Posterior-Weighted Annotation Enrichment at a Locus

Why Weight Annotation by Fine-Mapping Posterior Probability?

A locus can contain many variants that overlap a functional annotation, but those variants need not be equally plausible causal candidates. Counting overlaps treats a variant with negligible fine-mapping support like one with most of the posterior probability. Posterior-weighted annotation instead asks how much of the locus’s causal-variant probability lies on annotated variants, then compares that amount with an appropriate background expectation.

The comparison depends on the background. A regulatory annotation may cover a modest fraction of the locus in base pairs but contain a large fraction of its observed variants. Comparing posterior mass with the base-pair fraction would then ignore variant density. A useful locus-level background is built from the same eligible variant opportunities used in fine-mapping, or from a carefully matched baseline when those opportunities differ in relevant ways.

Definition and Estimand

The posterior probabilities should cover the complete variant set represented by the fine-mapping model, so that \(\sum_j\pi_j=1\). The background weights also sum to one, but they describe a different distribution: the reference distribution of variant opportunities under the chosen comparison. With a uniform background over the \(M\) eligible variants, \(b_j=1/M\), so \(p_0\) is the fraction of those variants that are annotated. With nonuniform weights, the background can reflect prespecified differences in variant opportunity or baseline causal probability.

The ratio \(E\) compares two proportions on the same scale. For example, \(E=1.5\) means posterior probability mass on the annotation is 1.5 times its prevalence under the specified background. The difference \(q-p_0\) is an absolute contrast in probability points. Neither quantity, by itself, is a p-value or proof that the annotation caused the association.

Deriving the Comparison

The weighting has a direct probability interpretation. If \(C\) denotes the causal variant, then \(a_C\) is one when the causal variant has the annotation and zero otherwise. Averaging that indicator over the posterior distribution gives the probability that the causal variant is annotated, conditional on the data and fine-mapping model. Averaging it over the background distribution gives the reference prevalence.

Thus the enrichment ratio is the posterior probability of an annotated causal variant divided by the corresponding background probability. This derivation also shows why raw counts in a credible set are not the target: they assign equal weight to variants even when their posterior probabilities differ.

Under a uniform background, the relationship to variant density is explicit:

The denominator counts eligible variant sites, not bases. If an annotation occupies 20% of the locus but contains 80% of its eligible variants, its uniform-variant background prevalence is 80%, not 20%. A different background may be preferable if uniform sampling of eligible variants is not a defensible reference; the key is to state what distribution the weights represent.

Choose a Background That Reflects Variant Opportunity

The background defines the question being answered. For a locus-level comparison, a natural starting point is the set of all variants eligible for fine-mapping in the locus, not only the variants selected into a reported credible set. The eligible set should use compatible coordinate, allele, and quality rules, and should reflect the variant calls or reference panel underlying the analysis as closely as possible. Otherwise, apparent enrichment can arise because the annotation and background were evaluated over different sets of opportunities.

A uniform distribution over eligible variant sites answers: “How concentrated is posterior mass on annotated variants relative to the fraction of eligible variants that are annotated?” It automatically gives more representation to regions with more observed variant sites. It does not automatically account for every possible difference in opportunity, such as substantial variation in callability or a deliberately nonuniform pre-data prior. If such differences matter to the scientific comparison, define weights \(b_j\) that represent the relevant baseline and explain how they were set.

A genome-wide base-pair fraction is usually not an adequate substitute for a variant-level background. If the annotation covers a small share of bases in a variant-dense region, its share of eligible variants can be much larger than its share of bases. Conversely, a broad annotation in a variant-poor region can have high base-pair coverage but low variant prevalence. The comparison should use the scale of the question: annotation among variant opportunities, not annotation among genomic bases.

Nonuniform Background Weights

Uniform weights are not mandatory. If the intended null comparison gives some eligible variants more baseline opportunity than others, encode that distribution explicitly. For example, a prespecified baseline might use weights adjusted for relevant variant-opportunity strata. The weights must be nonnegative and normalized to sum to one. They should not be chosen after inspecting which choice makes the enrichment largest.

When \(b_j\) represents a baseline distribution for the causal variant, it is useful to distinguish it from \(\pi_j\). The posterior \(\pi_j\) incorporates the observed association data and the fine-mapping model; \(b_j\) is the reference distribution used for comparison. A background constructed from the posterior itself would generally answer a different question and can make the contrast circular.

Credible-Set Coverage and Model Dependence

A credible set often contains less than 100% of the locus posterior probability. If a reported set contains total posterior mass \(c\), summing \(\pi_j a_j\) only over that set gives the annotated posterior mass included in the set, not necessarily the full-locus value of \(q\). If the annotation status of omitted variants is known, include their contributions. If only the credible-set members are available, dividing their annotated mass by \(c\) gives the posterior probability of annotation conditional on the causal variant being in that set; it is not the unconditional locus-wide \(q\). State which quantity is reported and use a background matching that conditional comparison if calculating enrichment from the restricted set.

The posterior is also conditional on the fine-mapping model. If annotation information was used to construct variant priors, then the posterior mass on that annotation may partly reflect that input. The resulting \(q\) remains a model-based posterior quantity, but it is not independent confirmation of the annotation. For an evaluation intended to assess annotation support without feeding that annotation into the result, use a fine-mapping analysis that does not use it as a prior input, or clearly disclose the dependence.

If \(p_0=0\), the ratio \(q/p_0\) is undefined; report \(q\) and explain that the chosen background contains no annotated opportunities rather than reporting an infinite fold enrichment. If \(p_0\) is very small, the ratio can also be unstable or strikingly large, so report \(q\), \(p_0\), and the difference alongside the ratio. These quantities describe concentration relative to the selected background; they do not establish that the annotation is causal or that the enrichment would replicate.

Common Mistakes and Interpretation

• Counting variants equally. The wrong reasoning is that three annotated variants among six candidates means 50% of causal probability is annotated. The correction is to calculate \(\sum_j\pi_j a_j\); in the first example, three annotated variants carry 59% of posterior mass.
• Using base-pair coverage as variant prevalence. The wrong reasoning is to use the fraction of locus length annotated as the null expectation for variant overlap. The correction is to calculate prevalence across eligible variant sites, or use a justified weighted variant background.
• Using only selected credible-set members as the background. This can condition the reference on posterior selection and overstate contrast. Prefer the full eligible locus set, or explicitly define and interpret a conditional credible-set comparison.
• Calling an enrichment ratio a significance test. A ratio such as 1.18 does not itself provide a p-value, a probability that the annotation is causal, or evidence against a null hypothesis. It is a comparison of posterior mass with a specified reference prevalence.