Calculating Posterior Inclusion Probabilities

From Configuration Probabilities to Variant Probabilities

Fine-mapping considers which variants in a defined candidate set may be causal for a genetic association. A posterior probability can describe an entire proposed causal configuration, such as the configuration in which variant A is included and variant B is not. Often the question is instead about one particular variant: what is the posterior probability that this variant is included? The answer is its posterior inclusion probability, or PIP.

A PIP is obtained by adding the posterior probabilities of every configuration that includes the variant. This is a marginal probability: it summarizes the posterior uncertainty about one variant while leaving the identities of the other variants unspecified. The calculation is straightforward, but it depends on having a well-defined set of configurations and correctly normalized posterior probabilities.

Definition: Let $C$ denote a causal configuration drawn from a specified set of configurations, and let $q(C)=P(C\mid D)$ be its posterior probability given data $D$. For candidate variant $j$, its posterior inclusion probability is $$ \operatorname{PIP}_j=P(j\in C\mid D)=\sum_{C:\,j\in C}q(C). $$ The sum includes every configuration containing $j$, and excludes every configuration not containing $j$.

The configuration probabilities must refer to mutually exclusive possibilities: in a given model and analysis, the realized configuration is one configuration in the stated set. For the sum to represent the posterior probability over that model, the set must cover all allowed configurations, including an empty or null configuration if the model permits no causal variant. The probabilities must be nonnegative and sum to one.

A configuration can be described by a set of included variants. If the model allows only one causal variant, the configurations are the null configuration and the configurations containing one variant each. In that special case, a variant’s PIP is just the posterior probability of its own single-variant configuration. When configurations can contain different combinations of variants, the PIP must instead collect the posterior mass from all configurations containing the variant. This tutorial concerns that aggregation; it does not specify how a model creates or estimates the configuration probabilities.

Calculating the Inclusion Probability

If posterior probabilities are already available, identify the configurations containing the variant and add their probabilities. If the analysis supplies unnormalized posterior weights, first normalize them. For example, suppose $w_C$ is the posterior weight for configuration $C$, proportional to its prior probability times its Bayes factor relative to a common null reference. Then

q(C)=\frac{w_C}{\sum_{C'}w_{C'}}, \qquad \operatorname{PIP}_j= \frac{\sum_{C:\,j\in C}w_C}{\sum_{C'}w_{C'}}.

The second expression follows by substituting the normalized probabilities into the definition. The denominator includes the weights of all allowed configurations, not only those containing $j$. Variant-specific prior probabilities affect the weights through the model’s prior specification; the PIP is calculated from the resulting posterior probabilities, not from the prior alone.

Worked Example: Normalizing Weights in a Single-Variant Model

Suppose a hypothetical analysis has three candidate variants, A, B, and C, and allows either no causal variant or exactly one causal variant. The prior probabilities for the null, A, B, and C configurations are respectively 0.40, 0.20, 0.20, and 0.20. Their Bayes factors relative to the null are respectively 1, 5, 2, and 1. Calculate the PIP for each variant.

The unnormalized weights are prior probability multiplied by Bayes factor:

w_{\varnothing}=0.40(1)=0.40,\quad w_A=0.20(5)=1.00,\quad w_B=0.20(2)=0.40,\quad w_C=0.20(1)=0.20.

The weights sum to $0.40+1.00+0.40+0.20=2.00$. Normalizing gives posterior probabilities 0.20 for the null configuration, 0.50 for A, 0.20 for B, and 0.10 for C. Since each non-null configuration contains only its named variant, the PIPs are $0.50$ for A, $0.20$ for B, and $0.10$ for C. The remaining posterior probability, 0.20, is assigned to the null configuration; it is not a PIP for any candidate variant.

Worked Example: Summing Across Configurations

In a second hypothetical analysis, the posterior probabilities for the allowed configurations are supplied directly. They are: empty configuration, 0.10; $\{A\}$, 0.25; $\{B\}$, 0.20; $\{A,B\}$, 0.30; and $\{B,C\}$, 0.15. Calculate the PIPs.

First check that the probabilities sum to one: $0.10+0.25+0.20+0.30+0.15=1.00$. Variant A occurs in $\{A\}$ and $\{A,B\}$, so

\operatorname{PIP}_A=0.25+0.30=0.55.

Variant B occurs in $\{B\}$, $\{A,B\}$, and $\{B,C\}$, giving $\operatorname{PIP}_B=0.20+0.30+0.15=0.65$. Variant C occurs only in $\{B,C\}$, so $\operatorname{PIP}_C=0.15$. Each PIP is between zero and one. Their sum is $0.55+0.65+0.15=1.35$, which is not an error: configurations in this example can contain more than one variant, so the sum of variant-level inclusion probabilities need not equal one.

Why the Sum Gives the Marginal Probability

The key justification is the addition rule for disjoint events. The event that variant $j$ is included can be divided into the separate events that the realized configuration is $C$, for each allowed $C$ containing $j$. These events are mutually exclusive because only one configuration is realized. Their union is exactly the event $j\in C$. Therefore, adding their posterior probabilities gives the probability of inclusion.

Identify the event.
For variant $j$, the event of interest is that the realized configuration contains $j$.

Partition the event.
List every allowed configuration containing $j$. Each is a distinct, mutually exclusive way for $j$ to be included.

Add posterior mass.
By additivity over mutually exclusive events, sum $q(C)$ over that list. This sum is the PIP.

Justification: The configurations containing $j$ form a disjoint partition of the inclusion event. Hence $$ P(j\in C\mid D) =P\left(\bigcup_{C:\,j\in C}\{C\}\,\middle|\,D\right) =\sum_{C:\,j\in C}P(C\mid D). $$

Inclusion Is Not the Same as Effect Significance

A PIP answers a model-based probability question: given the specified data, candidate variants, likelihood, and prior assumptions, what posterior probability is assigned to configurations including this variant? A frequentist association p-value answers a different question: under a specified null hypothesis and sampling model, how unusual would data at least as extreme as those observed be? A p-value is not the probability that a variant is causal, and it cannot be substituted for a PIP.

Nor does a PIP directly describe the size or precision of a variant’s effect. A variant may have substantial posterior probability of inclusion while its effect-size estimate remains uncertain. Conversely, a small p-value from an association test does not, by itself, say how posterior probability should be divided among correlated candidate variants. The result depends on the fine-mapping model and its priors as well as the observed evidence.

Worked Example: Reading a PIP Beside Association P-Values

Suppose a hypothetical fine-mapping analysis reports posterior probabilities 0.50 for the null configuration, 0.48 for $\{A\}$, and 0.02 for $\{B\}$. Separate association tests report p-values 0.001 for A and 0.04 for B. Calculate the PIPs, then interpret both kinds of result.

The posterior probabilities sum to $0.50+0.48+0.02=1.00$. Under these single-variant configurations, $\operatorname{PIP}_A=0.48$ and $\operatorname{PIP}_B=0.02$. Thus the model assigns 48% posterior probability to configurations including A and 2% to configurations including B. The p-values describe the association tests under their respective null hypotheses; at a conventional 0.05 threshold, both reported p-values are below 0.05. That does not make either PIP equal to $1-p$, nor does it mean the posterior probability of inclusion is $1-p$. In particular, the reported p-value for A does not turn its PIP of 0.48 into a probability of 99.9% that A is causal.

The different numbers are not contradictory: the p-values and PIPs are different summaries with different definitions. The PIPs are conditional on the fine-mapping model and its prior specification, while the p-values are calibrated under their association-test null models. Neither number alone gives the magnitude of A’s or B’s effect.

Diagnostics, Comparisons, and Limitations

Several checks can catch errors in PIP calculations. Confirm that configuration posterior probabilities sum to one, allowing only for small rounding discrepancies. For each variant, verify that the sum includes every configuration containing that variant exactly once. Each result must lie between zero and one. If configurations are mutually exclusive and exhaustive, the PIP cannot exceed one because it is a sum over only part of the total posterior probability.

When every allowed configuration contains at most one variant, the sum of all variant PIPs equals the posterior probability that some candidate variant is included; it is less than one if the null configuration has positive posterior probability. If configurations can contain multiple variants, the sum of PIPs instead equals the posterior expected number of included variants. This follows from writing the number included in a configuration as a sum of variant-specific indicators. In that setting, a sum greater than one is possible and does not indicate a normalization failure.

A PIP differs from a configuration posterior probability because the PIP combines multiple configurations whenever the variant appears in more than one. It also differs from a set-level posterior probability: a set-level probability concerns an event about the set as a whole, whereas a PIP concerns inclusion of one named variant. These quantities answer different questions and should not be used interchangeably.

The calculation is only as meaningful as its inputs. If allowed configurations are missing, the reported probabilities do not describe a complete posterior distribution over the intended model. If weights were normalized using only configurations containing the target variant, the result is a conditional probability among those configurations, not the PIP. If prior probabilities or likelihoods are inappropriate, correct addition cannot repair the resulting model-based uncertainty. A PIP is not a guarantee of biological causality outside the assumptions and candidate set used in the analysis.

Common Mistakes

Using only one configuration. If a variant appears in several configurations, using the probability of just one of them omits posterior mass. Add all configuration probabilities containing the variant.
Normalizing only within the included configurations. This removes the posterior probability of configurations that exclude the variant. Normalize weights over the full allowed configuration set before calculating a PIP.
Calling a PIP a p-value. A PIP is posterior probability of inclusion under a specified model; a p-value is a tail probability under a null sampling model. They are not interchangeable.
Assuming PIPs must sum to one. That is generally true only in a model that assigns exactly one included variant whenever a non-null configuration occurs, with the null probability accounted for separately. With multi-variant configurations, the sum can exceed one.
Interpreting inclusion as a precise effect estimate. A PIP concerns whether the variant is included, not the magnitude or precision of its effect.

Key takeaway: A variant’s PIP is the total posterior probability of all configurations that contain it. Calculate it by summing normalized configuration probabilities, and interpret it as model-conditional inclusion probability—not as a p-value, a probability derived from a p-value, or an effect-size summary.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Three mutually exclusive configurations have posterior probabilities 0.15 for the empty configuration, 0.35 for $\{A\}$, and 0.50 for $\{B\}$. What are the PIPs for A and B?
Configurations $\{A\}$, $\{B\}$, and $\{A,B\}$ have posterior probabilities 0.20, 0.30, and 0.50. Calculate both PIPs. Must they sum to one?
Why is a PIP of 0.70 not the same statement as a p-value of 0.30?
A calculation gives a PIP of 1.08. Give two checks that should be made before reporting it.

Solutions. (1) A occurs only in $\{A\}$, so $\operatorname{PIP}_A=0.35$; B occurs only in $\{B\}$, so $\operatorname{PIP}_B=0.50$. The remaining 0.15 is posterior probability for no included variant. (2) $\operatorname{PIP}_A=0.20+0.50=0.70$, and $\operatorname{PIP}_B=0.30+0.50=0.80$. They sum to 1.50 because the configuration $\{A,B\}$ includes both variants. (3) A PIP is posterior probability of inclusion under a specified fine-mapping model; a p-value is a tail probability under a null sampling model. They concern different events and cannot be converted into each other by taking a complement. (4) Check that the configuration probabilities were normalized over the full allowed set and that every included configuration was counted once. Also confirm that the configurations are mutually exclusive and exhaustive: a valid PIP formed by summing a subset of normalized configuration probabilities cannot exceed one.