Choosing Priors For Colocalization Analysis

Why Colocalization Priors Matter

Colocalization asks whether genetic association with a phenotype and genetic association with gene expression are consistent with a shared causal variant. The data provide evidence through Bayes factors, but the posterior also depends on how plausible the competing association patterns were before examining that evidence. In particular, the prior probability of a shared association is often much smaller than the prior probability of association with either trait alone. A numerically small shared-association prior can therefore have a substantial effect on posterior support.

The goal is not to choose values that produce a preferred conclusion. Priors should represent plausible per-variant association rates, be set independently of the locus-specific evidence being evaluated, and be reported so readers can judge their influence. The earlier material on posterior odds for shared versus distinct signals establishes the comparison; here the emphasis is on what the prior quantities mean and how to reason about their scale.

From Per-Variant Priors to the Five Hypotheses

Consider a region containing \(M\) candidate variants. The five hypotheses distinguish no association (H0), GWAS association only (H1), expression association only (H2), associations with distinct causal variants (H3), and a shared causal variant (H4). The per-variant priors describe how likely the relevant association states are before the observed association evidence is considered. They do not by themselves determine the posterior probability of any hypothesis.

For each variant \(i\), let \(BF_{1i}\) and \(BF_{2i}\) be Bayes factors for association with the GWAS and expression traits, respectively, relative to no association. Under the standard single-causal-variant-per-trait model, with small per-variant priors, the unnormalized posterior weights can be written as:

The posterior probability for a hypothesis is its weight divided by the sum of all five weights. H3 sums over ordered pairs of different variants: the GWAS association is assigned to \(i\), the expression association to \(j\), and \(i\ne j\). H4 sums products of evidence at the same variant. The factor \(p_1p_2\) in H3 represents the approximate prior weight for associations at two distinct variants; \(p_{12}\) represents the prior weight for one variant associated with both traits.

These expressions are rare-prior, sparse-model weights. They treat the candidate variants and Bayes factors as given and assume the model’s causal-variant structure is appropriate. Because the events are rare, the weight for H0 is taken as 1 on this relative scale. The formulas are not a claim that the five prior probabilities are exactly 1, \(Mp_1\), or \(Mp_{12}\); exact normalization and corrections depend on the chosen prior model. The weights make clear, however, how changing \(p_1\), \(p_2\), or \(p_{12}\) changes the posterior comparison.

Choosing the Three Prior Quantities

The priors have different roles and should not be treated as interchangeable tuning knobs.

• \(p_1\), GWAS-only association: represents the prior probability that a candidate variant is associated with the GWAS trait but not expression. Its scale should reflect how sparse GWAS associations are over the modeled candidate set, under the model being used.
• \(p_2\), expression-only association: represents the corresponding probability for the expression trait but not the GWAS trait. It need not equal \(p_1\): the traits can differ in sample size, genetic architecture, and expected association frequency.
• \(p_{12}\), shared association: represents the prior probability that one candidate variant is associated with both traits. It should generally be smaller than either single-trait prior, since joint association is a more specific event. It should not be set high merely because a gene is biologically interesting.

All three are defined relative to a candidate-variant universe and a model. Changing the number of variants under consideration can change the expected number of associations in the region even when the per-variant probabilities stay fixed. For example, if \(M=10{,}000\) and \(p_1=10^{-4}\), the approximate expected number of GWAS-only associated variants is \(Mp_1=1\). If the candidate set is ten times larger, the same per-variant value implies about ten such variants. This is one reason to state the variant universe and the assumptions behind prior values rather than reporting numbers without context.

Prior values can be informed by broad knowledge of association frequency, prior analyses, or a prespecified convention for the model. The relevant information should be external to the locus-specific evidence being tested. Biological knowledge may motivate a higher shared prior in a well-justified analysis, but a prior should not be raised simply because the observed locus appears to colocalize; doing so reuses the evidence and can inflate posterior support.

Why the Shared Prior Can Change the Result

The H4 weight is proportional to \(p_{12}\), while H1, H2, and H3 depend on \(p_1\), \(p_2\), or their product. Consequently, increasing \(p_{12}\) increases H4’s weight relative to the other weights. It does not multiply the final posterior probability by a fixed factor, because the posterior denominator changes too. If H4 is already strongly supported by the Bayes factors, a plausible range of \(p_{12}\) may leave the qualitative conclusion intact. If evidence is moderate or H3 is competitive, the same range can change the conclusion substantially.

A Short Derivation of the Shared-versus-Distinct Prior Effect

The main prior comparison is visible by taking the ratio of H4 and H3 weights. Write \(C=\sum_i BF_{1i}BF_{2i}\) for same-variant evidence and \(D=\sum_{i\ne j}BF_{1i}BF_{2j}\) for distinct-variant evidence. If both are positive, then:

This follows by substituting the definitions of \(W_4\) and \(W_3\) and cancelling no terms other than the displayed factors. The posterior odds \(P(H4\mid\text{data})/P(H3\mid\text{data})\) equal this ratio because the common posterior normalizing constant cancels. Thus the evidence ratio \(C/D\) favors shared over distinct association, while \(p_{12}/(p_1p_2)\) is the prior-odds contribution under this approximation. A small \(p_{12}\) can still imply substantial prior odds for H4 relative to H3 if \(p_1p_2\) is even smaller. This ratio does not compare H4 with H0, H1, or H2; a favorable H4:H3 comparison alone does not guarantee a high H4 posterior.

Diagnostics, Alternatives, and Limitations

A useful prior diagnostic is a sensitivity analysis over a defensible range, not a search for the value that yields the desired result. Recalculate all hypothesis weights and posterior probabilities while varying one prior at a time, then consider plausible joint changes if the prior assumptions are related. Report the baseline values, the rationale for them, the alternatives examined, and whether the substantive interpretation changes. Pay particular attention to cases where support moves across a decision threshold or where H4 and H3 exchange rank.

Check that the priors are small enough for the sparse-association approximation and that their interpretation matches the model. For instance, if \(M p_1\) or \(M p_2\) is much larger than the number of causal variants the model allows, the per-variant priors and the single-causal-variant assumptions are not aligned. A posterior that changes sharply under small prior perturbations is not necessarily computed incorrectly, but it is prior-sensitive and should be described as such. Extreme posterior support can also arise from misspecified Bayes factors or an inadequate causal model; changing priors cannot repair those problems.

The main alternative to a single chosen prior is to report a structured sensitivity analysis, such as a grid of \(p_{12}\) values with \(p_1\) and \(p_2\) fixed at justified values. A more elaborate alternative is to average over a distribution on priors, but this adds assumptions and should be explained rather than presented as assumption-free. Neither approach removes the need to justify the prior range. The analysis also remains conditional on the candidate variants, the association evidence, and the model structure; priors do not establish that the shared variant causes a downstream molecular or clinical effect.

Practical Reporting

A clear report states the numerical values of \(p_1\), \(p_2\), and \(p_{12}\), identifies them as per-variant priors, and explains their scale or source. It should also state the modeled region or candidate set and the causal-variant assumptions needed to interpret the calculation. Presenting posterior probabilities for all five hypotheses helps distinguish “H4 exceeds H3” from “H4 has high overall posterior support.” If conclusions change across reasonable priors, that sensitivity is part of the result, not a nuisance to hide.

Before interpreting a colocalization posterior, ask whether the prior values were chosen independently of the observed locus evidence, whether they imply a plausible number of associations over the candidate set, and whether the conclusion survives reasonable alternatives. These checks make the posterior easier to interpret and harder to overstate.