Prior Sensitivity Analysis for Fine-Mapping

Why Examine Prior Sensitivity?

Fine-mapping combines the observed association evidence with prior probabilities over which variants may be causal and how many causal variants may be present. A posterior result is therefore conditional on both the data model and the priors. Prior sensitivity analysis asks a focused question: do the main fine-mapping conclusions persist across plausible choices of those priors?

The analysis is informative only if the comparison is structured. Keep the summary statistics, LD input, candidate variants, likelihood specification, and computational procedure fixed while changing the prior under study. First vary variant priors while holding the causal-count prior fixed; then vary the causal-count prior while holding variant priors fixed. A final set of joint scenarios can show whether plausible changes in both prior components reinforce or offset one another.

This is not a procedure for selecting whichever prior produces a preferred variant. Nor does stability prove that the model is correct. It describes how conclusions respond to specified prior changes, conditional on the analysis being held fixed.

Configuration Priors and Posterior Summaries

Definition: Let $J$ be the set of candidate variants and let $C\subseteq J$ denote a causal configuration, with $k=|C|$ causal variants. Let $p_k=P(|C|=k)$ be the prior probability of causal count $k$. Give variant $j$ a positive prior weight $w_j$. Conditional on $k$, assign a configuration probability proportional to the product of its variant weights: $$ P(C\mid |C|=k)=\frac{\prod_{j\in C}w_j}{e_k(w)},\qquad e_k(w)=\sum_{\substack{A\subseteq J\\|A|=k}}\prod_{j\in A}w_j. $$ For $k=0$, the empty product and $e_0(w)$ equal 1. Thus $P(C)=p_k\prod_{j\in C}w_j/e_k(w)$. This convention separates the prior distribution on causal count from relative variant weights within each count.

This is one explicit, coherent way to define the priors used in the examples. Other implementations, including independent Bernoulli inclusion priors, can induce different distributions over configurations. Do not mix conventions within a sensitivity comparison: state the prior construction and use it consistently.

Let $D$ denote the observed fine-mapping data and $BF_C$ the Bayes factor for configuration $C$ against the null configuration, with $BF_{\varnothing}=1$. Under a fixed likelihood model, the posterior probability of $C$ is

P(C\mid D)= \frac{BF_C P(C)} {\sum_{A\subseteq J}BF_A P(A)}.

The posterior inclusion probability (PIP) for variant $j$ is $PIP_j=\sum_{C:j\in C}P(C\mid D)$. It is a marginal posterior probability that variant $j$ is included in the causal configuration; across variants, PIPs need not sum to one when multiple causal variants are allowed.

A Justification for Comparing Scenarios

For any two configurations $C$ and $A$, Bayes' rule gives the cross-multiplied identity

P(C\mid D)BF_A P(A)=P(A\mid D)BF_C P(C).

When both prior probabilities are positive and $BF_A>0$, division gives the posterior-odds form

\frac{P(C\mid D)}{P(A\mid D)} = \frac{BF_C}{BF_A}\frac{P(C)}{P(A)}.

Thus a change in relative prior probability changes posterior odds unless the likelihood evidence or other prior terms offset it. The cross-multiplied form avoids dividing by a zero Bayes factor. For a sensitivity analysis, the important practical consequence is to recompute the normalized posterior under each scenario rather than infer PIPs from prior weights alone.

Worked Examples

Worked Example: Variant Weights Change a 95% Credible Set

Suppose a hypothetical region has three candidate variants and the model allows exactly one causal variant. The configuration Bayes factors are 20 for variant 1, 12 for variant 2, and 2 for variant 3. Compare equal variant weights with weights $w=(2,1,1)$. Because exactly one variant is causal, the conditional prior probabilities are the normalized weights.

Equal weights. Each variant has prior probability $1/3$. The unnormalized posterior weights are $20/3,12/3,2/3$, whose exact sum is $34/3=11.3333\ldots$. The common factor $1/3$ cancels, so the PIPs are $20/34=0.5882$, $12/34=0.3529$, and $2/34=0.05882$. The two largest sum to $32/34=0.941176\ldots$, or 0.9412. A PIP-ranked 95% credible set therefore needs all three variants.

Weights $w=(2,1,1)$. The prior probabilities are $1/2,1/4,1/4$. The unnormalized posterior weights are $20(1/2)=10$, $12(1/4)=3$, and $2(1/4)=0.5$, with total 13.5. The PIPs are $10/13.5=0.7407$, $3/13.5=0.2222$, and $0.5/13.5=0.03704$. The top two sum to $13/13.5=0.9630$, so the 95% set contains variants 1 and 2.

The rank order is unchanged, but the credible set shrinks from $\{1,2,3\}$ to $\{1,2\}$. Their intersection-over-union is $2/3=0.6667$. This is a substantive set change despite the same leading variant.

Worked Example: A Causal-Count Prior Changes PIPs

Consider a hypothetical two-variant region allowing zero, one, or two causal variants. The Bayes factors for the empty, variant-1-only, variant-2-only, and two-variant configurations are 1, 8, 4, and 10. Use equal variant weights. Conditional on one causal variant, each singleton has prior probability $1/2$; the two-variant configuration is the only configuration at count two.

Scenario A: let $p_0=0.2,\ p_1=0.6,\ p_2=0.2$. The unnormalized posterior weights for the four configurations are $0.2,\ 0.6(1/2)(8)=2.4,\ 0.6(1/2)(4)=1.2,$ and $0.2(10)=2$. Their total is 5.8. The posterior configuration probabilities are therefore $0.03448,\ 0.4138,\ 0.2069,$ and $0.3448$, respectively. The PIPs are $0.4138+0.3448=0.7586$ for variant 1 and $0.2069+0.3448=0.5517$ for variant 2.

Scenario B: change only the count prior to $p_0=0.6,\ p_1=0.3,\ p_2=0.1$. The unnormalized weights are $0.6,\ 1.2,\ 0.6,\ 1.0$, with total 3.4. The PIPs are $(1.2+1.0)/3.4=0.6471$ for variant 1 and $(0.6+1.0)/3.4=0.4706$ for variant 2. The leading variant is stable, but both marginal inclusion probabilities decrease. The posterior probability of two causal variants also changes, from $2/5.8=0.3448$ to $1/3.4=0.2941$. This comparison isolates the effect of changing the causal-count prior.

Worked Example: Rank Shifts and Set Overlap Across Variant Priors

For a hypothetical single-causal analysis of four variants, suppose the Bayes factors are 16, 12, 4, and 2. Compare equal weights with $w=(1,2,1,1)$, which gives variant 2 twice the relative prior weight. Since exactly one variant is causal, posterior probabilities are proportional to Bayes factor times weight.

With equal weights the total is $16+12+4+2=34$, giving PIPs $0.4706, 0.3529, 0.1176,$ and $0.05882$, in that order. Under $w=(1,2,1,1)$, the weighted values are $16,24,4,2$, totaling 46; the PIPs are $0.3478, 0.5217, 0.0870,$ and $0.04348$. Variant 1 moves from rank 1 to rank 2 (a rank shift of $+1$, where larger rank numbers mean lower position); variant 2 moves from rank 2 to rank 1 (shift $-1$).

At an 85% cumulative-PIP threshold, the equal-weight set needs the first three variants: the top two sum to $28/34=0.8235$, while the top three sum to $32/34=0.9412$. Under the alternative weights, variants 2 and 1 sum to $40/46=0.8696$, so the set is $\{1,2\}$. The intersection-over-union is $2/3$. These are single-causal credible sets, where cumulative PIP has the usual direct interpretation as posterior mass on the causal variant being in the set.

A Structured Sensitivity Workflow

Define a baseline.
Record the candidate variants, likelihood and LD inputs, variant-weight convention, causal-count distribution, and baseline posterior summaries.

Specify plausible alternatives in advance.
Choose a small set of defensible variant-weight and count-prior scenarios. State which prior component changes in each scenario and why it is plausible.

Recompute all posterior quantities.
For every scenario, normalize configuration probabilities and recalculate PIPs, posterior ranks, and the same defined credible-set procedure.

Summarize changes and limits.
Report rank shifts, PIP differences, and overlap of matched sets. Separate conclusions that persist from those that depend on prior choice.

For two sets $S_1$ and $S_2$, a simple overlap summary is the Jaccard index $|S_1\cap S_2|/|S_1\cup S_2|$. It ranges from zero (no shared variants) to one (identical sets). When sets are defined for distinct signals, compare corresponding signals rather than matching sets solely by their position in a reordered list. For multiple-causal models, state the credible-set construction and signal assignment; a cumulative sum of marginal PIPs is not automatically a 95% probability that the set contains at least one causal variant.

Diagnostics, Alternatives, and Limitations

Large rank reversals, substantial PIP changes, or low set overlap indicate that the reported resolution is prior-sensitive. A useful diagnostic table includes each scenario's assumptions, PIPs, ranks, set membership, and overlap with the baseline. Also inspect whether conclusions change only for nearly tied variants or whether the leading candidates change substantially. Verify that every scenario has normalized priors and that PIPs are calculated from unrounded posterior configuration probabilities.

A common mistake is to interpret a high PIP under one favored prior as evidence that the variant is robustly identified. The correction is to distinguish the posterior conditional on that prior from the conclusion that remains stable across plausible priors. Another mistake is to change the causal-count prior and variant weights simultaneously, then attribute the result to one component; use one-at-a-time comparisons before joint scenarios.

Prior sensitivity analysis differs from checking alternative likelihood or LD inputs: here the likelihood and LD are held fixed by design. It also differs from posterior uncertainty within one specified analysis, because it compares results across prior specifications rather than summarizing uncertainty under a single specification. It cannot repair a misspecified likelihood, omitted variants, or inaccurate LD. Nor is there a universal threshold for an acceptable rank shift or overlap: report the numerical changes and explain their consequences for the scientific claim.

Key takeaway: A transparent fine-mapping sensitivity analysis states exactly which prior changes, holds the data model fixed, recomputes posterior summaries, and reports both stable findings and prior-dependent ones.

Check Your Understanding

Answer each question, then compare with the solutions below.

With one causal variant and Bayes factors 6 and 3, equal prior weights are changed to weights 2 and 1. What are the posterior probabilities under each scenario?
Why should variant weights and the causal-count prior first be varied separately?
Two matched credible sets are $\{a,b,c\}$ and $\{b,c,d\}$. Calculate their Jaccard index.
In a model allowing multiple causal variants, why is a PIP-ranked cumulative set not automatically a 95% probability set for containing at least one causal variant?

Solutions. 1. Equal weights give probabilities $6/(6+3)=2/3=0.6667$ and $3/(6+3)=1/3=0.3333$. Weights 2 and 1 give weighted evidence 12 and 3, so the probabilities are $12/15=0.8000$ and $3/15=0.2000$. 2. Separate changes identify which prior component accounts for a posterior change; changing both at once confounds their effects. 3. The intersection has size 2 and the union $\{a,b,c,d\}$ has size 4, so the Jaccard index is $2/4=0.5$. 4. PIPs are marginal inclusion probabilities and can count posterior mass from configurations containing different numbers of causal variants; their cumulative sum does not directly equal the probability that at least one member of the set is causal.