Deriving Posterior Probabilities for Colocalization

Why Colocalization Uses Five Posterior Probabilities

At a genomic region associated with both a phenotype and a molecular trait, the association results alone do not establish whether the same variant underlies both signals. Colocalization addresses that question by comparing five hypotheses: neither trait is associated, only the first trait is associated, only the second is associated, both are associated through distinct variants, or both share one causal variant. The posterior probabilities quantify support for these alternatives after combining the association evidence with prior weights.

The calculation has two ingredients. For each candidate variant, a Bayes factor summarizes how well that variant’s causal model explains the evidence for one trait relative to that trait’s null model. Prior weights then represent the relative plausibility of the hypotheses and variant configurations before observing the association evidence. The resulting weights are normalized to sum to one.

Here, trait 1 and trait 2 refer to the two traits being compared, and the candidate variants are indexed \(i=1,\ldots,m\). The five hypotheses are as follows. H0 means neither trait has a causal variant in the region. H1 means only trait 1 is associated. H2 means only trait 2 is associated. H3 means both traits are associated, but their causal variants are distinct. H4 means both traits are associated and share the same causal variant.

The derivation below uses the single-causal-variant assumption: each trait has either no causal variant in the region or exactly one. It also assumes the two traits’ evidence can be factored into trait-specific evidence once a causal configuration is specified. The candidate variants must refer to the same set of genomic positions for both traits, and the Bayes factors must be calculated against their respective trait-specific null models.

From Variant Evidence to Hypothesis Weights

Let \(p_1\) and \(p_2\) be the per-variant prior weights for a causal effect on trait 1 and trait 2, respectively. Let \(p_{12}\) be the prior weight for a shared causal variant. These are small prior weights, not posterior probabilities. Take the all-null configuration as the reference with weight 1. A configuration in which variant \(i\) is causal only for trait 1 has prior weight \(p_1\); its evidence relative to the all-null configuration is \(BF_{1i}\). Thus its unnormalized posterior weight is \(p_1 BF_{1i}\). Summing over possible causal variants gives the H1 weight. The other hypotheses follow by the same logic.

For H3, the indices must differ: if \(i\) is causal for trait 1, then \(j\ne i\) is causal for trait 2. The prior weight for that pair is \(p_1p_2\), and the joint evidence factor is \(BF_{1i}BF_{2j}\). For H4, the same variant \(i\) is causal for both traits, so the evidence factor is \(BF_{1i}BF_{2i}\), with shared-variant prior weight \(p_{12}\). This distinction between matching and different indices is the central calculation in the posterior derivation.

The H3 sum can also be evaluated using two simpler sums. Every product formed by selecting one variant for trait 1 and one for trait 2 is either a same-variant product or a distinct-variant product. Therefore,

Rearranging gives the usual compact expression for the H3 evidence term. The subtraction removes the same-variant configurations, which belong to H4 rather than H3.

Finally, the \(W_k\) values are unnormalized weights. Bayes’ rule says that the posterior probability of a hypothesis is proportional to its prior weight times the evidence under that hypothesis. Dividing each weight by their sum yields probabilities that add to one:

where \(D\) denotes the association evidence for both traits. This normalization is necessary: a large H4 weight by itself does not equal the H4 posterior probability, because the other four hypotheses also contribute to the denominator.

Worked Calculations

Interpretation, Assumptions, and a Common Pitfall

A colocalization posterior is conditional on the candidate region, the evidence summarized by the Bayes factors, the prior weights, and the model assumptions. In particular, the single-causal-variant assumption matters: if either trait has multiple causal variants in the region, the five-hypothesis calculation may not represent the data-generating possibilities adequately. The calculation also depends on having aligned variant identities across the two traits; otherwise, same-index products do not refer to the same genomic variant.

The posterior \(P(H_4\mid D)\) is the probability assigned to the shared-variant hypothesis under the specified model and priors, after observing the evidence. It is not the probability that the traits are causally related to one another, and it does not show that changing the molecular trait would change the phenotype. Colocalization concerns whether the association signals share a variant, not whether one trait mediates the effect of the variant on the other.

A common mistake is to calculate H3 as \(p_1p_2(\sum_i BF_{1i})(\sum_j BF_{2j})\), then also calculate H4 separately. That reasoning counts same-variant configurations in H3 and again in H4, inflating the total evidence for joint association. The correction is to subtract the matching-index sum from the product of aggregate sums before multiplying by \(p_1p_2\). Another mistake is to call the largest unnormalized weight a posterior probability; normalize all five weights first.