Bayesian Trans-Ancestry Posterior Probabilities

From Ancestry-Specific Evidence to Posterior Probabilities

At a locus, a variant can appear associated in more than one ancestry because it is causal, because it is correlated through linkage disequilibrium (LD) with a causal variant, or both. Trans-ancestry fine-mapping compares candidate causal variants using the association evidence and LD structure in each ancestry. The result is not merely a list of the strongest association statistics: it is a probability distribution over causal hypotheses, conditional on the data and the model.

A key modeling distinction is between sharing a causal variant and sharing its effect. A model can assert that the same variant is causal across ancestries while allowing its effect size to differ. This tutorial constructs posterior probabilities under that distinction, then shows how a prior over variants and the option of no causal variant affect the calculation.

Likelihood, LD, and the Shared-Variant Hypothesis

Suppose \(A\) ancestries have harmonized summary statistics for the same \(m\) candidate variants. For ancestry \(a\), let \(z_a\) be the vector of association \(z\)-statistics, \(R_a\) the ancestry-specific LD correlation matrix, and \(N_a\) the sample size. Assume genotypes and the phenotype have been standardized in a way that supports the usual approximate summary-statistic likelihood. Under the null of no effects at the locus, take \(z_a\sim N(0,R_a)\). Under a vector of standardized genetic effects \(\beta_a\), use the approximation

This approximation requires correctly aligned alleles and variants, appropriate ancestry-matched LD, and a sufficiently accurate summary-statistic model. The calculation below also assumes that ancestry-specific samples are independent, so their likelihoods multiply. If samples overlap, their cross-ancestry covariance must be modeled rather than ignored. For simplicity, assume \(R_a\) is positive definite in the derivation; the implications of singular LD matrices are noted below.

For a single shared causal variant \(j\), set \(\beta_a=\theta_a e_j\), where \(e_j\) is the vector with 1 in position \(j\) and 0 elsewhere. The causal indicator is shared, but the effect \(\theta_a\) may vary across ancestries. A simple prior is \(\theta_a\sim N(0,W_a)\), independently across ancestries. The variance \(W_a\) expresses the prior scale for effects in ancestry \(a\); it is not estimated from the observed \(z\)-statistic in this calculation.

The likelihood and prior yield an ancestry-specific Bayes factor \(BF_{aj}\): the marginal likelihood under the hypothesis that \(j\) is causal, divided by the null marginal likelihood. With standardized genotypes, \(R_{a,jj}=1\), and the factor is

The shared-variant model has \(BF_j=\prod_{a=1}^{A}BF_{aj}\) under the independence assumptions. LD is part of the likelihood: \(R_a\) determines the covariance of the vector of statistics and how effects at a variant produce association patterns across correlated variants. In the single-variant Bayes-factor ratio above, this dependence simplifies to the statistic at \(j\), because the model uses the \(j\)th LD column for the mean and the same \(R_a\) under the null and alternative. That simplification does not mean that LD is irrelevant or that the \(z\)-statistics can be interpreted as independent across variants.

Deriving the Single-Variant Bayes Factor

The formula can be justified by integrating over the uncertain effect size rather than plugging in an estimated effect. For ancestry \(a\), under candidate \(j\), integrating \(\theta_a\sim N(0,W_a)\) gives a normal distribution with mean zero and covariance \(R_a+N_aW_a R_a e_j e_j^{T}R_a\). Under the null, the covariance is \(R_a\). The matrix determinant lemma gives the ratio of determinants as \(1+N_aW_a\), since \(e_j^TR_ae_j=R_{a,jj}=1\).

For the quadratic form in the density ratio, the rank-one inverse identity gives

Consequently, the alternative-minus-null log density is

Exponentiating yields the stated \(BF_{aj}\). This argument uses \(R_a^{-1}\), so positive definiteness is required for this full-rank derivation. A correlation matrix with unit diagonal can still be singular. If LD is singular, do not apply the inverse formula as written: formulate the likelihood on its support using a reduced-rank representation or suitable pseudoinverse, and define the likelihood ratio on the common support of the null and alternative. In practice, the likelihood and any handling of near-singular LD must be specified consistently.

From Bayes Factors to PIPs

Let \(C=j\) denote the hypothesis that candidate \(j\) is the shared causal variant, and let \(\pi_j=P(C=j)\) be its prior probability. Bayes' rule gives

The denominator shown assumes the model assigns probability one across the listed single-causal hypotheses. If there may be no causal variant in the region, include a null hypothesis with prior probability \(\pi_0\) and Bayes factor \(BF_0=1\). Then the denominator is \(\pi_0+\sum_k\pi_kBF_k\). Variant priors need not be equal: annotation-informed priors can give higher prior weight to variants with specified functional features. Such weights affect posterior probabilities, but do not turn an annotation into proof of causality.

When more than one causal variant is allowed, the hypothesis is a configuration \(C\), a set of included variants. Give configurations prior probabilities \(\pi(C)\), calculate their marginal likelihoods relative to a null configuration, and normalize:

The second expression is the PIP for variant \(j\): it adds the posterior probabilities of every configuration containing that variant. The configuration Bayes factor must use the joint LD-aware likelihood; multiplying single-variant Bayes factors is generally not valid when causal effects and correlated signals are modeled jointly.

What the Probabilities Depend On

A trans-ancestry PIP is conditional on a model, not an unconditional probability that a variant is biologically causal. Its value can be affected by the LD matrices, sample sizes, effect-size prior variances, variant priors, candidate set, and assumptions about effect sharing and sample independence. A shared causal indicator does not require identical effects: here, \(\theta_a\) can differ by ancestry. Conversely, if a model requires a common effect, that is an additional restriction and should not be silently substituted for the shared-variant assumption.

LD differences matter because the same causal variant can generate different patterns of marginal associations across ancestries. Joint modeling can use those patterns to distinguish candidates that are difficult to distinguish in one ancestry alone. However, if a candidate is absent, poorly measured, or inconsistently harmonized in one ancestry, the apparent cross-ancestry evidence may not represent comparable data. The likelihood, variant set, allele orientation, and LD references must correspond to the same candidate-variant definitions.

Common Mistakes

• Equating shared causality with an identical effect. The shared-variant hypothesis specifies which variant is causal; it does not, by itself, require equal effect sizes across ancestries.
• Multiplying \(z\)-statistics or treating correlated variants as independent. The likelihood uses ancestry-specific LD covariance. Independent ancestry samples justify multiplying ancestry likelihoods, not treating variants within an ancestry as independent.
• Leaving out a plausible null state without noticing. If the candidate hypotheses do not exhaust the prior probability, include the null or other relevant configurations in the normalization.
• Assuming the prior has no effect. Posterior odds equal prior odds multiplied by Bayes factors. A strong prior can shift the posterior even when the data are unchanged.
• Inverting a singular LD matrix as if it were positive definite. A unit diagonal does not guarantee invertibility. Use a likelihood defined on the appropriate support or another explicitly justified treatment.