Posterior Inference Over Multiple-Causal Configurations

Why Compare Configurations?

When variants are correlated through linkage disequilibrium (LD), several can show association with the same underlying signal. If more than one variant may contribute, the inferential target is not just a ranking of variants: it is the probability assigned to each possible set of causal variants under a specified model. A configuration might contain one variant, two variants, or none. Comparing configurations requires accounting for both the observed joint effects and the fact that correlated estimates do not provide independent evidence.

The calculations below use joint effect estimates and their sampling covariance. This is important when marginal signals overlap: an association estimate for one variant can reflect effects at correlated variants, whereas a joint model represents effects conditional on the other variants included in the analysis. The posterior is conditional on the likelihood, effect-size prior, and configuration prior that have been specified. It is not a data-only probability that a variant is causal.

Joint Estimates, LD, and Configuration Likelihoods

Let \(\hat{\boldsymbol{\beta}}\) be the vector of joint effect estimates for \(m\) variants. In the normal summary-statistic model used here, conditional on the true joint effects \(\boldsymbol{\beta}\),

The sampling covariance matrix \(V\) has sampling variances on its diagonal and covariances between estimates off the diagonal; standard errors are the square roots of the diagonal entries. LD affects these covariances. Thus, even if two estimates have similar standard errors, they need not represent independent evidence. This normal likelihood is an assumption of the analysis; its adequacy depends on the summary estimates and covariance being appropriate for the variants and sample under study.

For a configuration \(C\), let \(S_C\) be the \(m\)-by-\(|C|\) matrix that places the effects for variants in \(C\) into the full vector of effects. For example, with three variants, \(C=\{1,3\}\) gives \(\boldsymbol{\beta}=S_C\boldsymbol{\theta}_C=(\theta_1,0,\theta_3)^\mathsf{T}\). Use a zero-centered multivariate normal effect prior,

A common specification takes \(W_C=\tau^2 I\), giving independent effects with prior variance \(\tau^2\) for the variants in the configuration. The prior describes plausible effect sizes before observing the estimates; it is not the configuration prior. The latter, denoted \(\pi(C)=P(C)\), assigns prior probability to each set of variants. These are distinct modeling choices.

Integrating the effect sizes out gives a marginal likelihood for each configuration. Since a linear transformation of a normal vector is normal, and independent normal vectors add to a normal vector, the likelihood after integration is

The null configuration has covariance \(V\). The configuration Bayes factor \(BF_C\) compares the marginal likelihood under \(C\) with that under the null:

The determinant term and the quadratic-form term both matter. The determinant accounts for how much probability the model spreads over possible estimates; the quadratic form reflects how well the observed joint estimates fit that model. A larger estimated effect can improve fit, but a configuration is not automatically favored simply because it contains more variants: its fit is judged alongside the prior effect scale and the uncertainty represented by \(V\).

From Bayes Factors to Posterior Probabilities

Given a set of mutually exclusive configurations covering the possibilities being considered, Bayes’ rule gives

The sum in the denominator is over the configurations in the model. It normalizes their prior-weighted evidence so the posterior probabilities sum to one. A configuration posterior therefore depends on its Bayes factor and its prior probability. If configurations are restricted, for example to a fixed causal count, the posterior is conditional on that restriction.

A variant-level posterior inclusion probability (PIP) can be calculated from the configuration probabilities by summing the probabilities of all configurations that contain the variant:

This sum is useful for describing marginal support for including a variant across the configurations in the model. It does not say which other variants accompany it, so it cannot replace the configuration posterior when the combinations themselves matter.

A Short Derivation of the Integrated Likelihood

The configuration Bayes factor follows directly by integrating the likelihood with respect to the effect prior. Under configuration \(C\), write \(\hat{\boldsymbol{\beta}}=S_C\boldsymbol{\theta}_C+\boldsymbol{\varepsilon}\), where \(\boldsymbol{\varepsilon}\sim N(\boldsymbol{0},V)\) is independent of \(\boldsymbol{\theta}_C\sim N(\boldsymbol{0},W_C)\). It follows that \(S_C\boldsymbol{\theta}_C\) has covariance \(S_CW_CS_C^\mathsf{T}\). Adding the independent error gives

Consequently, the marginal density under \(C\), evaluated at the observed estimate vector, is the multivariate normal density with that covariance. Dividing it by the null density, which has covariance \(V\), cancels the common factor \((2\pi)^{-m/2}\) and yields the determinant ratio and difference of quadratic forms shown above. This argument also clarifies why the diagonal of \(V\) must contain variances: the normal density and its quadratic form use a covariance matrix, not a matrix of standard errors.

Configuration Priors and Interpretation

Configuration priors can encode beliefs about how many variants are causal as well as how probability is distributed among variants at a given count. For example, a prior on causal count can assign probability to \(K=|C|\), then distribute that probability across configurations of that size. If all configurations with a given count are treated equally, then each receives the count prior divided by the number of such configurations. The resulting posterior can differ from one based on equal prior probability for every configuration, because the latter generally gives different total prior mass to different causal counts.

The same distinction matters in interpreting a PIP. In the first example, the PIP for variant 1 is the posterior probability of \(\{1\}\) plus that of \(\{1,2\}\), approximately \(0.1166+0.0541=0.1707\). This summarizes the model’s posterior support for configurations containing variant 1. It is not the posterior probability that variant 1 is biologically causal without qualification: it depends on the candidate variants, likelihood, effect prior, and configuration prior.

Common Mistakes and a Practical Check

• Using standard errors as covariance-matrix entries. A standard error is the square root of a sampling variance. Put variances on the diagonal of \(V\), and covariances—not correlations alone—off the diagonal.
• Ignoring LD in the joint likelihood. Treating correlated joint estimates as independent can misstate the evidence for a set of variants. Use a covariance matrix appropriate to the joint estimates and the analysis sample.
• Equating a Bayes factor with a posterior probability. A Bayes factor compares marginal likelihoods; posterior probabilities also depend on the prior probabilities of the configurations and normalization across the modeled set.
• Interpreting an excluded configuration as having no posterior support in reality. If a model restricts the causal count, its posterior is conditional on that restriction. It cannot assess configurations the model left out.
• Reading a high configuration posterior as proof of biological causation. The posterior describes support under a specified statistical model and candidate-variant set. It does not establish a mechanism or rule out model misspecification.

A useful check is to verify that the configuration priors sum to one, that all covariance matrices used in the normal likelihood are valid, and that the normalized configuration posteriors sum to one. Also confirm whether the posterior is conditional on a fixed causal count or compares counts. These checks make the assumptions behind a multi-variant conclusion explicit.