Interpreting High Posterior Inclusion Probabilities

What a High PIP Says

Fine-mapping aims to distinguish variants that could explain an observed genetic association. A posterior inclusion probability (PIP) is a variant-level summary from a Bayesian fine-mapping analysis. Its meaning is precise but conditional: it is the posterior probability that the variant is included in the causal configuration, given the observed data, the specified model, and the prior distributions.

That conditioning matters. A PIP of 0.95 is strong model-based support for inclusion, but it does not establish with 95% certainty that the variant is the biological cause in an unrestricted scientific sense. The model may omit the actual causal variant, use an inaccurate likelihood or linkage disequilibrium (LD) structure, or impose an unsuitable prior. The PIP answers a probability question inside the analysis; evaluating whether that analysis represents the biological situation is a separate task.

Definition: Let $D$ denote the observed genetic and phenotype data, and let $M$ denote a specified fine-mapping model, including its candidate variants, likelihood, and prior distributions. A configuration $\gamma$ is one possible set of included causal variants. For candidate variant $j$, its posterior inclusion probability is $$ \operatorname{PIP}_j = P(\gamma_j=1\mid D,M) =\sum_{\gamma:\,j\in\gamma}P(\gamma\mid D,M), $$ where $\gamma_j=1$ means that variant $j$ is included. The posterior probabilities of all configurations allowed by $M$, including the empty (null) configuration if it is allowed, sum to 1.

The configuration probabilities containing variant $j$ and those excluding it are mutually exclusive and together include every allowed configuration. Therefore, $1-\operatorname{PIP}_j$ is the posterior probability of configurations that exclude $j$, including the null configuration when the null is allowed. A PIP is not the probability that the locus has any association, nor is it a measure of effect size.

Calculating and Interpreting Configuration Probabilities

The sum in the definition is straightforward, but interpretation depends on using a coherent set of configurations. Each allowed configuration must be counted once, and their posterior probabilities must sum to 1. A common source of arithmetic error is to treat the null configuration as a separate category from all configurations that exclude a variant. It is already one of those configurations.

Worked Example: Summing Posterior Mass for One Variant

Suppose a hypothetical fine-mapping model allows two candidate variants, A and B, and produces these posterior probabilities: the empty configuration has probability 0.05, $\{A\}$ has probability 0.70, $\{B\}$ has probability 0.10, and $\{A,B\}$ has probability 0.15. These are all the allowed configurations.

First check the total: $0.05+0.70+0.10+0.15=1.00$. The configurations containing A are $\{A\}$ and $\{A,B\}$, so $$ \operatorname{PIP}_A=0.70+0.15=0.85. $$

Thus, under this model and its priors, the posterior probability of configurations including A is 0.85. The configurations excluding A are the empty configuration and $\{B\}$, with total probability $0.05+0.10=0.15$. Notice that the probability of a non-null configuration excluding A is 0.10, not 0.15: the null probability is part of the total probability excluding A. The PIP does not say that A has an 85% probability of being the only included variant; the configuration $\{A,B\}$ contributes to it.

A high PIP supports inclusion, not exclusivity. If the scientific question is whether a variant is the sole causal variant, the relevant posterior event is the probability of configurations containing that variant and no others. That is a different sum from the PIP. Likewise, a variant can have high inclusion probability while another variant also has substantial support, as the example illustrates.

Why the Probability Is Conditional on a Model

A fine-mapping model defines what “included” means and what possibilities are considered. Its candidate-variant list sets the scope of inference. Its likelihood describes how genotypes and variant effects relate to the observed phenotype data. Its priors assign probability to configurations and effect parameters before observing those data. The posterior combines these ingredients. A high PIP is interpretable as causal support only to the extent that inclusion in the model corresponds to the causal event of scientific interest and the model is a reasonable representation of the data-generating setting.

For example, an analysis can assign high probability to one candidate because that candidate is a strong proxy for a causal variant that was not included. The posterior calculation can be internally correct for the restricted model while its causal interpretation is wrong. The earlier material on fine-mapping with an omitted causal variant illustrates why the candidate set itself matters; the point here is that a high PIP does not repair a misspecified set.

Worked Example: High PIP for a Proxy When the Causal Variant Is Omitted

Consider a hypothetical locus where the true biological causal variant is W, but W is absent from the analysis candidate list. The model includes only X and Y, which are strongly correlated with W through LD. Suppose its posterior probabilities are 0.01 for the empty configuration, 0.90 for $\{X\}$, 0.03 for $\{Y\}$, and 0.06 for $\{X,Y\}$. The probabilities sum to $0.01+0.90+0.03+0.06=1.00$.

The model-based PIP for X is $$ \operatorname{PIP}_X=0.90+0.06=0.96. $$ The computation supports X strongly among the configurations the model allows. But every configuration excludes W because W is not in the candidate set. Thus, 0.96 is not evidence that the analysis has compared X directly with W; it has not. In this hypothetical setting, X can receive high posterior inclusion probability as the best available proxy even though W is the true causal variant.

The correct interpretation is conditional: given this candidate set, data, likelihood, and priors, configurations containing X receive posterior probability 0.96. A causal claim about X requires confidence that the candidate set and model are adequate, not merely a large numerical PIP.

How Priors Contribute to a High PIP

A PIP reflects both data and prior assumptions. In a simple comparison between inclusion and exclusion, posterior odds equal prior odds multiplied by a Bayes factor. Here the Bayes factor is the ratio of the marginal likelihood under inclusion to the marginal likelihood under exclusion; any nuisance parameters or other configurations must be integrated or summed over as specified by the model. This identity explains why identical data evidence can yield different posterior inclusion probabilities under different prior inclusion probabilities.

Result: For a comparison of inclusion $I$ against exclusion $E$, $$ \frac{P(I\mid D,M)}{P(E\mid D,M)} = \frac{P(D\mid I,M)}{P(D\mid E,M)} \frac{P(I\mid M)}{P(E\mid M)}. $$ The first ratio is the Bayes factor for inclusion versus exclusion; the second is the prior odds. The posterior probability of inclusion is the posterior odds divided by one plus the posterior odds.

This result follows directly from Bayes’ rule. Write $P(I\mid D,M)=P(D\mid I,M)P(I\mid M)/P(D\mid M)$, and write the corresponding expression for $E$. Dividing the first equation by the second cancels the common marginal probability $P(D\mid M)$, leaving the displayed odds identity. In a multi-variant analysis, inclusion and exclusion each aggregate multiple configurations; their marginal likelihoods and prior probabilities must reflect those configurations. One must not treat a single configuration’s probability as the total probability of inclusion without summing the relevant configurations.

Worked Example: The Same Bayes Factor Under Two Prior Choices

Suppose, hypothetically, the Bayes factor for including a variant rather than excluding it is 100. In a one-variant comparison, take first a prior inclusion probability of 0.10. The prior odds are $0.10/0.90=1/9$, so the posterior odds are $100(1/9)=100/9$. Converting odds to probability gives $$ P(I\mid D,M)=\frac{100/9}{1+100/9}=\frac{100}{109}=0.9174. $$

Now use the same Bayes factor but a prior inclusion probability of 0.01. The prior odds are $0.01/0.99=1/99$, and the posterior odds are $100/99$. Therefore, $$ P(I\mid D,M)=\frac{100/99}{1+100/99}=\frac{100}{199}=0.5025. $$

The evidence ratio is unchanged, yet the posterior inclusion probability differs substantially because the prior odds differ. This is not an arithmetic defect: posterior probabilities combine prior and likelihood information. It does mean that a PIP near a threshold should be interpreted with awareness of the prior choices. The earlier material on prior sensitivity analysis for fine-mapping develops ways to examine this dependence.

Comparison, Diagnostics, and Limits

A PIP is not interchangeable with an association-test p-value. A p-value assesses how incompatible the observed data are with a specified null hypothesis under a sampling model; it is not the probability that a variant is causal. A PIP is a posterior probability for a defined inclusion event, conditional on a Bayesian model and prior. Association evidence can motivate fine-mapping, but it does not by itself distinguish a causal variant from correlated candidates.

A PIP is also different from a credible set. A PIP summarizes posterior support for one variant. A credible set is a collection of variants intended to capture a specified amount of posterior support for a signal under the analysis’s construction. A high PIP for one member may make that variant a prominent candidate, but it does not by itself establish the set’s coverage or show that other signals and configurations have been handled correctly. The earlier material on credible-set purity and LD and on calibration under model misspecification provides relevant context for those distinct questions.

Before emphasizing a high PIP, check whether the analysis’s assumptions are plausible: Is the causal variant represented in the candidate list? Is the LD information appropriate for the analyzed samples? Does the likelihood reflect the phenotype and sampling design? Is the allowed number and structure of causal variants reasonable? Are the priors defensible? These checks do not turn a posterior into certainty; they help judge whether its conditional interpretation is useful. Comparing analyses under plausible prior and LD choices can reveal fragility, but no single sensitivity analysis guarantees that every source of model error has been found.

Warning signs include a high PIP that changes sharply under plausible prior choices, candidate variants missing from the analyzed region, or LD estimates that poorly match the study population. A high PIP can also be misleading if a model forces a choice among candidates when the true variant is absent. In those cases, report the result as support conditional on the model, not as proof of biological causality. A low PIP is not proof that a variant has no biological role either: the model may have little power to distinguish it, or may not represent the relevant mechanism.

A Practical Interpretation

State the event.
Identify which variant’s inclusion is represented by the PIP and what inclusion means in the model.

Check the probability calculation.
Sum the posterior probabilities of every allowed configuration containing that variant, and verify that all configuration probabilities sum to 1.

Keep the conditioning visible.
Interpret the result given the data, candidate set, likelihood, LD information, and priors used.

Separate support from certainty.
State that a high PIP is strong posterior support for inclusion under the specified model, then consider whether that model adequately represents the scientific question.

Common mistake: “The PIP is 0.95, so there is a 95% chance this variant is truly causal, regardless of the analysis.” That statement erases the model conditioning and treats inclusion under the model as an unconditional statement about biological reality. A more accurate interpretation is: “Under the specified fine-mapping model, data, and priors, the posterior probability of configurations including this variant is 0.95.” Whether this supports a biological causal claim depends on the adequacy of the model and the evidence for its assumptions.

Key takeaway: A high PIP is a high posterior probability of inclusion in configurations defined by a particular fine-mapping model. It is meaningful evidence within that model, not a guarantee that the candidate set is complete or that the variant is certainly causal in nature.

Check Your Understanding

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

A model assigns probability 0.62 to configurations containing X, 0.28 to non-null configurations excluding X, and 0.10 to the null configuration. What is the total posterior probability of configurations excluding X?
In a model with configurations $\varnothing,\{A\},\{B\},\{A,B\}$, which configurations contribute to $\operatorname{PIP}_B$?
A variant has PIP 0.98, but the candidate list excludes a plausible causal variant in strong LD with it. What does the PIP establish, and what does it not establish?
In a one-variant comparison, the Bayes factor for inclusion is 9 and the prior inclusion probability is 0.20. Calculate the posterior inclusion probability.
In one sentence, distinguish a PIP from an association-test p-value.

Solutions. 1. The total probability excluding X is $0.28+0.10=0.38$; the null configuration is included among configurations excluding X. 2. The configurations $\{B\}$ and $\{A,B\}$ contribute, because both contain B. 3. It establishes 0.98 posterior probability for configurations including that variant, conditional on the stated model; it does not show that the omitted variant was compared or rule out that the reported variant is only a proxy. 4. Prior odds are $0.20/0.80=0.25$; posterior odds are $9(0.25)=2.25$, so the posterior probability is $2.25/(1+2.25)=0.6923$. 5. A PIP is a model-conditional posterior probability of variant inclusion, whereas a p-value measures how incompatible data are with a specified null under a sampling model and is not the probability that the variant is causal.