Choosing Effect-Size Priors for Fine-Mapping

Why the Effect-Size Prior Matters

Fine-mapping compares evidence for candidate variants, but that evidence depends not only on the observed association statistics. It also depends on what effect sizes the model regards as plausible before seeing those statistics. A prior that concentrates very near zero and a broad prior make different predictions for the observed estimate, so they can produce different Bayes factors and posterior inclusion probabilities.

Here the focus is the effect-size prior conditional on a variant being causal. This is distinct from the prior probability that the variant is causal at all. The earlier material on Bayes factors and prior odds establishes how these two ingredients combine; changing one should not be mistaken for changing the other.

Definition: For a candidate variant with effect $\beta$, a zero-centered normal effect-size prior with variance $W$ is $\beta\mid\text{causal}\sim N(0,W)$. Its prior standard deviation is $\tau=\sqrt{W}$. The prior is symmetric, assigns equal probability to positive and negative effects of the same magnitude, and expresses uncertainty about effect size conditional on causality.

The meaning of $W$ depends on the scale of $\beta$. If $\beta$ is a change in a quantitative trait measured in standard-deviation units per allele, then $\tau=0.2$ means a prior standard deviation of 0.2 trait standard deviations per allele. If the effect is on a log-odds scale, the same numerical value has a different interpretation. A prior scale should therefore be reported with its effect units, not just as a number.

Bayes Factors Under a Normal Effect Prior

Let $\hat\beta$ be the estimated effect for a candidate variant, with standard error $SE$, and let $V=SE^2$. Use the normal summary-statistic likelihood $\hat\beta\mid\beta\sim N(\beta,V)$. Under the null hypothesis of no effect, $\beta=0$; under the alternative, $\beta\sim N(0,W)$. The Bayes factor $BF_{10}$ is the likelihood of the observed summary statistic integrated over this alternative prior, divided by its likelihood under the null.

This formulation assumes that the summary estimate has an approximately normal likelihood, that $SE$ adequately represents its sampling uncertainty, and that the effect prior is on the same scale as $\hat\beta$. For a given variant, the same observed $\hat\beta$ and $SE$ are used when comparing prior scales.

Result: With $z=\hat\beta/SE$, define $r=W/(V+W)$. Under the normal likelihood and zero-centered normal effect prior, $$ BF_{10}=\sqrt{\frac{V}{V+W}}\exp\!\left(\frac{\hat\beta^2}{2}\left(\frac{1}{V}-\frac{1}{V+W}\right)\right) =\sqrt{1-r}\exp\!\left(\frac{z^2r}{2}\right). $$

The factor $\sqrt{1-r}$ penalizes a more dispersed alternative prior: its probability density is spread over a wider range of possible estimates. The exponential factor rewards a prior scale that gives substantial probability to effects compatible with the observed $z$-statistic. The balance between these factors is why Bayes factors need not increase as the prior variance increases.

Why the Bayes Factor Can Prefer an Intermediate Scale

The scale dependence can be seen directly from the formula. As $W$ approaches zero, $r$ approaches zero and $BF_{10}$ approaches 1: the alternative prior collapses toward the null, so it becomes difficult to distinguish the hypotheses. As $W$ grows without bound, $r$ approaches 1; for a fixed finite $z$, the square-root factor approaches zero while the exponential factor remains finite. The resulting Bayes factor also approaches zero. An extremely broad prior can therefore make the observed data look relatively unsurprising under the null.

For $z^2>1$, the scale giving the maximum Bayes factor in this one-variant calculation can be found by differentiating its log with respect to $r$:

\log BF_{10}=\frac{1}{2}\log(1-r)+\frac{z^2r}{2}, \qquad \frac{d}{dr}\log BF_{10}=-\frac{1}{2(1-r)}+\frac{z^2}{2}.

Setting the derivative to zero gives $r=1-1/z^2$, which lies between zero and one when $z^2>1$. The second derivative is $-1/[2(1-r)^2]$, which is negative, so this stationary point is a maximum. This is a mathematical description of the scale that best matches this particular $z$-statistic under the stated model; it is not a rule for choosing a prior after looking at the data. Choosing a scale to maximize the observed Bayes factor would make the reported evidence data-dependent and would ignore prior justification.

Worked Example: Comparing Three Prior Variances

Suppose a hypothetical candidate variant has an estimated effect of $0.60$ standard deviations per allele and $SE=0.20$ standard deviations per allele. Thus $z=0.60/0.20=3$, and $z^2=9$. Compare causal-effect priors with standard deviations $0.10$, $0.20$, and $0.40$ standard deviations per allele. The corresponding variances $W$ are $0.01$, $0.04$, and $0.16$; the sampling variance is $V=SE^2=0.04$.

For $W=0.01$, $r=0.01/(0.04+0.01)=0.20$. Therefore, $$ BF_{10}=\sqrt{1-0.20}\exp(9(0.20)/2) =\sqrt{0.80}\exp(0.90) \approx 2.200. $$

For $W=0.04$, $r=0.04/(0.04+0.04)=0.50$, so $$ BF_{10}=\sqrt{0.50}\exp(9(0.50)/2) =\sqrt{0.50}\exp(2.25) \approx 6.709. $$

For $W=0.16$, $r=0.16/(0.04+0.16)=0.80$, giving $$ BF_{10}=\sqrt{0.20}\exp(9(0.80)/2) =\sqrt{0.20}\exp(3.60) \approx 16.37. $$

For these data, the broader of the three priors gives the largest Bayes factor. This does not establish that the broadest prior is generally best: the result depends on the observed $z$-statistic, and a still broader prior could reduce the Bayes factor. The comparison is meaningful only if all three prior standard deviations are defensible on the trait-effect scale.

From a Bayes Factor to a Posterior Inclusion Probability

For a single candidate variant, let $\pi$ be its prior probability of being causal and assume the noncausal state has the null effect. The posterior odds equal the prior odds multiplied by $BF_{10}$. Equivalently, the posterior inclusion probability is

P(\text{causal}\mid\text{data}) =\frac{\pi BF_{10}}{(1-\pi)+\pi BF_{10}}.

This conversion requires both ingredients: the Bayes factor, which depends on the effect-size prior, and the prior causal probability $\pi$. A large Bayes factor does not by itself determine a posterior probability. The following calculation uses the single-causal-variant setting and formula already introduced in the earlier material on posterior inclusion probabilities.

Worked Example: Converting a Bayes Factor to a Posterior Probability

In a hypothetical analysis, suppose $z^2=16$, and the chosen effect-size prior gives $r=0.50$. Then $$ BF_{10}=\sqrt{1-0.50}\exp(16(0.50)/2) =\sqrt{0.50}\exp(4) \approx 38.61. $$ Assume the variant’s prior probability of being causal is $\pi=0.01$. The prior probability and the effect-size prior are separate assumptions.

Substitution into the posterior formula gives $$ P(\text{causal}\mid\text{data}) =\frac{38.61(0.01)}{38.61(0.01)+0.99} =\frac{0.3861}{1.3761} \approx 0.2806. $$

The posterior inclusion probability is about $28.1\%$ under these specified assumptions. It is not the probability that the null hypothesis is true, nor does it establish that the variant is causal in reality. It summarizes uncertainty under the model and its priors.

Calibrating a Scale and Checking Sensitivity

A useful prior scale begins with an effect that subject-matter knowledge regards as plausible, expressed in the model’s units. For a zero-centered normal prior, about 95% of its mass lies between $-1.96\tau$ and $1.96\tau$. This provides a direct way to state what a proposed standard deviation means. It is an elicitation aid, not a claim that all causal effects must fall within that range.

Worked Example: Translating a Plausible Effect Range Into a Prior

Suppose a hypothetical study analyzes a standardized quantitative trait, with $\beta$ measured in trait standard deviations per allele. Investigators consider effects between about $-0.49$ and $0.49$ standard deviations to be a reasonable central 95% prior range. Since $1.96\tau=0.49$, the prior standard deviation is $$ \tau=0.49/1.96=0.25, \qquad W=\tau^2=0.0625. $$

Now suppose a hypothetical variant has $\hat\beta=1.50$ standard deviations per allele and $SE=0.50$ standard deviations per allele. Then $V=0.25$, $z=1.50/0.50=3$, and $$ r=\frac{0.0625}{0.25+0.0625}=0.20. $$ Its Bayes factor under this prior is $$ BF_{10}=\sqrt{0.80}\exp(9(0.20)/2) =\sqrt{0.80}\exp(0.90) \approx 2.200. $$

The calculation shows how to translate a stated plausible range into $W$ and then evaluate its evidential consequence. The estimated effect is larger than the prior’s central 95% range, but the likelihood is uncertain; the Bayes factor integrates over effects rather than simply checking whether $\hat\beta$ falls inside that range. The range and resulting prior should be assessed for scientific plausibility before interpreting the evidence.

A sensitivity analysis asks whether conclusions are robust to reasonable alternatives for $W$. It should vary the effect-size scale while making clear which other assumptions are held fixed. For example:

Specify a primary prior standard deviation and one or more alternatives, all in the same effect units.
For each scale, recompute the Bayes factors and posterior inclusion probabilities using the same data and causal prior probabilities.
Compare which variants receive substantial posterior support and whether the posterior concentration or credible-set membership changes.
Report the scale choices and the extent of any changes, rather than presenting only the most favorable result.

In a multi-variant analysis, the same effect prior can affect relative support across variants when their estimates or standard errors differ. Thus sensitivity is not only about whether an individual Bayes factor changes; it is also about whether the resulting allocation of posterior probability among candidates changes. Interpretation remains conditional on the model, including its likelihood and variant set.

Common Pitfall and Key Takeaway

A common mistake is to reason, “A broader prior includes more possible effects, so it must give stronger evidence for association.” This overlooks the density penalty: spreading prior probability across a wider range can reduce the integrated likelihood for the observed data. Conversely, a very narrow prior may barely distinguish the alternative from the null. The correction is to compute or evaluate the Bayes factor across scientifically defensible scales; neither “broader is better” nor “narrower is safer” is generally valid.

Another important distinction is between the effect-size prior and the prior probability of causality. Altering $W$ changes the likelihood averaged under the causal alternative and therefore changes $BF_{10}$. Altering $\pi$ changes the prior odds. Both can affect the posterior inclusion probability, but they represent different assumptions and should be reported separately.

Key takeaway: A normal effect-size prior is a substantive assumption about plausible causal effects, on a specified scale. Its variance can change fine-mapping Bayes factors and posterior inclusion probabilities, so justify the scale in context and check whether conclusions persist across reasonable alternatives.

Check Your Understanding

Use the model assumptions and formulas in this tutorial to answer the following questions.

What does $W$ describe, and why must its units match the scale of $\beta$?
For fixed $z$, what happens to $BF_{10}$ as $W$ approaches zero? Explain using $r$.
With $z^2=9$ and $r=0.50$, calculate $BF_{10}$ from the formula.
Why can a broader effect-size prior reduce rather than increase a Bayes factor?
In a sensitivity analysis, why should changes to $W$ be distinguished from changes to the prior causal probability $\pi$?