Sensitivity to Effect-Size Priors

Why Examine Effect-Size Priors?

Fine-mapping compares how well alternative causal explanations account for the observed association evidence. That comparison depends not only on the data, but also on assumptions about plausible effect sizes. A prior concentrated near zero favors small effects; a more dispersed prior assigns more probability to larger effects. Consequently, the same association statistic can yield different Bayes factors under different effect-size priors, and those differences can change posterior support for variants.

The purpose of this tutorial is to examine that dependence directly. In each comparison, keep the variant-specific prior probabilities and the causal model fixed, and change only the prior distribution for the effect size. This isolates the question of whether a fine-mapping conclusion relies on a particular effect-size assumption. It does not determine which prior is correct; that requires scientific justification and, where possible, external information or calibration.

How the Prior Enters a Bayes Factor

For a single variant, let \(\hat{\beta}\) be its estimated effect, with estimated sampling variance \(V\), and let \(z=\hat{\beta}/\sqrt{V}\). Under the null of no effect, suppose \(\hat{\beta}\mid \beta=0\sim N(0,V)\). Under the causal alternative, use the normal prior \(\beta\sim N(0,W)\), where \(W\) is the prior variance of the effect. Assume the estimate is approximately normal with variance \(V\), and that this likelihood and prior are appropriate for the effect scale being analyzed.

Integrating over the prior gives the alternative marginal distribution \(\hat{\beta}\sim N(0,V+W)\). The Bayes factor \(BF\) is the alternative marginal density divided by the null density. Taking the ratio of the two normal densities gives:

Equivalently, with \(r=W/V\), the prior variance relative to the sampling variance:

The first term penalizes a more dispersed alternative prior because it spreads probability over a wider range of possible estimates. The second term rewards a prior that can accommodate the observed association. Thus, increasing \(W\) does not always increase the Bayes factor: the balance depends on the observed \(z\)-statistic. The prior variance \(W\) is measured in squared units of the effect, while \(V\) is the sampling variance in those same units. A numerical value for \(W\) is meaningful only in relation to the phenotype and effect scale.

With variant-specific prior probabilities \(\pi_j\), posterior probabilities are obtained by combining each variant’s Bayes factor with its prior probability, together with the probability assigned to the null or other allowed explanations. If the causal model and the \(\pi_j\) values are fixed, changing the effect-size prior changes the Bayes factors, not the prior probabilities themselves.

Changing the Shape of the Prior

Sensitivity is not limited to changing \(W\) in a single normal prior. A prior may, for example, put most of its probability on small effects but retain a smaller component for larger effects. Such a mixture can represent uncertainty about the scale of plausible effects. If the prior assigns weight \(a_k\) to a normal component with variance \(W_k\), where the nonnegative weights sum to one, the marginal likelihood averages the component marginal likelihoods. Therefore, the mixture Bayes factor is the weighted average of the component Bayes factors, not the exponential of a weighted average of their log Bayes factors.

Posterior Effect Estimates Also Depend on the Prior

Prior sensitivity can affect effect estimates as well as variant support. Under the normal likelihood and normal prior above, the posterior distribution for the effect is normal, with mean \(W\hat{\beta}/(V+W)\) and variance \(VW/(V+W)\). This follows by multiplying the likelihood density in \(\beta\), proportional to \(\exp\{-(\hat{\beta}-\beta)^2/(2V)\}\), by the prior density, proportional to \(\exp\{-\beta^2/(2W)\}\), and completing the square in \(\beta\).

In the previous example, the single prior \(W=1\) gives posterior mean \(1(2)/(1+1)=1.00\) effect units. In the mixture, the narrow and wide components give posterior means \(0.10(2)/1.10=0.182\) and \(4(2)/5=1.60\). Averaging these component means using the updated weights gives approximately \(0.674(0.182)+0.326(1.60)=0.645\) effect units. This is an illustration of model-conditional shrinkage, not a claim that the more strongly shrunk estimate is necessarily closer to the unknown true effect.

A Practical Comparison and Its Diagnostics

A focused comparison changes one effect-size assumption at a time. For instance, compare a set of plausible prior variances on the same effect scale, then compare a single normal prior with a scientifically defensible mixture. Keep the observed summary statistics, LD information, variant-specific prior probabilities, and causal model unchanged. For every fit, record the Bayes factors, posterior inclusion probabilities (PIPs), and the leading variants. A useful report states both what remains stable and what changes.

Large changes in PIPs across reasonable priors are a warning that variant-level conclusions depend on the assumed effect-size distribution. They may be especially consequential when variants have similar association evidence: modest changes to relative Bayes factors can alter their posterior ranking. Conversely, stable PIPs across a meaningful range of effect priors provide evidence of robustness to this particular assumption, not proof that all model assumptions are correct.

Check that the effect scale is consistent across analyses, that \(W\) is expressed in squared units matching \(V\), and that mixture weights sum to one. Inspect whether conclusions change smoothly as \(W\) varies or jump because of a coding, numerical, or model-comparison error. Confirm that a mixture Bayes factor was computed by averaging Bayes factors, and that posterior probabilities were normalized over the same set of hypotheses in every comparison. If changing the effect prior also changes the variant priors, LD inputs, or causal model, the comparison no longer isolates effect-size sensitivity.

Limitations and a Common Mistake

The normal approximation and the Bayes factor formula require a suitable approximately normal effect estimate with a valid sampling variance, and a prior defined on the same effect scale. The approximation may be unreliable if the summary statistics are poorly calibrated, the sampling variance is wrong, or the likelihood is materially non-normal. A normal prior also may not represent scientific knowledge about effect sizes. A mixture broadens the range of possibilities, but its component variances and weights still require justification.

Common mistake: “The more dispersed prior produced the larger Bayes factor, so larger effect sizes must be more plausible and the variant is more likely causal.” This reasoning confuses a model comparison conditional on a chosen prior with evidence that the prior is true. The Bayes factor can change because the prior redistributes probability across effect sizes; the posterior also depends on prior causal odds and the other candidate explanations. The correction is to report the prior assumptions, compare defensible alternatives, and describe any resulting PIP change as sensitivity rather than as confirmation.

Effect-size prior sensitivity is distinct from changing how many causal variants are allowed or how prior probability is allocated among variants. Those changes answer different modeling questions. Keep them separate when interpreting an analysis so that the source of a changed posterior is identifiable.