The Single-Causal-Variant Posterior

From Variant Evidence to a Posterior

Fine-mapping asks which candidate variant, if any, accounts for an association signal. This tutorial considers a deliberately restricted model: the region contains exactly one causal variant, and that variant is among the $m$ candidates under consideration. The goal is to combine the evidence in the observed data with prior probabilities for those candidates. The resulting probabilities are conditional on this model; they do not by themselves establish that the model is true.

Let $D$ denote the observed data for the region, such as summary statistics together with the information needed for their likelihood. For candidate $j$, define $H_j$ to mean that candidate $j$ is the sole causal variant and all other candidates are noncausal. These hypotheses are mutually exclusive and, under the exactly-one assumption, exhaustive. Write $\pi_j=P(H_j)$ for the prior probability of $H_j$, with $\sum_{j=1}^m\pi_j=1$. Let $L_j=p(D\mid H_j)$ be the likelihood evidence for that hypothesis. For Bayesian comparison, $L_j$ is the marginal likelihood: nuisance parameters, including an unknown effect size when appropriate, are integrated over their specified prior distributions.

Definition: Under the model that exactly one of $m$ candidate variants is causal, the posterior probability for candidate $j$ is $$ P(H_j\mid D)=\frac{\pi_j L_j}{\sum_{k=1}^{m}\pi_k L_k}. $$ The numerator is the prior probability times the evidence under $H_j$; the denominator sums these quantities over all mutually exclusive candidates.

The data likelihoods for the candidate hypotheses can be obtained from the Gaussian summary-statistic likelihood developed in the earlier material on that subject. Here, the task is to combine those variant-specific likelihoods rather than derive their sampling distribution. The posterior depends on both evidence and prior probability: a candidate with the largest likelihood need not have the largest posterior if its prior is sufficiently smaller than a competitor's.

Bayes Factors and Prior Odds

It is often more convenient to express evidence as a likelihood ratio. Relative to a chosen reference hypothesis $H_r$, the Bayes factor for candidate $j$ is $BF_{jr}=L_j/L_r$. A Bayes factor greater than one means the data are more probable under $H_j$ than under $H_r$, while a value less than one favours the reference over $H_j$. A Bayes factor is not itself a posterior probability.

Dividing the posterior probability for $H_j$ by that for $H_r$ gives the posterior-odds identity:

\frac{P(H_j\mid D)}{P(H_r\mid D)} = \frac{\pi_j}{\pi_r}\frac{L_j}{L_r} = \frac{\pi_j}{\pi_r}BF_{jr}.

Thus evidence multiplies prior odds. For computation, choose any candidate as the reference and assign its Bayes factor the value one. The posterior weights are proportional to $\pi_j BF_{jr}$, then divided by their sum. The choice of reference changes the scale of all weights by the same factor, not the posterior probabilities.

Worked Example: Combining Three Likelihoods and Unequal Priors

Suppose a hypothetical region has three candidates, $H_1,H_2,H_3$, with marginal likelihoods $L_1=0.12$, $L_2=0.03$, and $L_3=0.05$. Suppose the prior probabilities are $\pi_1=0.50$, $\pi_2=0.30$, and $\pi_3=0.20$. The priors are nonnegative and sum to one, as required under the exactly-one model.

Compute the unnormalized posterior weight for each candidate: $w_1=\pi_1L_1=(0.50)(0.12)=0.060$, $w_2=(0.30)(0.03)=0.009$, and $w_3=(0.20)(0.05)=0.010$. The normalizing constant is $0.060+0.009+0.010=0.079$. Therefore,

P(H_1\mid D)=\frac{0.060}{0.079}=0.7595,\quad P(H_2\mid D)=\frac{0.009}{0.079}=0.1139,\quad P(H_3\mid D)=\frac{0.010}{0.079}=0.1266.

The probabilities sum to one, up to rounding. Under the stated candidate set, priors, likelihoods and exactly-one assumption, candidate 1 has posterior probability about 0.760 of being the sole causal variant. Candidate 3 has a larger likelihood than candidate 2, and also a larger posterior here; the prior weighting increases candidate 1's advantage. These probabilities are not probabilities that an association exists in the population independently of the model.

Why Normalization Works

The posterior formula follows directly from Bayes' theorem and the fact that the candidate hypotheses partition the assumed model space. For each $j$, Bayes' theorem gives $P(H_j\mid D)=p(D\mid H_j)P(H_j)/p(D)$. Since exactly one of the hypotheses holds within this model, the law of total probability gives $p(D)=\sum_{k=1}^{m}p(D\mid H_k)P(H_k)$. Substitution yields the normalized expression.

P(H_j\mid D) =\frac{L_j\pi_j}{p(D)} =\frac{L_j\pi_j}{\sum_{k=1}^{m}L_k\pi_k}.

This argument also explains why likelihoods may be replaced by Bayes factors relative to a common reference. Since $L_j=L_r BF_{jr}$, every unnormalized weight contains the same multiplier $L_r$. It cancels between numerator and denominator:

P(H_j\mid D) =\frac{\pi_j BF_{jr}}{\sum_{k=1}^{m}\pi_k BF_{kr}}, \qquad BF_{rr}=1.

The cancellation requires that all likelihoods refer to the same observed data and are computed under comparable model definitions. A likelihood evaluated at a fitted effect size is not generally the same quantity as a marginal likelihood; substituting a profile likelihood ratio and calling it a Bayes factor does not follow from this derivation.

Worked Example: Updating Prior Odds With Bayes Factors

For a hypothetical three-candidate region, take candidate 3 as the reference. Suppose the Bayes factors are $BF_{13}=12$, $BF_{23}=3$, and $BF_{33}=1$, and the prior probabilities are $(\pi_1,\pi_2,\pi_3)=(0.10,0.20,0.70)$. These values describe evidence and priors under the same exactly-one model.

The prior-weighted Bayes-factor weights are $w_1=(0.10)(12)=1.20$, $w_2=(0.20)(3)=0.60$, and $w_3=(0.70)(1)=0.70$. Their sum is $2.50$, so the normalized posterior is

P(H_1\mid D)=\frac{1.20}{2.50}=0.480,\quad P(H_2\mid D)=\frac{0.60}{2.50}=0.240,\quad P(H_3\mid D)=\frac{0.70}{2.50}=0.280.

Candidate 1 has the strongest evidence relative to candidate 3, but its posterior probability is 0.480, not 12. The factor 12 is a Bayes factor; it multiplies the prior odds. In particular, the posterior odds for candidates 1 and 3 are $0.480/0.280=1.714$, while their prior odds were $0.10/0.70=0.1429$; multiplying the prior odds by 12 gives approximately 1.714.

Assumptions, Diagnostics, and Failure Modes

The posterior calculation is exact for the specified finite model, but its scientific interpretation depends on assumptions that should be checked rather than hidden. The candidate list must cover the plausible causal variants for the conclusion to refer to the actual causal variant. The exactly-one assumption must be reasonable for the region and analysis; if there are no causal variants or multiple causal variants, the hypotheses do not describe the relevant possibilities. Likelihoods must use a suitable sampling model and consistent data, including compatible allele coding and linkage-disequilibrium information where required. Priors must represent the intended assumptions before observing the data.

A useful diagnostic is to examine the ingredients and the normalized weights separately. Check that prior probabilities are nonnegative and sum to one, that Bayes factors are relative to a common reference, and that posterior probabilities sum to one within rounding. Compare rankings by likelihood, prior probability and posterior probability; a large change in ranking identifies a result that is sensitive to prior weighting. Also inspect whether highly correlated or nearly indistinguishable candidates receive similar likelihood evidence. If they do, the data may not separate them, even if the model assigns different posterior probabilities because of different priors.

A zero prior probability forces a zero posterior probability, regardless of the data, because $\pi_jL_j=0$. This is appropriate only when the candidate is intentionally excluded from the model. Very small positive priors can also require strong evidence to yield a large posterior. At the other extreme, if all likelihoods are numerically tiny, direct multiplication may underflow in computation. A stable calculation uses log weights, $\log(\pi_j)+\log(L_j)$, subtracts the largest log weight, exponentiates, and normalizes. This changes no mathematical result.

Worked Example: Similar Evidence Does Not Guarantee Resolution

Consider three hypothetical candidates with Bayes factors relative to candidate 3 of $1$, $0.95$, and $0.05$, respectively. Let their priors be $0.45$, $0.45$, and $0.10$, which sum to one. The first two candidates have nearly equal evidence; the third has much less evidence relative to the reference.

The weights are $w_1=(0.45)(1)=0.450$, $w_2=(0.45)(0.95)=0.4275$, and $w_3=(0.10)(0.05)=0.005$. Their sum is $0.8825$. Normalizing gives $P(H_1\mid D)=0.450/0.8825=0.5099$, $P(H_2\mid D)=0.4275/0.8825=0.4844$, and $P(H_3\mid D)=0.005/0.8825=0.005666$.

The posterior is split almost evenly between candidates 1 and 2 because their priors and evidence are similar. The small posterior for candidate 3 follows from its small prior-weighted evidence. These values do not show that candidate 1 has been distinguished reliably from candidate 2: their posterior probabilities remain close. In a region where candidates are difficult to distinguish because of linkage disequilibrium, reporting only the top-ranked candidate would conceal this uncertainty.

How This Posterior Differs From Other Summaries

With equal priors, posterior probabilities are proportional to the likelihoods, so ranking candidates by posterior is equivalent to ranking them by likelihood. With unequal priors, the posterior ranking can differ. Equal priors are a modeling choice, not a neutral mathematical default; nonuniform priors can encode pre-existing biological or technical knowledge, but their rationale should be stated.

A likelihood ranking reports which candidate fits the data best under the specified likelihoods but does not quantify normalized probability across the candidate set. A p-value addresses a null-distribution tail probability for a test statistic; it is not the probability that a particular variant is causal. The posterior instead describes relative uncertainty among the hypotheses in the specified model, conditional on the data, priors, likelihood assumptions and exactly-one restriction.

The main limitation is therefore not the normalization formula but the scope of the model. If the region could contain no causal variant, the candidate hypotheses are not exhaustive; if it could contain several, they are not an adequate set of alternatives. In either situation, a posterior that sums to one across the single-variant candidates can still be precisely computed but can misrepresent the scientific uncertainty. The assumption should be justified for the intended analysis, and conclusions should be qualified by it.

Common Mistake

Common mistake: “Candidate 1 has a Bayes factor of 12, so there is a 12-fold probability—or a 92% probability—that it is causal.” A Bayes factor is a ratio of marginal likelihoods, not a probability. The correction is to multiply it by prior odds to obtain posterior odds, then normalize across all candidate hypotheses. A posterior probability is conditional on the specified model and data.

Check Your Understanding

Answer each question using the exactly-one-causal-variant model. Brief solutions are included for checking.

Two candidates have equal likelihoods and prior probabilities 0.25 and 0.75. What are their posterior probabilities? Solution: The unnormalized weights are $0.25L$ and $0.75L$; after normalization the posterior probabilities are 0.25 and 0.75.
For two candidates, the prior odds $H_1:H_2$ are 1:4 and $BF_{12}=8$. What are the posterior odds? Solution: Posterior odds equal prior odds times the Bayes factor, so they are $(1/4)(8)=2:1$.
Why does a candidate with prior probability zero retain posterior probability zero? Solution: Its posterior numerator is $\pi_jL_j=0$, so normalization cannot give it positive probability.
Three candidates have prior-weighted likelihoods 0.2, 0.3 and 0.5. What is the posterior for the second candidate? Solution: The weights sum to 1.0, so the second candidate's posterior is $0.3/1.0=0.30$.
What does it mean if two highly correlated candidates have similar posterior probabilities? Solution: Under the specified model, posterior support is divided between them; the data have not clearly distinguished which one is causal. It does not establish that either probability is an unconditional probability of causality.