Functional Annotation Priors for Causal Variants

Why Use Functional Annotations as Priors?

Fine-mapping compares candidate variants using association evidence, but candidates need not be equally plausible before that evidence is considered. A variant in a protein-altering sequence or a relevant regulatory element may be assigned a higher prior probability of being causal than a variant with no such annotation. The key is to use annotation as prior information, then update that information with the association data. An annotation is not a demonstration that a variant causes the trait.

Let \(\mathcal V=\{1,\ldots,m\}\) be the set of candidate variants at a locus. Let \(A_i\) denote the functional annotation vector for variant \(i\), and let \(D\) denote the association data used for fine-mapping. A prior describes relative plausibility before considering \(D\); a Bayes factor summarizes how well the data support one causal explanation relative to a reference explanation. Keeping these roles separate makes the analysis interpretable.

From Annotation Scores to Variant Priors

A common model maps each variant's annotations to a score. For example, let \(A_i\) contain indicators for a coding consequence and for overlap with a relevant regulatory element. A linear score is \(\eta_i=\gamma_0+\gamma^\mathsf{T}A_i\), where \(\gamma\) contains coefficients describing how annotations relate to prior causal plausibility. Transforming the score with an exponential gives a positive weight \(w_i=\exp(\eta_i)\). A positive coefficient raises the weight, relative to otherwise comparable variants; a negative coefficient lowers it.

For a model that assumes exactly one causal variant in the candidate set, normalize the weights to obtain prior probabilities. The normalization ensures that probabilities sum to one. The coefficients should be chosen or estimated using a defensible source of prior information; they are not automatically learned from the association evidence at the locus being fine-mapped.

Here \(C=i\) means variant \(i\) is the causal variant, and \(A\) denotes the annotation information for the candidate set. The prior odds between two variants are \(\pi_i/\pi_j=w_i/w_j=\exp(\eta_i-\eta_j)\). In this single-causal model, adding the same constant to every \(\eta_i\) multiplies all weights by the same amount and leaves the normalized priors unchanged. Thus, only relative scores matter.

Updating Annotation Priors With Bayes Factors

Under the single-causal-variant model, let \(BF_i\) be the Bayes factor for the hypothesis that variant \(i\) is causal, relative to a common reference model. The Bayes factor is the likelihood under that causal hypothesis divided by the likelihood under the reference model. Posterior probabilities are proportional to the product of prior probability and Bayes factor.

The second equality follows because the common prior-weight normalization \(\sum_j w_j\) cancels from numerator and denominator. Consequently, the posterior odds for two variants satisfy \(P(C=i\mid D,A)/P(C=j\mid D,A)=(BF_i/BF_j)(\pi_i/\pi_j)\). The association evidence changes the prior odds by the Bayes-factor ratio; neither the prior weights nor the Bayes factors alone are the posterior probabilities.

Annotation Priors for Multiple-Causal Configurations

When a model permits multiple causal variants, the object receiving a prior is a configuration \(C\), a subset of the candidate variants. One way to use annotation weights is to specify the prior conditional on the number of causal variants \(K=|C|\). For a fixed \(K=k\), assign a configuration a weight equal to the product of the weights of its included variants, then normalize across all size-\(k\) configurations. The normalizing constant is the sum of those products, denoted \(e_k(w)\).

For \(k=0\), the only configuration is the empty set, and its product and normalizing constant are both defined as \(1\). If the model also specifies a prior \(P(K=k)\) on causal count, then \(P(C\mid A)=P(K=|C|)P(C\mid K=|C|,A)\). This construction changes which variants are favored within a given causal count while leaving the specified distribution of causal counts unchanged. Other prior constructions are possible; the chosen one must be stated because different constructions imply different prior probabilities.

Let \(BF(C)\) be the Bayes factor for configuration \(C\) relative to a common reference model. The posterior over configurations is \(P(C\mid D,A)\propto BF(C)P(C\mid A)\), normalized over the configurations allowed by the model. In general, \(BF(C)\) must come from the likelihood for the whole configuration. It should not be assumed to equal the product of single-variant Bayes factors, particularly when variants are correlated.

Choosing and Interpreting Annotation Models

Annotation effects should have a defensible interpretation. A coefficient may be estimated from previously characterized variants, chosen from external evidence, or assigned through a prespecified model. Whichever approach is used, the annotation definitions, coefficient values, and normalization scheme affect the posterior and should be reported. If annotation effects are estimated, uncertainty in those estimates may also matter; treating estimated coefficients as fixed can understate uncertainty when the training evidence is limited.

Annotations should be defined consistently for the variants being compared. For example, a regulatory annotation relevant to one tissue may not be informative for a trait acting in another tissue. Missing or uncertain annotations also require an explicit treatment rather than silently being interpreted as evidence of no function. These choices concern the prior model; they do not replace checking that the likelihood and candidate set are appropriate for the fine-mapping analysis.

For a prior to represent information separate from the association likelihood, avoid using the same locus-level association result both to choose annotation weights and then again in the Bayes factor without a model that accounts for that reuse. Otherwise, evidence can be counted twice. External functional evidence can inform prior plausibility, but it still does not establish that a particular variant is the causal one for the trait under study.

Common Mistake: Treating Annotation as Proof

• Wrong reasoning: “This variant overlaps a functional element, so it must be causal.” An annotation can be imperfect, context-dependent, or shared by many variants.
• Correction: The annotation changes a prior probability according to the specified model. The posterior also depends on the association likelihood, other candidate variants, and model assumptions.
• Wrong reasoning: “The variant has the largest prior weight, so it should have the largest posterior probability.”
• Correction: Posterior comparisons use \(BF_iw_i\), not \(w_i\) alone. A sufficiently larger Bayes factor for another variant can reverse the prior ranking.
• Wrong reasoning: “A posterior probability of 0.8 means there is an 80% chance the annotation is correct.”
• Correction: It is the model-based posterior probability of the stated causal hypothesis, conditional on the data, candidate set, likelihood, and prior specification. It is not an accuracy probability for the annotation itself.