Prior Sensitivity And Posterior Diagnostics

Why Examine Prior Sensitivity?

A colocalization analysis does not produce a posterior probability from association data alone. It combines the data’s relative support for competing explanations with prior probabilities for association at the variants under study. In particular, the posterior probability assigned to a shared causal variant can change when the prior probability of association with both traits changes. A strong-looking posterior under one setting is therefore not, by itself, evidence that the conclusion is insensitive to prior choice.

This tutorial focuses on prior sensitivity and posterior diagnostics under the single-causal-signal model. The five hypotheses are: \(H_0\), neither trait is associated in the region; \(H_1\), only trait 1 is associated; \(H_2\), only trait 2 is associated; \(H_3\), both traits are associated but through distinct variants; and \(H_4\), both traits share a causal variant. The assumptions of that model matter: if either trait has multiple causal signals in the region, a five-hypothesis single-signal analysis may not represent the data-generating situation.

From Priors to Posterior Mass

For each variant \(i\), let \(BF_{1i}\) and \(BF_{2i}\) be the Bayes factors comparing association with no association for trait 1 and trait 2, respectively. A Bayes factor summarizes the likelihood evidence for one explanation relative to another; it is not itself a posterior probability. Under the usual rare-association-prior approximation, the unnormalized weights for the five hypotheses are:

The shared-variant terms in \(W_4\) pair evidence at the same variant; the distinct-variant terms in \(W_3\) pair evidence at different variants. The approximation treats the variant-level association events as sufficiently rare that the displayed prior products are appropriate. Prior values should be chosen coherently: in particular, a joint association prior \(p_{12}\) should not exceed either marginal association prior \(p_1\) or \(p_2\). The weights must be nonnegative, and the resulting posterior probabilities sum to one.

Changing a prior changes the relevant weights and, through the common denominator, can also change every posterior probability. It does not change the Bayes factors when the observed data and likelihood model are fixed. A useful diagnostic is therefore to inspect both the posterior for \(H_4\) and the competing posterior mass, especially \(H_3\), rather than treating a single posterior probability as a standalone measure of data strength.

A Useful Sensitivity Result

The effect of changing \(p_{12}\) can be derived directly. Let \(R=W_0+W_1+W_2+W_3\), and suppose \(W_4\) is its baseline value. If \(p_{12}\) is multiplied by a positive factor \(c\), then the new \(H_4\) weight is \(cW_4\), while \(R\) is unchanged. Therefore:

For \(R>0\) and \(W_4>0\), differentiating gives:

So increasing \(p_{12}\) increases \(PP.H4\) when the other weights are fixed. This is a mathematical consequence of the normalization, not new evidence from the data. In odds form, the same result is especially transparent:

The posterior odds of \(H_4\) against all alternatives scale in direct proportion to \(p_{12}\), conditional on the other priors and the evidence being held fixed. This explains why prior-sensitivity results should report the prior settings alongside the posterior probabilities.

Varying Marginal Association Priors

Sensitivity analysis should not automatically vary only \(p_{12}\). The marginal priors \(p_1\) and \(p_2\) affect the weights for association with one trait, and both enter the weights for \(H_3\). For fixed Bayes factors, \(W_1\) is proportional to \(p_1\), \(W_2\) to \(p_2\), and \(W_3\) to their product \(p_1p_2\). \(W_4\) is proportional to \(p_{12}\), not to \(p_1p_2\), in this prior parameterization. Consequently, changing marginal priors can alter the balance among \(H_1\), \(H_2\), and \(H_3\), as well as the normalized posterior for \(H_4\).

Practical Diagnostics and Reporting

A defensible sensitivity analysis uses a set of prior settings chosen for a reason, such as a plausible range supported by subject-matter knowledge or an established analysis convention. The analysis should keep the data, variant set, likelihood specification, and other modeling choices fixed while changing the prior component under investigation. If several priors are varied, make clear which changed together; otherwise it is difficult to attribute a posterior shift to a particular choice.

For each setting, inspect \(PP.H0\) through \(PP.H4\). A large \(PP.H0\) indicates that the model retains substantial support for neither trait being associated. Large \(PP.H1\) or \(PP.H2\) indicates support for association with only one trait. Large \(PP.H3\) means the analysis favors associations with distinct variants over a shared variant. \(PP.H4\) is the posterior probability of the shared-variant hypothesis within the specified model and prior. A result is prior-sensitive if its substantive interpretation changes across the plausible prior settings, not merely because the last decimal place changes.

Useful reporting includes the prior values or ranges, the posterior probabilities for all five hypotheses, and a brief statement about whether the interpretation changed. If the analysis favors \(H_4\) across the chosen settings, that is evidence of robustness to those settings, not proof that the model is correct or that the variants cause a trait through a particular mechanism. If posterior mass shifts substantially between \(H_3\) and \(H_4\), the shared-versus-distinct conclusion is especially dependent on the modeling assumptions and priors.

An additional diagnostic is the distribution of support across variants conditional on \(H_4\). The terms \(BF_{1i}BF_{2i}\) show how the shared-hypothesis evidence is distributed across candidate variants. If support is concentrated on one variant, the shared signal has a clearer variant-level basis under the model; if it is spread across many variants, the identity of the shared variant remains uncertain. This conditional distribution does not replace the five-hypothesis posterior profile.

Comparisons, Assumptions, and Common Pitfalls

A Bayes factor comparison and a posterior probability answer related but different questions. The Bayes factor for \(H_4\) relative to \(H_3\) summarizes the data’s relative evidence for sharing versus distinct variants under the specified likelihood model. Posterior odds also incorporate prior odds:

This ratio is useful for focusing on the sharing-versus-distinct comparison, but it does not describe uncertainty across all five hypotheses. Conversely, \(PP.H4\) does reflect all five hypotheses through its denominator, but its value depends on all the priors. Reporting both the hypothesis profile and the prior settings is more informative than reporting only a thresholded label such as “colocalized.”

The key assumptions include a correctly specified single-signal model for each trait, comparable and correctly aligned association evidence, and a suitable likelihood calculation for the data and linkage disequilibrium structure. Prior sensitivity cannot repair a mismatch in alleles, a poorly defined region, or a model that excludes relevant signals. If the single-signal assumption is doubtful, posterior probabilities may be misleading even when they are stable over the examined priors. Diagnostics of prior sensitivity address sensitivity to priors, not every source of model uncertainty.

• Mistake: “A high \(PP.H4\) proves sharing, so priors do not matter.” The correction is that a posterior probability combines data evidence and prior probabilities. Recompute it over plausible priors and report whether the conclusion changes.
• Mistake: “Only \(PP.H4\) needs to be reported.” The correction is to inspect all five hypotheses. A substantial \(PP.H3\) can signal uncertainty between sharing and distinct signals, while \(PP.H0\), \(PP.H1\), or \(PP.H2\) can indicate uncertainty about association itself.
• Mistake: “A stable posterior proves the model assumptions.” The correction is that stability over a prior grid shows robustness only to the choices varied. It does not validate the single-signal assumption, the summary statistics, or other model components.
• Mistake: “Choose the prior that gives the desired conclusion.” The correction is to specify plausible prior settings for scientific or methodological reasons and show the range of results, rather than selecting a prior after seeing which one produces the preferred posterior.