Variant-Specific Prior Probabilities

Why Assign Different Prior Probabilities to Variants?

In fine-mapping, several candidate variants can be compatible with the observed association evidence. When their likelihood evidence is similar, the assumptions made before examining that evidence can have a visible effect on their ranking. A prior probability records how plausible each candidate is before the likelihood evidence being analyzed is used. Variant-specific priors allow that plausibility to differ across candidates, for example because of external functional information. They do not turn a functional annotation into proof of causality.

The setup here is a single-causal-variant model over a specified set of \(m\) candidates. Assume exactly one candidate in that set is causal, the likelihood or Bayes factor for each candidate is computed under a consistent model, and prior information is specified without reusing the association evidence that will enter the likelihood. The earlier material on the single-causal-variant posterior and Bayes factors supplies the likelihood comparison; the focus here is how to specify and interpret the prior probabilities used in that comparison.

The probabilities are conditional on the candidate set and the stated model: one of these \(m\) variants is causal. Thus, changing the candidate set can change the meaning and normalization of the prior. If the scientific model allows no causal variant in the region, a separate null hypothesis must receive its own prior probability; the variant probabilities alone cannot represent that possibility.

From Relative Weights to Prior Probabilities

A relative weight expresses how much prior mass one candidate receives compared with another. The values need not add to one. For example, weights \(1,2,7\) say that the third candidate receives seven times the prior weight of the first, not that it has a 70% probability until the weights are normalized. Multiplying all weights by the same positive constant leaves the normalized probabilities unchanged.

Weights may be informed by prior biological knowledge, such as functional annotations or independent experimental evidence. But an annotation score is not automatically a prior probability. A score needs a defensible mapping to relative weights, and the resulting probabilities need to be checked for their implications. If, for example, an annotation score is converted to \(q_j=\exp(\gamma a_j)\), then the choice of the score \(a_j\) and scaling parameter \(\gamma\) determines how strongly the prior favors differences in annotation. Those choices should be justified rather than treated as innocuous defaults.

A prior probability of zero excludes a variant from being causal under the model, regardless of its likelihood evidence. That can be appropriate only when exclusion is scientifically justified. Very small probabilities may also strongly suppress a candidate, so it is useful to inspect the normalized probabilities rather than relying only on the original weight scale.

How Priors and Bayes Factors Determine Rankings

Let \(BF_j\) be the Bayes factor for \(C_j\) relative to a common reference model, with larger values representing stronger evidence for \(C_j\). Under the single-causal-variant model, Bayes' rule gives posterior probabilities proportional to prior probability times Bayes factor. Normalizing these products yields the probability for each candidate after the evidence is incorporated.

where \(D\) denotes the data used to compute the Bayes factors. Since the denominator is the same for every candidate, the posterior ranking is the ranking of \(\pi_j BF_j\). Equivalently, for two candidates \(j\) and \(k\), their posterior odds are their prior odds multiplied by their Bayes-factor ratio:

This relation makes the role of the prior explicit. If two Bayes factors are close, prior odds can be consequential. If one candidate has much stronger likelihood evidence, a moderate prior difference may not change their order. Neither statement means that the prior changes the data or its likelihood; it changes the starting odds to which the evidence is applied.

Prior Sensitivity and Practical Diagnostics

A ranking based on nearly equal likelihood evidence deserves particular scrutiny: it may be driven mainly by the prior. One useful diagnostic is to compare posterior rankings under the stated variant-specific prior and a uniform prior, where \(\pi_j=1/m\). This is a sensitivity comparison, not a rule that the uniform prior must be correct. If the ranking changes, report that dependence and explain the basis for the nonuniform prior. If it does not, the ranking is less sensitive to this particular prior contrast, though it is not thereby guaranteed to be robust to every modeling choice.

Several checks help reveal problems before interpreting a ranking:

• Check the normalization and scope. Confirm that all included candidates have been accounted for, that weights are nonnegative with a positive total, and that the normalized probabilities sum to one. State whether the model assumes one causal variant among the listed candidates.
• Check prior provenance. An annotation or external dataset used to set weights should be described. If the same association evidence informs both the weights and the Bayes factors, that evidence may be counted twice unless the joint modeling explicitly accounts for it.
• Inspect extreme or zero probabilities. Determine whether a very small weight reflects a defensible prior belief or a scaling choice that overstates annotation differences. A zero weight is an exclusion, not merely a strong preference for another candidate.
• Compare rankings and probabilities under reasonable alternatives. Uniform priors provide a useful reference; plausible alternative annotation mappings can reveal whether conclusions depend on arbitrary scaling. This check does not prove any prior is correct.
• Check variant identity and annotation alignment. The weights must refer to the same variant alleles and candidate set as the likelihood analysis. Misaligned identifiers or annotations can assign prior preference to the wrong candidate.

For many candidates or very different Bayes factors, computation on the log scale can avoid numerical overflow. Since multiplying all scores by the same positive constant leaves their normalization unchanged, subtract the largest log score before exponentiating:

This calculation requires positive \(\pi_j\) and \(BF_j\); a zero prior probability remains an exclusion and must be handled as such rather than by taking its logarithm.

Comparing Prior Strategies and Their Limits

A uniform prior is often a useful baseline, but it is not a claim that variants have identical biological plausibility in every scientific setting. Conversely, a biologically motivated prior does not guarantee better localization: annotations can be incomplete, poorly calibrated, or irrelevant to the trait. When priors are learned from separate data, those data and the estimation procedure should be described; when they are learned using the same observations, the dependence must be handled in the statistical model.

Finally, these calculations are conditional on the single-causal-variant assumption and the specified candidate set. If multiple causal variants are possible, or if the null hypothesis must be represented, the hypotheses and their prior probabilities need to be expanded accordingly. Those changes cannot be achieved merely by adjusting weights among the current variants. The central practice remains the same: state what the prior means, show how it was normalized, and assess whether the resulting ranking is sensitive to plausible alternatives.

• Common mistake: “Variant \(V_3\) has the largest posterior probability, so the data alone identify it as causal.”
• Correction: The posterior ranking combines the Bayes factors with the stated priors and is conditional on the model and candidate set. When likelihood evidence is similar, prior choices can materially affect the ranking; report and examine that dependence.