Averaging Over Uncertainty in the Number of Causal Variants

Why Average Over the Causal Count?

Fine-mapping asks which variants in a locus are causal, given the observed data and a specified model. A fixed-count analysis answers that question under a particular assumption: exactly \(K\) variants are causal. But the count itself is often uncertain. If the data do not clearly distinguish one causal variant from two, reporting results only under one chosen count can conceal that uncertainty and make variant probabilities depend heavily on the choice.

Posterior model averaging treats the causal count as unknown. It combines the results from candidate counts, weighting each count by its posterior probability given the data. This does not mean that counts are equally plausible, or that one should average results without regard to evidence. The weights arise from the prior on the count and the marginal evidence for the data under each count.

Definition: Let \(D\) denote the fine-mapping data, let \(C\) be a configuration (a set of causal variants), and let \(K=|C|\) be its causal count. A fixed-count analysis conditions on one specified value of \(K\). Posterior model averaging instead computes posterior quantities by averaging over the possible values of \(K\), using \(P(K\mid D)\) as the weight for count \(K\).

The candidate variants, likelihood, effect-size prior, and configuration prior must be specified consistently across the counts being compared. Suppose \(\pi(K)\) is the prior probability of count \(K\), and \(\pi(C\mid K)\) is a prior over configurations of that count, normalized so that its probabilities sum to one. Let \(m_C(D)\) be the likelihood integrated over the effect parameters for configuration \(C\). The count-specific marginal evidence is

Z_K=P(D\mid K)=\sum_{C:\,|C|=K}\pi(C\mid K)m_C(D).

The evidence averages over configurations of count \(K\), while accounting for their prior probabilities. The posterior probability of the count is then proportional to its prior probability times this evidence:

P(K\mid D)= \frac{\pi(K)Z_K} {\displaystyle\sum_{k=0}^{K_{\max}}\pi(k)Z_k}.

Here \(K_{\max}\) is the largest count included in the analysis. A count prior may assign zero probability to some counts. In practice, the included range and prior should be chosen to represent plausible causal counts; posterior averaging cannot account for counts excluded from the model space.

From Count-Specific Results to Variant Probabilities

For a variant \(j\), define its posterior inclusion probability (PIP) as the posterior probability that \(j\) belongs to the causal configuration. Within a fixed count \(K\), its conditional inclusion probability is \(P(j\in C\mid K,D)\). The model-averaged PIP follows from the law of total probability: partition the posterior according to the possible values of \(K\), then add the conditional probabilities weighted by the posterior count probabilities.

\operatorname{PIP}_j =P(j\in C\mid D) =\sum_{K=0}^{K_{\max}} P(j\in C\mid K,D)P(K\mid D).

For \(K=0\), every conditional inclusion probability is zero. The formula therefore combines evidence about which variants are included within each count with evidence about which counts are plausible. It also shows why a model-averaged PIP need not match the PIP from the single count with the largest posterior probability.

Worked Example: Updating the Probability of the Causal Count

Suppose a hypothetical analysis considers \(K=0,1,2\). Let the prior count probabilities be \(0.20, 0.50, 0.30\), respectively. For illustration, suppose the count-specific marginal evidences, on a common relative scale, are \(1, 2, 1\). Calculate the posterior probability of each count.

The unnormalized posterior weights are the prior probabilities multiplied by the evidence:

\begin{aligned} w_0&=0.20(1)=0.20,\\ w_1&=0.50(2)=1.00,\\ w_2&=0.30(1)=0.30. \end{aligned}

Their sum is \(0.20+1.00+0.30=1.50\). Normalizing gives

P(K=0\mid D)=\frac{0.20}{1.50}=0.1333,\qquad P(K=1\mid D)=\frac{1.00}{1.50}=0.6667,\qquad P(K=2\mid D)=\frac{0.30}{1.50}=0.2000.

Thus, \(K=1\) has the largest posterior probability, but it does not have certainty: the posterior assigns \(0.3333\) probability to the other two counts combined. The posterior odds and probabilities here are conditional on the specified count prior, configuration priors, likelihood, and candidate count range.

A short derivation makes the averaging rule explicit. The posterior probability of a variant being included is the sum of the posterior probabilities of all configurations that contain it. Group those configurations by their count, and use the definition of conditional probability:

\begin{aligned} P(j\in C\mid D) &=\sum_K P(j\in C,\ K\mid D)\\ &=\sum_K P(j\in C\mid K,D)P(K\mid D). \end{aligned}

This is a probability identity, not a special approximation. Approximation can enter when a method estimates the count-specific probabilities or evidence, but the averaging formula itself is exact for the specified posterior model.

Worked Example: Averaging Can Change Variant Rankings

Consider three hypothetical variants, A, B, and C. Suppose the posterior probability of \(K=1\) is \(0.30\) and of \(K=2\) is \(0.70\). Within \(K=1\), the conditional PIPs for A, B, and C are \(0.80, 0.15, 0.05\). Within \(K=2\), suppose they are \(0.45, 0.80, 0.75\). These are valid inclusion probabilities for a two-variant configuration model: they sum to 2.

For A, the model-averaged PIP is

\operatorname{PIP}_A=0.30(0.80)+0.70(0.45)=0.240+0.315=0.555.

For B and C, the same calculation gives

\begin{aligned} \operatorname{PIP}_B&=0.30(0.15)+0.70(0.80)=0.045+0.560=0.605,\\ \operatorname{PIP}_C&=0.30(0.05)+0.70(0.75)=0.015+0.525=0.540. \end{aligned}

Under \(K=1\) alone, A ranks first. After averaging, B ranks first, followed by A and C. This change occurs because the posterior gives more weight to \(K=2\), under which B and C have higher inclusion probabilities. The sum of the averaged PIPs is \(0.555+0.605+0.540=1.700\), which equals the posterior expected causal count, \(0.30(1)+0.70(2)=1.700\). It is not required to equal one when multiple causal variants are possible.

Credible Sets When the Count Is Uncertain

In a single-causal-variant model, a credible set is commonly formed by ranking variants by posterior probability and including them until their cumulative posterior probability reaches a chosen target, such as 95%. With multiple causal variants, marginal PIPs remain useful for ranking, but the sum of PIPs in a set is not generally the posterior probability that the set contains at least one causal variant. A configuration can contain several members of the set, so summing their inclusion probabilities can count the same configuration more than once.

If the intended coverage event is “at least one causal variant is in the set \(S\),” its posterior probability is obtained directly from configurations:

P(C\cap S\ne\varnothing\mid D) =\sum_C \mathbf{1}(C\cap S\ne\varnothing)P(C\mid D),

where \(\mathbf{1}(\cdot)\) is 1 when the condition holds and 0 otherwise. This calculation automatically includes uncertainty in \(K\), because configurations of every allowed count contribute according to their posterior probability. Other coverage goals, such as including all causal variants, define a different event and require a different calculation. A report should state which event its credible-set coverage refers to.

Worked Example: Checking Coverage for a Set of Variants

In a hypothetical analysis, suppose \(P(K=1\mid D)=0.40\) and \(P(K=2\mid D)=0.60\). Under \(K=1\), the posterior configuration probabilities are \(P(\{A\}\mid K=1,D)=0.55\), \(P(\{B\}\mid K=1,D)=0.35\), and \(P(\{C\}\mid K=1,D)=0.10\). Under \(K=2\), they are \(P(\{A,B\}\mid K=2,D)=0.60\), \(P(\{A,C\}\mid K=2,D)=0.25\), and \(P(\{B,C\}\mid K=2,D)=0.15\). Each within-count distribution sums to 1.

Take \(S=\{A,B\}\). Under \(K=1\), the set contains a causal variant for configurations \(\{A\}\) and \(\{B\}\), giving probability \(0.55+0.35=0.90\). Under \(K=2\), every listed configuration contains A or B, giving probability 1. The model-averaged probability that \(S\) contains at least one causal variant is therefore

P(C\cap S\ne\varnothing\mid D) =0.40(0.90)+0.60(1)=0.36+0.60=0.96.

Thus this set has 96% posterior probability of containing at least one causal variant, under the stated model. It does not have 96% probability of containing every causal variant. For example, under \(K=2\), configuration \(\{A,C\}\) includes a causal variant outside the set.

The marginal PIPs in the same analysis are \(\operatorname{PIP}_A=0.40(0.55)+0.60(0.85)=0.73\), \(\operatorname{PIP}_B=0.40(0.35)+0.60(0.75)=0.59\), and \(\operatorname{PIP}_C=0.40(0.10)+0.60(0.40)=0.28\). Their sum is 1.60, the expected count \(0.40(1)+0.60(2)=1.60\), not the 0.96 set-coverage probability.

Assumptions and Practical Interpretation

Count averaging is meaningful only relative to the models being averaged. The candidate variants and data likelihood must be comparable across counts, and each count must have a defined prior over its configurations and effect parameters. The count prior matters: it influences posterior count probabilities, which in turn influence every model-averaged PIP. Marginal evidence also matters because it accounts for how well each count-specific model predicts the observed data after averaging over its parameters and configurations.

A posterior probability is conditional on those modeling choices. For example, a model-averaged PIP of 0.73 means that the variant has posterior probability 0.73 of being included in the causal configuration, given the data and specified model. It is not a probability that the variant is causal under every possible model, and it does not establish that the variant biologically causes the trait. Fine-mapping posterior results support prioritization and uncertainty description, not a causal conclusion independent of assumptions.

When reporting results, it is useful to present the posterior distribution of \(K\), model-averaged PIPs, and the definition of any reported set-level coverage. If conclusions change sharply across plausible count priors or count ranges, that sensitivity is part of the inferential result rather than a nuisance to hide.

Common Mistake

Wrong reasoning: “The most probable count is \(K=1\), so I can report the \(K=1\) PIPs as if the count were known.” A most probable count need not have posterior probability near 1.
Correction: Use the posterior probabilities of all included counts to calculate model-averaged PIPs, unless the analysis is explicitly intended to condition on a fixed count. Report the count posterior so readers can see how much uncertainty remains.
Wrong reasoning: “The PIPs in my multi-variant set sum to 0.95, so the set has 95% probability of containing a causal variant.”
Correction: In a multi-causal model, calculate the posterior probability of the stated set event from configuration probabilities. A sum of PIPs can count configurations more than once.

Key takeaway: Averaging over causal-count uncertainty means weighting each count-specific posterior by \(P(K\mid D)\). This yields model-averaged variant inclusion probabilities, while set-level coverage must be calculated for its explicitly stated event using posterior configuration probabilities.

Check Your Understanding

Try these questions before reviewing the calculations above.

Why does the posterior probability of a count depend on both its prior probability and its marginal evidence?
A variant has conditional PIPs of 0.20 under \(K=1\) and 0.80 under \(K=2\). If the posterior count probabilities are 0.25 and 0.75, what is its model-averaged PIP?
Why can the sum of model-averaged PIPs exceed 1? What quantity does their sum equal in expectation?
For a multi-causal model, why is the sum of the PIPs in a set not generally its probability of containing at least one causal variant?
What information should accompany a reported model-averaged PIP so its conditioning assumptions are clear?