Calibrating Colocalization Priors

Why Prior Calibration Matters

A colocalization posterior combines locus-level association evidence with prior weights assigned to different causal configurations. Even when the Bayes factors are fixed, changing those weights can change the posterior probability of shared causality. Prior calibration is therefore part of interpreting the result, not a cosmetic choice made after the analysis.

This tutorial uses the five hypotheses \(H_0\) through \(H_4\) and single-signal setup established in the earlier material on colocalization. There are \(M\) candidate variants. For each variant \(i\), let \(BF_{1i}\) and \(BF_{2i}\) be the Bayes factors for association with trait 1 and trait 2. The calculations below focus on how per-variant prior weights enter the posterior and how to check whether a conclusion depends strongly on their values.

Definition: Let \(p_1\) and \(p_2\) be the relative prior weights for a variant affecting trait 1 only or trait 2 only, respectively, and let \(p_{12}\) be the relative prior weight for the same variant affecting both traits. These are per-configuration weights relative to the null configuration, not prior weights for the aggregate hypotheses \(H_1\), \(H_2\), \(H_3\), and \(H_4\). In the standard rare-association approximation, the null configuration has weight 1, a single-trait configuration has weight \(p_1\) or \(p_2\), a distinct-variant pair has weight \(p_1p_2\), and a shared-variant configuration has weight \(p_{12}\).

The distinction between a per-configuration weight and an aggregate hypothesis weight is essential. There are \(M\) possible single-trait configurations, \(M(M-1)\) ordered pairs of distinct variants for \(H_3\), and \(M\) shared-variant configurations for \(H_4\). Thus the approximate aggregate prior weights are \(1\), \(Mp_1\), \(Mp_2\), \(M(M-1)p_1p_2\), and \(Mp_{12}\). But the evidence sums below already include those configurations. Multiplying an aggregate prior weight by a sum over configurations would count the number of configurations twice.

From Configuration Weights to Posterior Weights

For each causal configuration, the evidence relative to the null is the product of the relevant Bayes factors. Summing those evidence contributions over configurations gives:

L_1=\sum_{i=1}^{M}BF_{1i},\qquad L_2=\sum_{i=1}^{M}BF_{2i},

L_3=\sum_{\substack{i,j=1\\i\ne j}}^{M}BF_{1i}BF_{2j},\qquad L_4=\sum_{i=1}^{M}BF_{1i}BF_{2i}.

Here \(L_3\) sums over ordered pairs: variant \(i\) is the causal variant for trait 1 and a different variant \(j\) is the causal variant for trait 2. \(L_4\) sums over configurations in which the same variant is causal for both traits. Under the standard calculation, the unnormalized posterior weights are:

W_0=1,\qquad W_1=p_1L_1,\qquad W_2=p_2L_2,\qquad W_3=p_1p_2L_3,\qquad W_4=p_{12}L_4.

These formulas can be read directly as sums over configurations. For example, each distinct-variant configuration contributing to \(H_3\) has prior weight \(p_1p_2\), so summing its prior-weighted evidence gives \(W_3=p_1p_2L_3\). For \(M\ge 2\), equivalently, the aggregate prior weight \(M(M-1)p_1p_2\) is multiplied by the average evidence per distinct-variant configuration, \(L_3/[M(M-1)]\), not by \(L_3\) itself. For \(M=1\), \(H_3\) has no configurations and \(W_3=0\), so this average evidence is not defined. The same average-evidence logic applies to \(H_1\), \(H_2\), and \(H_4\).

The posterior probability for hypothesis \(H_k\) is its weight divided by the sum of all five weights:

P(H_k\mid D)=\frac{W_k}{W_0+W_1+W_2+W_3+W_4},

where \(D\) denotes the association data for both traits. This normalization makes the five posterior probabilities sum to 1. The priors affect the \(W_k\) values; the Bayes factors summarize how the observed data favor particular configurations relative to the null.

Worked Example: Calculating the Five Posterior Probabilities

Suppose a hypothetical locus has \(M=3\) variants, with Bayes factors \(BF_{1}=(20,2,1)\) and \(BF_{2}=(15,3,1)\). Use \(p_1=0.01\), \(p_2=0.02\), and \(p_{12}=0.001\).

First calculate the evidence sums. \(L_1=20+2+1=23\), and \(L_2=15+3+1=19\). For \(L_4\), multiply the Bayes factors at the same variant: \(L_4=(20)(15)+(2)(3)+(1)(1)=300+6+1=307\). For \(L_3\), sum the six products at distinct variants:

$$ L_3=(20)(3)+(20)(1)+(2)(15)+(2)(1)+(1)(15)+(1)(3) =60+20+30+2+15+3=130. $$

The unnormalized weights are therefore \(W_0=1\), \(W_1=(0.01)(23)=0.23\), \(W_2=(0.02)(19)=0.38\), \(W_3=(0.01)(0.02)(130)=0.026\), and \(W_4=(0.001)(307)=0.307\). Their total is \(1+0.23+0.38+0.026+0.307=1.943\).

The posterior probability of shared causality is \(P(H_4\mid D)=0.307/1.943=0.1580\), or about 15.8%. The posterior probability for distinct causal variants is \(P(H_3\mid D)=0.026/1.943=0.01338\), or about 1.34%. Although the same-variant evidence sum is large, the posterior for \(H_4\) is not close to 1 because the null and single-trait hypotheses also retain substantial weight. The result is a posterior distribution over the five hypotheses, not a standalone measure of how large one Bayes factor is.

What the Shared-Causality Prior Does

The parameter \(p_{12}\) enters \(W_4\) directly, while leaving the other four weights unchanged if \(p_1\), \(p_2\), and the Bayes factors are held fixed. Increasing \(p_{12}\) therefore increases the posterior weight of \(H_4\) and its share of the total. This change is a consequence of the assumed prior, not new information in the association data.

The scale of \(p_{12}\) should be considered relative to both the single-trait priors and the product \(p_1p_2\). A shared-variant configuration and a distinct-variant pair are different configurations, with different prior weights. It is not generally appropriate to assume that the shared prior is automatically equal to \(p_1p_2\), or to interpret \(p_{12}\) as a posterior probability.

Worked Example: Sensitivity to the Shared-Causality Prior

Use the evidence sums from the preceding hypothetical locus and hold \(p_1=0.01\) and \(p_2=0.02\) fixed. The weights other than \(W_4\) sum to \(W_0+W_1+W_2+W_3=1+0.23+0.38+0.026=1.636\). Since \(L_4=307\), \(W_4=307p_{12}\).

For \(p_{12}=0.0001\), \(W_4=307(0.0001)=0.0307\), so the total weight is \(1.636+0.0307=1.6667\), and \(P(H_4\mid D)=0.0307/1.6667=0.01842\), or 1.84%.

For \(p_{12}=0.001\), the total is \(1.636+0.307=1.943\), giving \(P(H_4\mid D)=0.307/1.943=0.1580\), or 15.8%. For \(p_{12}=0.005\), \(W_4=307(0.005)=1.535\), the total is \(1.636+1.535=3.171\), and \(P(H_4\mid D)=1.535/3.171=0.4841\), or 48.4%.

The data and Bayes factors did not change, yet the posterior for \(H_4\) ranged from 1.84% to 48.4% across these prior settings. This is a prior-sensitive result. A report should therefore identify the prior values and avoid presenting the posterior as if it were determined by the data alone.

Shared Versus Distinct Causality: An Odds Calculation

When both relevant weights are positive, the posterior odds of shared rather than distinct causality are the ratio of their posterior probabilities. The common normalizing denominator cancels, leaving:

\frac{P(H_4\mid D)}{P(H_3\mid D)} =\frac{W_4}{W_3} =\frac{p_{12}L_4}{p_1p_2L_3} =\frac{p_{12}}{p_1p_2}\frac{L_4}{L_3}.

This finite ratio formula requires \(p_1p_2L_3>0\). When it holds, the first factor is the relative prior weight of a shared configuration versus a distinct-variant pair, and the second factor compares the summed evidence for the two hypotheses. Because the sums include different numbers of configurations, the ratio can also be expressed using average evidence: \(L_4/M\) for \(H_4\) and \(L_3/[M(M-1)]\) for \(H_3\), alongside their aggregate prior weights.

Worked Example: Checking Shared-versus-Distinct Posterior Odds

For the hypothetical locus above, \(p_1p_2=(0.01)(0.02)=0.0002\), \(p_{12}=0.001\), \(L_3=130\), and \(L_4=307\). Since \(p_1p_2L_3=(0.0002)(130)=0.026>0\), the finite odds formula applies.

\frac{P(H_4\mid D)}{P(H_3\mid D)} =\frac{0.001(307)}{0.0002(130)} =\frac{0.307}{0.026} =11.81.

Thus the posterior odds of \(H_4\) versus \(H_3\) are about 11.8 to 1 under these priors and data. This comparison is conditional on these two hypotheses: it does not say that \(H_4\) has an 11.8-to-1 probability relative to all five hypotheses. In the full posterior, \(P(H_4\mid D)=15.8\%\) because \(H_0\), \(H_1\), and \(H_2\) also have weight.

For a further calibration check, consider the posterior odds within \(H_3\) and \(H_4\) alone. To make those odds exceed 9 to 1, the required condition is \(p_{12}L_4/(p_1p_2L_3)>9\). Solving for \(p_{12}\) gives \(p_{12}>9p_1p_2L_3/L_4\), or \(p_{12}>9(0.0002)(130)/307=0.0007622\). This threshold concerns the \(H_4\)-versus-\(H_3\) comparison only; it does not guarantee that \(P(H_4\mid D)>0.90\) across all five hypotheses.

A Practical Prior-Sensitivity Workflow

A useful sensitivity analysis varies scientifically plausible prior values while keeping the observed Bayes factors fixed. It should make clear which prior parameters changed and whether the locus-level interpretation changes with them. A compact calculation can follow these steps:

State the prior settings.
Report \(p_1\), \(p_2\), and \(p_{12}\), and specify that they are per-configuration relative weights under the calculation being used.

Compute evidence sums once.
Calculate \(L_1\) through \(L_4\) from the Bayes factors, checking that \(L_3\) excludes same-variant pairs and \(L_4\) includes same-variant products.

Recalculate and normalize.
For each prior setting, calculate all five \(W_k\) values and divide each by their total. Do not multiply an aggregate prior weight by a sum that already spans configurations.

Compare conclusions.
Examine the posterior for \(H_4\), the full five-hypothesis distribution, and—when defined—the \(H_4\)-versus-\(H_3\) odds. State whether the substantive conclusion is stable across the considered settings.

A sensitivity analysis is not a search for the prior that produces a preferred conclusion. The values considered should be justified by the analysis context, and the range should be shown rather than selectively reporting only one favorable setting. If all reasonable settings support the same broad interpretation, the result is less dependent on prior calibration. If the conclusion changes, that dependence is itself important to report.

Common Mistakes and Interpretation

Double-counting configurations. Multiplying \(Mp_1\) by \(L_1\), for example, counts the \(M\) single-trait configurations in both factors. Use \(p_1L_1\), or multiply the aggregate prior weight by the average evidence \(L_1/M\).
Calling a prior weight a posterior probability. \(p_{12}\) is an input to the calculation. The posterior probability \(P(H_4\mid D)\) depends on all five weights and the data.
Ignoring the competing hypotheses. A large \(L_4\) or a high \(H_4\)-to-\(H_3\) odds ratio does not by itself imply a high posterior for \(H_4\) across all hypotheses. Check the normalized posterior.
Using the odds ratio when its denominator is zero. If \(W_3=p_1p_2L_3=0\), the displayed finite ratio is not defined. If \(W_3=0\) and \(W_4>0\), the \(H_4\)-to-\(H_3\) odds are infinite in the extended sense, because \(H_3\) has zero posterior weight. If both weights are zero, the odds are undefined, as is a conditional comparison restricted to \(H_3\) and \(H_4\).
Overstating what colocalization establishes. A high posterior for shared causality is conditional on the model, candidate variants, data, and prior choices used. It is not proof of a molecular mechanism or of a causal effect of the molecular trait on a clinical trait.

Key takeaway: Colocalization posteriors combine evidence with per-configuration prior weights. Calculate each hypothesis weight by summing its configuration-level prior-weighted evidence, normalize across all five hypotheses, and test whether the locus-level interpretation is stable across defensible prior settings.

Check Your Understanding

Try these questions using the definitions and calculations above.

With \(M\) candidate variants, why is \(p_1L_1\) the standard \(H_1\) weight rather than \(Mp_1L_1\)?
Explain which variant pairs contribute to \(L_3\), and why the same-variant products contribute instead to \(L_4\).
If \(p_{12}\) is multiplied by 4 while all other inputs are fixed, which unnormalized weight changes? What happens to the normalized posterior probability for \(H_4\)?
State the condition needed for the finite formula for posterior odds \(H_4\) versus \(H_3\), and describe the case where \(W_3=0\) but \(W_4>0\).
Why can a large \(H_4\)-versus-\(H_3\) odds ratio coexist with a modest posterior probability for \(H_4\) across all five hypotheses?