Why High Inclusion Probability Does Not Prove Causality

Introduction

A fine-mapping analysis may assign one variant a posterior inclusion probability of 0.98. That is strong evidence within the fitted model, but it is not a guarantee that the variant is truly causal, nor that the model represents every relevant feature of the data. The number depends on the candidate variants, likelihood, prior assumptions, and linkage disequilibrium (LD) information used in the analysis.

The key interpretive task is to separate a posterior statement from a claim about reality. A posterior probability summarizes uncertainty conditional on the data and a specified model. It can be highly informative without certifying the model’s assumptions or establishing a biological mechanism. This distinction also matters for credible sets: in a multi-causal model, marginal inclusion probabilities alone do not generally determine the probability that a set contains a causal variant.

What a Posterior Inclusion Probability Says

Definition: Let $D$ denote the observed genetic-association data and $M$ a specified fine-mapping model. A causal configuration $\gamma$ is the set of candidate variants included as causal under that model. The posterior inclusion probability (PIP) for variant $j$ is $$ \operatorname{PIP}_j=P(j\in\gamma\mid D,M). $$ It is the posterior probability that $j$ is included in the causal configuration, conditional on the data and model.

The model $M$ includes assumptions that affect this probability: for example, which variants are considered, the summary-statistic likelihood and LD information, the prior on causal configurations, and the effect-size prior. The posterior averages over configurations allowed by that model, weighting each by its posterior probability. It does not automatically average over every plausible model, every missing variant, or every possible biological explanation.

Bayes’ rule makes the role of assumptions explicit. For two configurations $\gamma$ and $\gamma'$, their posterior odds are their likelihood ratio multiplied by their prior odds:

\frac{P(\gamma\mid D,M)}{P(\gamma'\mid D,M)} = \frac{P(D\mid\gamma,M)}{P(D\mid\gamma',M)} \frac{P(\gamma\mid M)}{P(\gamma'\mid M)}.

Thus, even when the data favor one configuration, the posterior comparison also depends on the prior comparison. A high PIP means the variant receives substantial posterior support relative to alternatives represented in $M$. It does not mean that the probability of a biological claim is high under all reasonable models.

Worked Example: Reading a High PIP Under a Single-Causal-Variant Model

Suppose a hypothetical analysis considers exactly three variants and assumes exactly one is causal. Under model $M$, the posterior probabilities that variants A, B, and C are the causal variant are 0.96, 0.03, and 0.01, respectively. These probabilities sum to 1, as required by the single-causal assumption.

For A, the PIP is $P(A\in\gamma\mid D,M)=0.96$. The correct interpretation is: conditional on this model, its candidate variants, and the observed data, the posterior probability that A is the single included causal variant is 0.96.

If the target is a 95% single-causal credible set, ranking variants by posterior probability gives A alone: its posterior mass is 0.96, which is at least 0.95. This set statement inherits the same conditioning on $M$. It is not a 96% probability that A has a causal biological mechanism regardless of whether the model assumptions hold.

The remaining posterior mass, 0.04, belongs to B or C within this model. It does not represent every possible source of uncertainty, such as an unobserved candidate variant or a misspecified LD reference.

Why High Posterior Support Is Not Causal Proof

A PIP is a model-based probability about inclusion in a causal configuration. A causal conclusion about the real biological system requires more than a large number: it requires that the model’s relevant assumptions and candidate space are adequate for the question. If the causal variant is absent from the analyzed set, for example, a model that must choose among the included variants can concentrate its posterior on the best available proxy. A high PIP then identifies the model’s preferred candidate, not necessarily the true causal variant.

LD creates another interpretive limitation. Variants in strong LD can have similar association patterns, so the data may distinguish them weakly. The posterior allocation between them can depend on the LD matrix and other model inputs. A PIP can still be high if the model’s likelihood and priors strongly favor one candidate, but that concentration should be examined rather than treated as independent confirmation of causality.

Finally, identifying a variant as likely causal under a fine-mapping model is not the same as identifying the molecular mechanism. The posterior inclusion event concerns the modeled variant’s role in the association. It does not, by itself, establish which gene, tissue, or molecular pathway mediates an effect.

Credible Sets and the Multi-Causal Distinction

In a single-causal-variant model, exactly one variant is included. The events that different variants are causal are mutually exclusive, so summing their PIPs over a set gives the posterior probability that the causal variant is in that set. This is the basis for forming a set whose cumulative posterior mass reaches a target such as 0.95.

In a multi-causal model, several variants can be included together. PIPs are marginal probabilities, and inclusion events are no longer mutually exclusive. Summing PIPs over a set gives the posterior expected number of included variants in the set, not generally the probability that the set contains at least one included variant. The distinction follows directly from indicator variables.

\sum_{j\in C}\operatorname{PIP}_j = \sum_{j\in C}E\!\left[I(j\in\gamma)\mid D,M\right] = E\!\left[\sum_{j\in C}I(j\in\gamma)\mid D,M\right].

The quantity inside the final expectation is a count. The probability that at least one variant in $C$ is included is instead the posterior probability of a union of events, computed from the joint posterior over configurations. Those quantities need not be equal.

Worked Example: Marginal PIPs Do Not Give Set-Level Coverage

Consider a hypothetical multi-causal model with three possible variants, A, B, and C. Its entire posterior distribution over configurations is:

Configuration $\gamma$	Posterior probability
{A, B}	0.475
{C}	0.525

The configuration probabilities sum to $0.475+0.525=1.000$. A is included exactly in the first configuration, so $\operatorname{PIP}_A=0.475$. The same reasoning gives $\operatorname{PIP}_B=0.475$, while $\operatorname{PIP}_C=0.525$.

For the set $C_0=\{A,B\}$, the sum of marginal PIPs is $0.475+0.475=0.950$. But the posterior probability that at least one variant in $C_0$ is included is only 0.475: that event occurs in the configuration $\{A,B\}$ and not in $\{C\}$. The sum 0.950 is the expected number of included variants in $C_0$: it is 2 with probability 0.475 and 0 otherwise, giving $2(0.475)+0(0.525)=0.950$.

If the target is the event that $C_0$ contains at least one included variant, the joint posterior gives probability 0.475, not 0.950. If the target is that $C_0$ contains all included variants, that event also has probability 0.475 in this example. These are distinct set-level statements, and neither should be inferred by simply adding marginal PIPs in a multi-causal model.

A careful multi-causal analysis therefore states what a set is intended to cover and calculates the corresponding event from the joint posterior configurations. For example, the posterior probability that at least one included variant lies in a set $C$ is:

P(\gamma\cap C\ne\varnothing\mid D,M) = \sum_{\gamma:\,\gamma\cap C\ne\varnothing}P(\gamma\mid D,M).

A different target, such as the probability that every included variant lies in $C$, requires summing over configurations satisfying $\gamma\subseteq C$. The appropriate target depends on the scientific question; the two probabilities are not interchangeable.

Sensitivity to Priors and LD

Since posterior probabilities combine likelihood and prior information, a high PIP should be accompanied by attention to the assumptions that helped produce it. Prior sensitivity is particularly transparent in a comparison of two mutually exclusive causal hypotheses. Holding the likelihood ratio fixed while changing prior odds can change posterior odds and hence the posterior probability.

Worked Example: Prior Odds Change the Posterior Inclusion Probability

Suppose a hypothetical two-variant model allows exactly one of A or B to be causal. The likelihood under A is nine times the likelihood under B, so the Bayes factor comparing A with B is 9. First use equal prior probabilities, giving prior odds for A versus B of $1:1$.

The posterior odds are $9\times1=9:1$. Therefore, $$ P(A\text{ causal}\mid D,M)=\frac{9}{9+1}=0.900. $$

Now keep the same data and likelihood ratio but use prior probabilities 0.90 for A and 0.10 for B. The prior odds are $0.90/0.10=9:1$, so the posterior odds are $9\times9=81:1$, and $$ P(A\text{ causal}\mid D,M)=\frac{81}{81+1}=\frac{81}{82}\approx0.9878. $$

Both calculations are internally consistent, but they answer the question under different priors. The increase from 0.900 to about 0.9878 is not new evidence from the data; it reflects the changed prior odds combined with the same Bayes factor. Reporting the high posterior without describing the prior would conceal an important part of its interpretation.

LD enters through the likelihood: association statistics for correlated variants are not independent pieces of evidence. A likelihood that uses an inaccurate LD matrix can rank competing configurations differently from one using a better-matched matrix. The following hypothetical calculation isolates the potential impact on a two-candidate comparison; the Bayes factors are assumed outputs of the specified likelihoods, not universal values for those LD settings.

Worked Example: A Candidate’s PIP Changes With the LD Model

Suppose variants X and Y are the only mutually exclusive causal candidates and have equal prior probabilities. Under one hypothetical LD reference, the likelihood gives a Bayes factor of 20 for X versus Y. The posterior odds are $20\times1=20:1$, so $$ P(X\text{ causal}\mid D,M_1)=\frac{20}{20+1}\approx0.9524. $$

Under an alternative, plausibly better-matched LD reference, suppose the likelihood ratio is 1.5 rather than 20. With the same equal prior odds, $$ P(X\text{ causal}\mid D,M_2)=\frac{1.5}{1.5+1}=0.6000. $$

The difference does not show that one answer is automatically correct; it shows that the posterior is conditional on the LD model. The analysis should investigate which LD reference is defensible for the data and report meaningful sensitivity, especially when correlated candidates are difficult to distinguish.

A Practical Interpretation Workflow

Before describing a high PIP as evidence for a causal variant, make the conditioning and the target explicit. A compact audit can prevent a numerical posterior from being mistaken for a model-free conclusion.

Name the posterior event.
State whether the probability concerns inclusion of one variant, at least one variant in a set, or a configuration-level event.

State the model boundary.
Identify the candidate set, causal-count or configuration assumptions, likelihood, and relevant prior choices that define $M$.

Check the joint posterior when needed.
For a multi-causal set-level question, sum probabilities of the configurations that satisfy the event rather than summing marginal PIPs.

Assess sensitivity.
Compare reasonable prior and LD choices, and note whether the leading candidate or set-level conclusion changes materially.

Limit the biological claim.
Describe posterior support under the model; do not treat inclusion as proof of a mechanism or as protection against model misspecification.

Common Mistake

Wrong reasoning: “The PIP is 0.98, so there is a 98% chance this variant is truly causal, and a 95% credible set is guaranteed to contain the causal variant.”

Correction: The PIP is a probability of inclusion conditional on the data and specified model. Its interpretation depends on the candidate set, likelihood, and priors. A credible-set probability has the intended coverage meaning only for the event it was constructed to represent; in a multi-causal analysis, adding marginal PIPs does not generally give the posterior probability that a set contains at least one causal variant. Use joint configuration probabilities for that event, and describe the model conditioning.

Key takeaway: High posterior inclusion probability is strong model-conditional evidence, not proof of causal truth. Interpret it alongside the model assumptions, prior and LD sensitivity, and the precise joint event represented by any credible set.

Check Your Understanding

Answer each question by distinguishing marginal from joint posterior statements.

In a single-causal-variant model, why can the PIPs of variants in a set be summed to obtain the posterior probability that the causal variant is in the set?
In a multi-causal model, what does the sum of PIPs over a set represent in general, and what posterior calculation gives the probability that at least one variant in the set is included?
A variant has PIP 0.97. Name two aspects of the model that condition this probability and explain why the number is not model-free proof of causality.
In the prior-odds example, why did the posterior probability increase when the prior changed even though the likelihood ratio stayed fixed?
What should be reported if changing a plausible LD reference substantially changes the leading variant’s PIP?