Posterior Mass and Credible-Set Coverage

What Does a Credible Set’s Posterior Mass Mean?

A fine-mapping credible set is often reported with a number such as 95%. That number is useful only if its probability statement is clear. It describes uncertainty about which variant is causal conditional on the observed data and the specified model. It does not, by itself, state how often a set-building procedure will contain the true causal variant across repeated studies.

Here, consider the single-causal-variant setting. Let \(S\) denote the index of the causal variant among the variants being analyzed, let \(D\) denote the observed data, and let \(M\) denote the specified statistical model, including its likelihood and prior assumptions. The posterior probability for variant \(j\) is \(P(S=j\mid D,M)\). In this setting these probabilities sum to one across the variants under consideration.

The conditioning is essential. The statement \(P(S\in C\mid D,M)=0.95\) is about the posterior distribution after observing \(D\) and adopting \(M\). It is not a claim that the model is certainly correct, nor does it remove uncertainty about model assumptions, the variant set, or the data. If any of these change, the posterior probabilities—and therefore the mass assigned to \(C\)—may change.

In the single-causal-variant setting, the posterior inclusion probability for variant \(j\) is \(P(S=j\mid D,M)\). Thus, to find the posterior mass of a set, add the inclusion probabilities of its members. A discrete credible set may have mass greater than its nominal target: adding the next variant can take a cumulative sum from below 0.95 to, for example, 0.97.

Calculating and Interpreting Posterior Mass

This interpretation concerns the event \(S\in C\), not a separate probability for every member of the set. For example, a set’s mass of 0.89 does not mean that each variant in it has an 89% probability of being causal. The individual probabilities are the posterior inclusion probabilities; their sum gives the probability assigned to the union of the mutually exclusive single-causal-variant events.

Posterior Mass Is Not Frequentist Coverage

Frequentist coverage describes the long-run behavior of a procedure under repeated sampling. Suppose the true causal variant is fixed at \(j_0\), and a procedure constructs a set \(C(D)\) from each study’s data. For a specified data-generating parameter \(\theta_0\), its coverage is \(P_{\theta_0}(j_0\in C(D))\). The probability is taken over repeated data sets generated under \(\theta_0\); the causal variant is not redrawn for each repetition.

A credible set can have posterior mass of at least 95% for a particular observed data set while the procedure’s frequentist coverage at a fixed causal variant is not 95%. These quantities answer different questions. Posterior mass asks, “Given these data and this model, how much posterior probability is assigned to variants in this set?” Coverage asks, “If data were repeatedly generated under this fixed truth, how often would the procedure’s set contain that truth?”

A Prior-Averaged Connection

Posterior mass and coverage are distinct, but they are related when data are averaged over the model’s prior predictive distribution. In the single-causal-variant setting, let \(\pi_j=P(S=j\mid M)\) be the prior probability that variant \(j\) is causal. For each observed data set \(D\), let \(C(D)\) be a set constructed from that data. The posterior mass is \(P(S\in C(D)\mid D,M)\).

Averaging that posterior mass over data generated from the model’s prior predictive distribution gives the prior-weighted average coverage. To see this, write the expectation as a sum over possible data sets and causal variants. (For continuous data, the sum over data sets is replaced by an integral.)

The second line follows from \(P(D\mid M)P(S=j\mid D,M)=\pi_jP(D\mid S=j,M)\), which is Bayes’ rule. The final line is an average of the procedure’s coverage conditional on each possible causal variant, weighted by the model’s prior probabilities \(\pi_j\). Therefore, if every data set receives a set with posterior mass at least \(q\), this prior-predictive average coverage is at least \(q\). The result is an average over the model’s prior distribution on causal variants—not a guarantee that coverage is at least \(q\) for each fixed variant.

This distinction matters in fine-mapping because a result is conditional on a chosen model and its represented variants. A posterior-mass statement should therefore be reported as such. It should not silently be converted into a repeated-study coverage claim.

Why the Model and Data Matter

Posterior mass is not a property of a variant set alone. It depends on the data and on the model used to turn data into posterior probabilities. The model includes, among other elements, the candidate variants and the assumptions used in the likelihood and prior. Different defensible specifications can assign different probabilities to the same variants, even when the observed data are unchanged.

The same conditionality applies when the data change. A new sample may alter the likelihood evidence, the posterior probabilities, and which variants enter a set. Consequently, a reported credible-set mass should be read together with the data analysis and model specification that produced it.

Common Mistake and Reporting

Wrong reasoning: “This is a 95% credible set, so it will contain the true causal variant in 95% of repeated studies.” This treats a posterior probability conditional on one data set and model as if it were a repeated-sampling frequency at a fixed truth.

Correction: State what the posterior mass means: “Under the specified model and observed data, the posterior probability that the causal variant lies in this set is at least 95%.” Make a separate coverage claim only when the repeated-sampling procedure and its data-generating conditions have been specified and its coverage evaluated.

A useful report names the posterior target, the set, and the conditioning context. For example: “Under the single-causal-variant model and the analyzed data, the variants in this set contain 96% of the posterior probability.” This wording neither asserts that the model is certainly correct nor promises 96% coverage across future studies.