Summarizing Annotation Support Across a Credible Set

Why Summarize Annotation Across a Credible Set?

A fine-mapping analysis can leave posterior support spread across several variants rather than concentrated on one. Annotation adds information about what those variants overlap or might affect, but a list of labels does not by itself summarize how much posterior support lies in each class. A useful first calculation is to weight each variant’s annotation by its posterior inclusion probability and sum within class.

That calculation is a summary of fine-mapping posterior support, not a direct measurement of biological activity. Its interpretation depends on the posterior model, the variants included, the annotation definitions, and whether the classes form a complete, non-overlapping partition. In particular, a large regulatory-class summary does not by itself establish that a regulatory mechanism is causal.

Definition: Let \(C\) be a set of fine-mapped variants, and let \(p_i=P(Z_i=1\mid D)\) be the posterior inclusion probability (PIP) for variant \(i\), where \(Z_i=1\) means that the variant is causal under the fine-mapping model and \(D\) denotes the association data and other inputs to that model. For an annotation class \(k\), let \(A_{ik}=1\) if variant \(i\) is assigned to class \(k\), and \(A_{ik}=0\) otherwise. The class posterior mass within \(C\) is \(M_k=\sum_{i\in C}p_iA_{ik}\). If \(T_C=\sum_{i\in C}p_i>0\), the normalized within-set fraction is \(F_k=M_k/T_C\).

The word “mass” is useful because \(M_k\) is a sum of posterior probabilities. With a single-causal-variant model, the PIPs across the full variant universe sum to one. A credible set selected to capture a target amount of posterior probability may have \(T_C\) below one because it omits variants outside the set. In a multi-causal-variant model, PIPs are marginal probabilities and may sum to more than one; their sum is an expected number of causal variants, not a probability. Thus \(M_k\) and \(F_k\) need to be interpreted in light of the fine-mapping model and the set being summarized.

Calculating Class Mass and Its Fraction

For a mutually exclusive, exhaustive annotation partition, each variant receives exactly one class, such as coding, regulatory, or other. In this case, the class masses sum to the total PIP in the set, and the normalized fractions sum to one. “Other” must be defined: it might include variants with other specified consequences, or variants not assigned to the coding or regulatory classes. Those choices change what the summary describes.

For example, if a variant overlaps both a coding consequence and a regulatory element, it does not automatically belong to a unique class. One can report overlapping class masses separately, or adopt a stated rule for assigning a single class. Without such a rule, the classes are not a partition, and their fractions need not sum to one. The choice should be visible rather than hidden in the arithmetic.

Worked Example: A Single-Causal-Variant Credible Set

Suppose a hypothetical fine-mapping analysis uses a single-causal-variant model. Its reported credible set contains four variants with PIPs 0.42, 0.28, 0.17, and 0.08. For this example, assume the annotations assign variant 1 to coding, variants 2 and 3 to regulatory, and variant 4 to other. These are mutually exclusive and exhaustive assignments for the set.

First calculate the total posterior mass in the set:

$$ T_C=0.42+0.28+0.17+0.08=0.95. $$

The class masses are the sums of the PIPs assigned to each class:

M_{\text{coding}}=0.42,\qquad M_{\text{regulatory}}=0.28+0.17=0.45,\qquad M_{\text{other}}=0.08.

Dividing each mass by \(T_C=0.95\) gives the fractions within this credible set:

F_{\text{coding}}=\frac{0.42}{0.95}=0.4421,\qquad F_{\text{regulatory}}=\frac{0.45}{0.95}=0.4737,\qquad F_{\text{other}}=\frac{0.08}{0.95}=0.0842.

The fractions sum to 1.0000, subject to rounding. Under the stated single-causal-variant model and the stated class assignments, the set contains 0.45 posterior mass assigned to regulatory variants; among the 0.95 mass represented in the set, 47.37% is assigned to that class. The raw class mass is not the same as the normalized fraction: the former retains the scale of the PIPs, while the latter conditions the summary on the mass included in \(C\). Neither calculation says that a regulatory mechanism has a 47.37% probability of being causal.

Why the Sum Has a Posterior Interpretation

The arithmetic follows from the definition of a PIP. Let \(Z_i\) be the indicator that variant \(i\) is causal, and suppose the class assignments are fixed for this calculation. The number of causal variants in class \(k\) and in set \(C\) is \(N_k=\sum_{i\in C}Z_iA_{ik}\). Taking its posterior expectation gives:

E(N_k\mid D) =E\left(\sum_{i\in C}Z_iA_{ik}\,\middle|\,D\right) =\sum_{i\in C}A_{ik}E(Z_i\mid D) =\sum_{i\in C}A_{ik}p_i =M_k.

This derivation uses linearity of expectation; it does not require the causal indicators \(Z_i\) to be independent. Therefore, in a multi-causal-variant model, \(M_k\) is the posterior expected number of causal variants in the class among the variants in \(C\). It is not generally the posterior probability that at least one variant in that class is causal. A sum of marginal probabilities does not, by itself, give the probability of a union of events.

There is an important special case. If the model specifies exactly one causal variant in the full variant universe and the annotations are mutually exclusive, then the probability that the causal variant belongs to class \(k\) is the sum of the PIPs for variants assigned to \(k\). If \(C\) omits variants, \(M_k\) is the probability mass for class \(k\) represented inside \(C\), not necessarily the class probability over the entire variant universe. The normalized fraction \(F_k\) describes the distribution of the included PIP mass across classes; it is a conditional-style summary within the chosen set, not a substitute for the full posterior.

Worked Example: Multi-Causal PIPs and Expected Counts

Consider a hypothetical locus analyzed with a multi-causal-variant model. A set \(C\) contains five variants with PIPs 0.40, 0.30, 0.25, 0.35, and 0.30. Suppose the first two are assigned to coding, the third and fourth to regulatory, and the fifth to other. These assignments are mutually exclusive and exhaustive.

The total PIP is:

$$ T_C=0.40+0.30+0.25+0.35+0.30=1.60. $$

The class masses and within-set fractions are:

M_{\text{coding}}=0.40+0.30=0.70,\qquad F_{\text{coding}}=\frac{0.70}{1.60}=0.4375; $$ $$ M_{\text{regulatory}}=0.25+0.35=0.60,\qquad F_{\text{regulatory}}=\frac{0.60}{1.60}=0.3750; $$ $$ M_{\text{other}}=0.30,\qquad F_{\text{other}}=\frac{0.30}{1.60}=0.1875.

The class masses sum to 1.60, and the fractions sum to 1.0000. In this multi-causal model, 0.60 is the posterior expected number of causal variants in the regulatory class among these five variants, and 37.50% is the regulatory share of their total expected causal count. The regulatory mass is not a 60% probability that at least one regulatory variant is causal. The denominator exceeds one because the sum of PIPs can exceed one when more than one causal variant may be present.

Why can marginal PIPs fail to determine the probability of any causal regulatory variant? Suppose, for illustration, that two regulatory variants each have PIP 0.40. One possible joint posterior assigns probability 0.40 to both being causal and 0.60 to neither being causal. The probability that at least one is causal is then 0.40, while the expected number is 0.80. Another joint posterior assigns probability 0.40 to only the first being causal, probability 0.40 to only the second being causal, and probability 0.20 to neither being causal. The two PIPs remain 0.40 each, but the probability that at least one is causal is now 0.80. The sum of PIPs is the same in both cases; the joint posterior structure differs.

Overlapping Labels and Unassigned Variants

Annotation tracks can overlap. A variant may have a coding consequence and also overlap a regulatory element; a regulatory annotation may itself include several kinds of evidence. If each class is summarized independently, a variant can contribute its PIP to more than one class. That can be a useful description of track overlap, but the class masses will not necessarily add to \(T_C\), and dividing each by \(T_C\) does not produce fractions that sum to one.

A different issue arises when some variants have no available annotation. If the intended comparison is a complete partition of the credible set, include an explicit unclassified or unannotated class. Otherwise, report that the listed classes cover only part of the set and state the denominator used. Treating missing annotation as “other” is defensible only if “other” is explicitly defined to include missing or unassigned variants; it should not be presented as an observed biological consequence.

Worked Example: Handling Overlap Transparently

Suppose a hypothetical set has three variants with PIPs 0.50, 0.30, and 0.20, so \(T_C=1.00\). Variant 1 has both a coding annotation and regulatory-element overlap; variant 2 has regulatory-element overlap only; variant 3 has neither of these annotations.

If coding and regulatory are reported as overlapping tracks, their masses are:

M_{\text{coding}}=0.50,\qquad M_{\text{regulatory}}=0.50+0.30=0.80.

The two reported fractions are 0.50 and 0.80 because \(T_C=1.00\), but they sum to 1.30. This is not an arithmetic error: variant 1 contributes to both tracks. The summaries describe separate annotation overlaps, not shares of one partition. The remaining 0.20 has neither of the two listed annotations.

If a single-class display is required, suppose the reporting rule gives coding priority whenever a variant has both labels. The assigned classes are then coding for variant 1, regulatory for variant 2, and other for variant 3. Their masses are 0.50, 0.30, and 0.20, respectively, and their fractions sum to one. This result depends on the priority rule. It is a convenient classification convention, not evidence that variant 1 has no regulatory overlap.

What These Summaries Do—and Do Not—Establish

A posterior-mass summary is a compact way to describe how fine-mapping support is distributed across annotation categories. It can help readers see whether the included posterior support is concentrated on variants with coding annotations, regulatory annotations, or neither of those specified labels. A clear report gives the class definitions, whether labels overlap, the posterior model’s causal-variant assumptions, the variants included, the total PIP \(T_C\), and the denominator for every fraction.

The summary is not an enrichment test: it does not compare the class’s posterior mass with how much of the relevant variant universe belongs to that class. It is also not a calibrated mechanism probability. An annotation class is a label attached to variants, whereas a causal mechanism is a biological explanation for how genetic variation affects a trait. A coding annotation may be ambiguous about its functional effect; a regulatory overlap does not establish that the element acts on the relevant gene or in the relevant tissue. Those distinctions matter even if the PIP calculations are exact.

Annotation uncertainty also matters. The formulas above treat \(A_{ik}\) as fixed. If an annotation is incomplete, uncertain, or dependent on a tissue-specific assay, the computed mass summarizes the chosen annotation calls, not all possible true annotations. State the annotation source and scope sufficiently for the class definitions to be understood; do not turn absence of a recorded label into evidence that a variant has no function.

Common Mistakes

Calling a class fraction the probability of a mechanism. A 60% regulatory fraction describes a share of included PIP mass under the stated coding of annotations. It does not establish a 60% probability that a regulatory mechanism causes the trait.
Assuming PIPs sum to one in every model. They sum to one across the full variant universe under a model with exactly one causal variant. In a multi-causal model, their sum can exceed one and represents an expected causal count.
Using “fraction” when classes overlap without explanation. Overlapping class summaries can exceed 100% in total. Label them as separate track summaries, or define a rule that makes the classes mutually exclusive.
Leaving the denominator implicit. A fraction using the total PIP in the credible set differs from a fraction using total PIP across all analyzed variants. State which quantity is used.
Equating unannotated with biologically inactive. An absent annotation can reflect the scope or availability of the annotation data. It is not proof that the variant has no functional effect.

Key takeaway: For each annotation class, sum the PIPs of its assigned variants. Report raw mass and, when useful, the fraction of an explicitly stated total PIP. Interpret these as summaries of posterior support attached to annotations—not, without additional assumptions and evidence, as probabilities of causal mechanisms.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following questions.

A credible set has total PIP 0.90, and coding variants contribute PIP 0.27. What are the coding class mass and the coding fraction within the set? State what each number summarizes.
Why can the sum of PIPs in a multi-causal credible set exceed one? What does that sum represent?
Two annotation classes overlap on one variant. Why might their normalized class summaries sum to more than one?
Under what assumptions can a class’s summed PIP be interpreted as the posterior probability that the single causal variant belongs to that class?
Explain why a regulatory-class fraction is not, by itself, the probability that a regulatory mechanism is causal.