Probability

Consider a hypothetical model that allows two causal variants, with PIPs 0.80 and 0.70 for two variants. Their sum is \(0.80+0.70=1.50\), which is possible because both variants may be causal. It cannot be the probability that at least one of them is causal: probabilities cannot exceed 1.

To see the difference, suppose the posterior probability that both are causal is 0.50. Then the probability that at least one is causal is \(0.80+0.70-0.50=1.00\), using inclusion-exclusion. The sum of PIPs, 1.50, is not that union probability. Thus sorting and accumulating these PIPs to a target such as 0.95 would not establish a 95% posterior probability that the set contains a causal variant. A model-specific set probability is needed for that interpretation.

For the single-causal construction, useful checks include confirming that the PIPs refer to the same candidate variants and model, checking their total before rounding, and recomputing the cumulative sums from unrounded values. If rounded values appear to sum to less than 1, inspect the stored probabilities rather than silently rescaling them. If the total differs substantially from 1, investigate whether a null or outside-candidate state is included, whether probabilities were conditioned on a different event, or whether variants were omitted or duplicated. Any renormalization changes the stated posterior conditioning and should be made explicit.

A related issue is whether the target is attainable. If the candidate variants do not collectively carry posterior mass \(q\) under the stated conditioning, no subset can meet the target. Do not report the full candidate set as a \(q\)-credible set unless its mass actually reaches \(q\). Resolve the model or conditioning issue, or report that the target was not achieved.

What a Credible Set Does—and Does Not—Say

Approach	What it selects	Important distinction
Smallest posterior-mass credible set	The fewest highest-PIP variants whose cumulative posterior mass reaches \(q\)	Targets set-level posterior mass under the model
Single top variant	The variant with the largest PIP	Does not generally reach a chosen set-level target
Fixed PIP cutoff	Every variant whose PIP exceeds a chosen value	Does not generally guarantee a specified cumulative mass or minimum size

The credible set is a summary of posterior uncertainty, not a declaration that every included variant is causal. Nor does a high-mass set establish that the causal variant has been identified precisely: a large set can indicate substantial uncertainty. The number of variants and their identities depend on the fitted model and candidate list. The set also does not by itself establish causal effects, functional consequences, or repeated-sampling calibration.

A common mistake is to treat each PIP as a standalone inclusion rule—for example, to include only variants with PIP above 0.10 and call the result a 95% set. That reasoning confuses an individual variant’s posterior probability with the sum of posterior probabilities across a set. The correction is to specify \(q\), rank the PIPs, calculate cumulative mass, and stop at the first rank that reaches \(q\), subject to the model assumptions above.

Key takeaway: Under a model with exactly one causal variant among the candidates, rank PIPs from largest to smallest and include variants through the first cumulative sum at least \(q\). This gives a minimum-cardinality set with posterior mass at least \(q\) under that model.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A single-causal-variant model gives PIPs 0.50, 0.25, 0.15, and 0.10. What is the smallest set reaching \(q=0.90\), and what is its mass?
Why does sorting by PIP give a minimum-cardinality set rather than merely one possible set?
For PIPs 0.40, 0.30, 0.20, and 0.10, what smallest set reaches \(q=0.70\)? Does the set need to have exactly 0.70 mass?
A model permits multiple causal variants, and the selected variants have PIPs 0.75 and 0.65. Why is their sum not the posterior probability that at least one is causal?

Solutions. (1) The first three PIPs sum to \(0.50+0.25+0.15=0.90\), so the smallest set contains those three variants and has mass 0.90. The first two sum to 0.75, below the target. (2) For any fixed number \(r\), the \(r\) largest PIPs have a sum at least as large as any other \(r\) PIPs. Thus if the top \(r\) fall short, every \(r\)-variant set falls short. (3) The first two sum to \(0.40+0.30=0.70\), so they form the smallest set and reach the target exactly; exact equality is not required in general, only mass at least \(q\). (4) With multiple possible causal variants, inclusion events can overlap. The sum \(0.75+0.65=1.40\) counts their marginal probabilities and can exceed 1, whereas the probability that at least one is causal is a union probability that accounts for overlap.