Bayes Factors and Prior Odds for Causal Variants

Why Prior Odds and Bayes Factors Go Together

Fine-mapping asks which genetic variant, if any, is responsible for an association signal. Evidence from the observed data matters, but it is not the only ingredient in a Bayesian comparison: the prior probabilities assigned to the competing hypotheses matter too. A Bayes factor describes how the data compare under two hypotheses; prior odds describe how plausible those hypotheses were before seeing those data. Their product gives the posterior odds.

This tutorial focuses on that update for hypotheses about causal variants. It does not derive a particular fine-mapping Bayes factor. Instead, it explains how to combine a Bayes factor already obtained from a valid comparison with stated prior odds, and how to interpret the result without treating it as proof of causality.

In genetic fine-mapping, \(H_1\) might state that a specified variant is causal for the locus-level association, while \(H_0\) might state that it is not causal. The precise hypotheses must be defined for the analysis at hand. In particular, “not causal” could mean no causal variant in the region, or it could mean another specified variant is causal. These are different comparisons and can have different Bayes factors.

The data probability \(P(D\mid H)\) is the probability of the observed data under a hypothesis. When a hypothesis includes unknown parameters, this is generally a marginal probability: the likelihood is averaged over the parameter values using the prior distribution specified under that hypothesis. The Bayes factor therefore depends on both the data and the model assumptions, including any parameter priors used to calculate the marginal probabilities.

Deriving the Posterior-Odds Update

Assume \(H_1\) and \(H_0\) are well-defined competing hypotheses with nonzero prior probabilities, and use the same observed data \(D\) to evaluate both. Bayes’ theorem gives \(P(H_i\mid D)=P(D\mid H_i)P(H_i)/P(D)\) for either hypothesis \(H_i\). Taking the ratio cancels the common marginal probability \(P(D)\):

This is the posterior-odds identity. It says that the data multiply, rather than replace, the prior odds. If the prior odds are 1:99 and the Bayes factor is 20, then the posterior odds are 20:99. The data have shifted the comparison toward \(H_1\), but the posterior odds are still less than 1:1 because \(H_1\) began with relatively small prior odds.

Odds and probabilities are related but are not interchangeable. If the posterior odds for \(H_1\) against \(H_0\) are \(O\), then the posterior probability of \(H_1\), in a comparison where those two hypotheses are exhaustive, is \(O/(1+O)\). Equivalently, if \(p=P(H_1)\) is the prior probability, then:

The probability conversion requires care about the comparison. If \(H_1\) and \(H_0\) are only two members of a larger set of hypotheses, their posterior odds can still be computed as a pairwise ratio, but that ratio alone does not give the overall posterior probability of \(H_1\). The other hypotheses also contribute to the total posterior probability.

Comparing Two Candidate Variants

The same update works when the hypotheses specify different candidate variants. Suppose \(H_A\) says variant A is causal and \(H_B\) says variant B is causal. The posterior odds for A against B equal their prior odds multiplied by the Bayes factor comparing A directly with B:

If Bayes factors are instead reported for each hypothesis against the same reference \(H_0\), their ratio supplies the pairwise Bayes factor: \(BF_{AB}=BF_{A0}/BF_{B0}\). This follows directly by dividing their likelihood ratios, so the shared reference cancels. It is important that the reference and data are genuinely the same in both comparisons.

How Prior Choices Affect the Update

A Bayes factor summarizes the relative support supplied by the data under a specified pair of models. It does not determine the posterior odds on its own. The same Bayes factor can produce quite different posterior conclusions when the prior odds differ. This is especially consequential when the prior probability for a particular variant being causal is small.

Prior probabilities should represent assumptions made before using the data being analyzed. If prior information is derived from independent biological or functional evidence, it may be reasonable to use it, provided the source and assumptions are stated. Using the same association data both to choose a prior and to calculate the Bayes factor can count that evidence twice unless the analysis explicitly accounts for the dependence.

Interpretation and a Common Pitfall

A Bayes factor is a ratio of data probabilities under hypotheses, not a probability that either hypothesis is true. For example, \(BF_{10}=20\) means that the observed data are 20 times as probable under \(H_1\) as under \(H_0\), given the specified models and their parameter priors. It does not mean that \(H_1\) has probability 20, nor that the data have a 20% or 95% probability of being causal evidence.

Likewise, posterior odds are a comparison conditional on the hypotheses and prior probabilities used. In a fine-mapping analysis, a large posterior preference for a causal-variant hypothesis remains conditional on the association model, the treatment of other variants and hypotheses, the data quality, and the prior specification. Statistical support does not by itself establish a biological mechanism or prove that changing the variant would change the trait.

Common mistake: “The Bayes factor is 20, so the posterior probability that the variant is causal is 20/21.” This reasoning silently assumes prior odds of 1:1 and an exhaustive two-hypothesis comparison. If the prior odds are instead 1:99, the posterior odds are 20:99 and the posterior probability is \(20/119\), about 0.1681. Always multiply the Bayes factor by the actual prior odds first; convert to a probability only afterward, and only with the comparison set clearly specified.