Posterior Inclusion Probabilities Versus Credible Sets

Two summaries, two different questions

Fine-mapping reports often place a variant’s posterior inclusion probability beside a credible set containing that variant. The numbers answer different questions. A posterior inclusion probability (PIP) describes one variant: how much posterior probability is assigned to configurations that include it as causal. A credible set describes a collection of variants: how much posterior probability is assigned to a specified event represented by that collection, under the model and construction used.

This distinction matters when no single variant dominates the evidence. A variant can have a moderate PIP and still belong to a set with high posterior mass, because the set’s mass is distributed across its members. Conversely, a large set-level probability does not imply that every member is individually likely to be causal.

A common construction ranks variants by posterior probability for a single causal signal and adds them until the cumulative posterior probability reaches a target such as 0.95. The resulting set is not a list in which each member has 95% probability of being causal. Rather, under the construction’s assumptions, the posterior probability that the causal variant is among the selected members is at least the target. Ties or discrete posterior masses can make the achieved probability exceed the target.

The exact event behind a credible set depends on the model. With exactly one causal variant and no separate null state, a set’s posterior mass is the sum of the PIPs of its members. With multiple causal variants, the event “at least one member of this set is causal” is generally not given by adding PIPs. Some methods instead construct a credible set for each modeled signal; that set’s target concerns that signal’s posterior distribution, not automatically the probability that the whole region contains a causal variant.

When set mass equals a sum of PIPs

Suppose the model specifies exactly one causal variant among \(m\) candidates. The events “variant 1 is causal,” “variant 2 is causal,” and so on are mutually exclusive and exhaustive. If \(S\) is any subset of candidates, the event that the causal variant lies in \(S\) is the disjoint union of the events that each particular member is causal. Therefore, its posterior probability is the sum of their PIPs.

This identity depends on the exactly-one-causal assumption. It is not a general rule for multi-causal models, because two or more members can be causal at the same time. In that setting, summing PIPs gives the posterior expected number of causal variants in \(S\), not the probability that \(S\) contains at least one.

Why PIPs do not add to a region probability with multiple causes

For a set \(S\), define \(I_j=1\) if variant \(j\) is causal and \(I_j=0\) otherwise. The number of causal variants in \(S\) is \(\sum_{j\in S}I_j\). Taking its posterior expectation gives the sum of the PIPs. But the event of interest for an at-least-one-causal probability is different: it occurs if this count is 1 or more. A count can exceed 1, so its expectation is not generally its probability of being positive.

For two variants \(A\) and \(B\), the distinction follows from inclusion-exclusion:

The joint probability is subtracted because outcomes in which both are causal are counted twice in the sum of the individual PIPs. With more variants, the same issue involves further overlaps. This is why the joint posterior over causal configurations matters when interpreting a region-level event.

Reading variant and set summaries together

A PIP is useful for comparing individual candidates: it records how posterior support is allocated to each variant. A credible set is useful for representing uncertainty about a signal when the data and model do not resolve one candidate. In a single-causal model, a high-mass set can contain several variants with modest PIPs because their probabilities accumulate. Reporting both summaries makes the distribution of support more visible than either alone.

In multi-signal analyses, check what the set was constructed to cover. A per-signal credible set may be formed from the posterior probabilities for one modeled effect. The overall PIP for a variant instead refers to whether it is included anywhere in the causal configuration. These quantities need not coincide: a variant can receive support across signals, and signal-specific summaries are not automatically interchangeable with overall inclusion probabilities.

Assumptions, diagnostics, and limitations

All of these interpretations are conditional on the model, including its candidate variants, causal-configuration assumptions, priors, and representation of the data. A posterior credible-set target is not by itself a guarantee of empirical coverage across repeated studies, nor does it establish that the model is correctly specified. The introductory material on credible-set probability and calibration distinguishes posterior mass conditional on a model from performance assessed across repeated data sets.

Before interpreting a reported set, identify its probability target and construction. Check whether it concerns one causal variant, one modeled signal, or the event that at least one member of a region is causal. Also check whether a no-causal-variant state is possible. If it is, posterior probabilities over candidate variants may sum to less than 1; a set can reach a target only under the method’s stated convention, such as conditioning on a signal being present. Do not silently treat the remaining probability as assigned to one of the listed variants.

A useful diagnostic is to compare the set’s reported cumulative mass with the probabilities actually used to construct it. In a single-causal model, these are the member PIPs. If the model permits multiple causal variants, do not expect the sum of overall PIPs to equal the at-least-one probability; calculate the latter from posterior configurations or a valid equivalent summary. A mismatch between a reported set interpretation and its construction is a warning about interpretation, even when the arithmetic is correct.

Credible sets also have limitations as summaries. The target alone does not show whether posterior mass is concentrated on a few members or spread across many, and the same target can produce sets of very different sizes. PIPs add useful variant-level detail, but neither a PIP nor set membership proves causality. Both are model-based measures of posterior uncertainty, not statements that an association is causal in the biological or experimental sense.