Posterior Inclusion Probabilities Under a Single-Causal-Variant Model

Why Posterior Inclusion Probabilities Need Normalization

Fine-mapping compares candidate variants as explanations for an observed association signal. A variant-specific Bayes factor summarizes how well the data are supported under one candidate causal variant relative to a common reference model. But a Bayes factor alone is not a posterior probability: posterior inference also uses prior probabilities, and the probabilities across candidate variants must be normalized.

This tutorial considers a deliberately specific model: exactly one variant in the candidate locus is causal. The candidate variants therefore define mutually exclusive explanations of the data. The posterior inclusion probability, or PIP, for a variant is the posterior probability that this variant is the causal one, conditional on the data and on this model.

Definition: Suppose a locus contains $m$ candidate variants. Let $H_j$ be the hypothesis that variant $j$ is the one causal variant, for $j=1,\ldots,m$. Under the single-causal-variant model, the hypotheses $H_1,\ldots,H_m$ are mutually exclusive and exhaustive. Let $\pi_j=P(H_j)$ be the prior probability of $H_j$, with $\sum_{j=1}^m \pi_j=1$. If $BF_j$ is the Bayes factor for $H_j$ relative to the same reference model, then the posterior inclusion probability is $$ PIP_j=P(H_j\mid D)=\frac{\pi_j BF_j}{\sum_{k=1}^m \pi_k BF_k}, $$ where $D$ denotes the observed data.

The prior probabilities describe how plausibly each candidate variant is considered before observing these data, within the specified model. The Bayes factors update those probabilities according to the relative support the data provide. The denominator sums the prior-weighted evidence across all candidate hypotheses, making the resulting PIPs sum to one.

Deriving the PIP Formula

For each candidate $j$, write $p(D\mid H_j)$ for the probability density or probability of the observed data under that hypothesis. A Bayes factor compares this quantity with the data probability under a shared reference model $H_0$:

BF_j=\frac{p(D\mid H_j)}{p(D\mid H_0)}.

The reference model is used to define the Bayes factors; it is not an additional candidate in the model considered here. Because the reference is the same for every $j$, it cancels when calculating posterior probabilities among the causal-variant hypotheses.

Apply Bayes’ theorem.
For each candidate, $P(H_j\mid D)=p(D\mid H_j)\pi_j/p(D)$, where $p(D)$ is the marginal probability of the data under the candidate model.

Sum over the possible causal variants.
Since the hypotheses are exhaustive, $p(D)=\sum_{k=1}^m p(D\mid H_k)\pi_k$.

Express likelihoods using Bayes factors.
Substitute $p(D\mid H_j)=BF_jp(D\mid H_0)$ into the numerator and denominator.

Cancel the shared reference likelihood.
The factor $p(D\mid H_0)$ appears in every term and cancels, leaving $P(H_j\mid D)=\pi_jBF_j/\sum_{k=1}^m\pi_kBF_k$.

This derivation also shows why Bayes factors must be comparable: each must use the same data and reference model, and each must represent the relevant candidate hypothesis in a consistent way. The derivation does not require the variants’ association statistics to be independent. It does require the $BF_j$ values to represent the evidence under the corresponding single-candidate hypotheses.

Worked Example: Equal Priors and Different Bayes Factors

Suppose a hypothetical locus has three candidate variants. The model assigns each variant prior probability $1/3$, and the variant-specific Bayes factors relative to a common reference model are 2, 10, and 4. Calculate the PIP for each variant.

The prior-weighted evidence values are $(1/3)(2)=2/3$, $(1/3)(10)=10/3$, and $(1/3)(4)=4/3$. Their sum is $16/3$. Thus:

PIP_1=\frac{2/3}{16/3}=\frac{2}{16}=0.125, \qquad PIP_2=\frac{10/3}{16/3}=\frac{10}{16}=0.625, \qquad PIP_3=\frac{4/3}{16/3}=\frac{4}{16}=0.250.

The PIPs sum to $0.125+0.625+0.250=1.000$, as required under a model with exactly one causal variant among these three candidates. Variant 2 has the largest posterior probability because it has the largest Bayes factor and the prior probabilities are equal. Its PIP is 0.625, not 1: the other two hypotheses retain posterior probability.

How Priors and Evidence Work Together

With equal prior probabilities, the prior terms cancel and each PIP is its Bayes factor divided by the sum of all Bayes factors. With unequal priors, however, the relevant quantity is the prior-weighted Bayes factor, $\pi_jBF_j$. Strong evidence can raise the posterior probability of a variant with a small prior, while a large prior does not guarantee a large PIP if the data provide little relative support.

Worked Example: A Lower Prior Offset by Stronger Evidence

Consider a hypothetical locus with three candidate variants. Their prior probabilities are $0.60$, $0.30$, and $0.10$. Their Bayes factors are $2$, $5$, and $20$, respectively, all relative to the same reference model. Calculate the posterior probabilities.

First multiply each Bayes factor by its prior probability:

w_1=0.60(2)=1.20,\qquad w_2=0.30(5)=1.50,\qquad w_3=0.10(20)=2.00.

The normalizing sum is $1.20+1.50+2.00=4.70$. Therefore:

PIP_1=\frac{1.20}{4.70}=0.2553,\qquad PIP_2=\frac{1.50}{4.70}=0.3191,\qquad PIP_3=\frac{2.00}{4.70}=0.4255.

The rounded PIPs sum to $0.9999$, with the small discrepancy due to rounding; the unrounded values sum to 1. Variant 1 had the largest prior probability, but variant 3 has the largest PIP because its Bayes factor is sufficiently larger. This is a comparison among hypotheses under the stated model, not evidence that the prior was incorrect.

A useful comparison is the posterior odds for two candidate variants. Dividing their PIPs cancels the common denominator:

\frac{PIP_j}{PIP_k} = \frac{\pi_jBF_j}{\pi_kBF_k} = \frac{\pi_j}{\pi_k}\frac{BF_j}{BF_k}.

Thus, posterior odds equal prior odds multiplied by the Bayes factor comparing the two candidates. For example, in the previous calculation, the posterior odds for variant 3 versus variant 1 are $0.4255/0.2553$, approximately $1.667$. The prior odds are $0.10/0.60=1/6$, while the Bayes-factor ratio is $20/2=10$; multiplying gives $10/6$, also approximately $1.667$.

Interpreting Similar Evidence

A large Bayes factor against a reference model does not necessarily mean a variant has a large PIP. PIPs compare each candidate’s prior-weighted evidence with the total across all candidates. If several variants have similarly strong evidence, the posterior probability is distributed among them. This can happen when candidate variants carry similar evidence under the specified analyses; linkage disequilibrium may make distinguishing nearby variants difficult, but it does not change the normalization formula.

Worked Example: Strong but Similar Bayes Factors

Suppose a hypothetical locus has three candidate variants with equal prior probabilities. Their Bayes factors are 30, 28, and 1, each relative to the same reference model. Compute the PIPs and interpret the result.

Because the priors are equal, the common prior factor cancels. The sum of the Bayes factors is $30+28+1=59$, so:

PIP_1=\frac{30}{59}=0.5085,\qquad PIP_2=\frac{28}{59}=0.4746,\qquad PIP_3=\frac{1}{59}=0.01695.

The first two variants each have strong support relative to the reference model, but neither receives a PIP close to 1 because their evidence is similar. Together they account for approximately $0.9831$ of the posterior probability under this model. The calculation does not establish which of those two variants is causal; it quantifies the model’s relative support given the data, priors, and candidate set.

Assumptions, Scope, and a Common Pitfall

The formula is simple, but its interpretation depends on the model being used. In particular, the single-causal-variant assumption says that exactly one candidate in the specified set is causal. The hypotheses $H_j$ must therefore cover the possibilities considered, and their prior probabilities must sum to one. The Bayes factors must use a consistent reference and model the data under each corresponding hypothesis. If approximate Bayes factors are used, as in the earlier material on Wakefield’s approximate Bayes factor, the resulting PIPs inherit the approximation.

Under this model, $PIP_j$ is a probability conditional on the observed data, the assigned priors, the candidate variants, and the assumption that exactly one candidate is causal. It is not a frequentist p-value, and it is not a probability that the model’s assumptions are correct. Nor does a high PIP by itself establish biological causality; it reports posterior support within the specified analysis.

Common pitfall: “Variant 1 has the largest Bayes factor, so its PIP must be its Bayes factor, or nearly 1.” This is wrong because a Bayes factor is evidence relative to a reference model, not a posterior probability. The correction is to multiply every candidate’s Bayes factor by its prior probability and divide by the sum of those products. A candidate’s PIP depends on the other candidates as well as on its own evidence.

A related modeling distinction is important: this tutorial assumes exactly one causal variant among the listed candidates. If a model instead allows no causal variant in the locus, then the no-causal hypothesis must also receive prior probability and contribute evidence to the posterior denominator. The PIPs among variants would then generally sum to less than one. That is a different model, not a different interpretation of the formula under the exactly-one assumption.

Key takeaway: Under a single-causal-variant model, each PIP is the candidate’s prior probability multiplied by its Bayes factor, divided by the sum of these prior-weighted Bayes factors across all candidates. The probabilities sum to one because the candidate hypotheses are mutually exclusive and exhaustive.

Check Your Understanding

Use the single-causal-variant assumptions stated above when answering.

Two variants have prior probabilities $0.25$ and $0.75$, with Bayes factors 8 and 2. Calculate both PIPs.
Why must the prior probabilities sum to one in the model used here?
Two candidates have equal priors and Bayes factors 40 and 39. Explain why the first candidate’s PIP is not close to one.
State one condition that must hold for variant-specific Bayes factors to be combined in the PIP denominator.
In context, what does a PIP of 0.70 mean, and what does it not establish?