The Fine-Mapping Posterior Target

What Is the Fine-Mapping Posterior About?

A locus may contain several associated variants that are correlated through linkage disequilibrium (LD). Association results describe patterns in the data, but fine-mapping asks a different question: which set of variants is treated as causal under a specified statistical model? The object being assigned posterior probability is a possible set, not merely a list of single-variant association statistics.

This distinction matters because correlated variants can produce similar patterns of association. A posterior distribution makes the uncertainty over possible causal sets explicit, but it does not remove the dependence of the answer on the data, LD information, and modeling assumptions.

Definition: Let \(V\) be the finite set of candidate variants in a locus. A causal configuration is a subset \(c\subseteq V\), representing the variants treated as causal under a specified model. The empty set, \(c=\varnothing\), represents the configuration in which no candidate variant in \(V\) is causal. A configuration posterior is the probability distribution over these possible sets, conditional on the association data, the LD information, and the statistical model.

For example, if \(V=\{A,B\}\), the possible configurations in a model that allows any subset are \(\varnothing\), \(\{A\}\), \(\{B\}\), and \(\{A,B\}\). They are distinct hypotheses about the set of causal variants. A model may restrict which configurations are allowed; in that case, its posterior is distributed over the allowed configurations, not over configurations excluded by the model.

The word “causal” here is model-relative. It describes a variant’s role in the specified genetic model for the trait, rather than asserting that an observed association alone establishes biological causation. The posterior is also conditional on the candidate set: a variant omitted from \(V\) cannot receive probability as a member of a configuration.

Definition of the Posterior Target

Write \(D\) for the association data, \(R\) for the LD information used in the analysis, and \(M\) for the statistical model. The data \(D\) might consist of individual-level observations or association summaries, depending on the analysis. The model specifies, among other things, how a configuration relates to the data and how unknown parameters such as effect sizes are handled.

Let \(\pi(c\mid M)\) denote the prior probability assigned to configuration \(c\) under model \(M\). Let \(m(D\mid c,R,M)\) denote the model-based marginal likelihood: the probability density or probability of the observed data under configuration \(c\), given the LD information and model, after averaging over unknown parameters according to the model’s specified parameter distributions. If the relevant parameters are fixed rather than averaged over, the corresponding likelihood is used instead. The configuration posterior is

P(C=c\mid D,R,M) = \frac{m(D\mid c,R,M)\,\pi(c\mid M)} {\displaystyle\sum_{c'\in\mathcal{C}}m(D\mid c',R,M)\,\pi(c'\mid M)}.

Here \(C\) is the unknown configuration and \(\mathcal{C}\) is the set of configurations allowed by \(M\). The denominator sums over every allowed configuration. This normalization ensures that the posterior probabilities sum to one. When nuisance parameters are integrated out, that integration is part of the marginal likelihood; it is not an optional step that can be ignored while still claiming to have computed the same posterior.

Key interpretation: \(P(C=c\mid D,R,M)\) is the posterior probability of the exact configuration \(c\), conditional on the analyzed data, LD input, model, and configuration prior. It is not a probability that an association result is true in isolation, nor a model-free probability that a variant is biologically causal.

Why the Formula Has This Form

The posterior formula follows from the definition of conditional probability and the fact that the configurations form a mutually exclusive, exhaustive set under the model. For any allowed configuration \(c\), the joint probability of the data and that configuration, conditional on \(R\) and \(M\), is the probability of the data under \(c\) multiplied by the prior probability of \(c\):

P(D,C=c\mid R,M) = m(D\mid c,R,M)\,\pi(c\mid M).

The probability of observing \(D\), conditional on \(R\) and \(M\), is obtained by summing this joint probability over all possible configurations:

P(D\mid R,M) = \sum_{c'\in\mathcal{C}}P(D,C=c'\mid R,M) = \sum_{c'\in\mathcal{C}}m(D\mid c',R,M)\,\pi(c'\mid M).

Dividing the joint probability for \(c\) by this total probability gives the conditional probability of \(c\) given the data. Substituting the two expressions yields the configuration posterior formula above. Summing that formula over \(c\in\mathcal{C}\) gives one, because its numerator terms sum to its denominator.

Worked Posterior Calculations

Worked Example: Normalizing Probabilities for Three Configurations

Suppose a hypothetical locus has two candidate variants, \(A\) and \(B\), and the model allows only three configurations: \(c_0=\varnothing\), \(c_A=\{A\}\), and \(c_B=\{B\}\). Thus, this particular model excludes a configuration with both variants causal. Assume the prior probabilities and marginal likelihoods below have been specified under one statistical model and one LD input. The values are illustrative, not results from a real study.

Configuration	Prior probability	Marginal likelihood
\(c_0=\varnothing\)	0.50	0.020
\(c_A=\{A\}\)	0.30	0.100
\(c_B=\{B\}\)	0.20	0.040

The unnormalized weights are the products of prior probability and marginal likelihood:

w_0=0.50(0.020)=0.010,\qquad w_A=0.30(0.100)=0.030,\qquad w_B=0.20(0.040)=0.008.

Their sum is \(0.010+0.030+0.008=0.048\). Dividing each weight by this sum gives the posterior distribution:

P(C=c_0\mid D,R,M)=\frac{0.010}{0.048}=0.2083, \quad P(C=c_A\mid D,R,M)=\frac{0.030}{0.048}=0.6250, \quad P(C=c_B\mid D,R,M)=\frac{0.008}{0.048}=0.1667.

The probabilities sum to 1.000. Under this hypothetical model and its inputs, \(\{A\}\) has the largest posterior probability among the three allowed configurations. That does not mean the posterior probability is a model-independent claim that \(A\) is causal; it is conditional on the specified configuration set, prior, likelihood, LD input, and model.

Worked Example: How the LD Input Can Change the Posterior

Consider a second hypothetical analysis with the same candidate variants and three allowed configurations: \(c_A=\{A\}\), \(c_B=\{B\}\), and \(c_{AB}=\{A,B\}\). Suppose their prior probabilities are 0.40, 0.40, and 0.20, respectively. For each LD input below, assume the same data and the same statistical model apart from the specified LD matrix, and use the illustrative marginal likelihoods shown.

Configuration	Prior	Likelihood using \(R_1\)	Likelihood using \(R_2\)
\(c_A\)	0.40	0.120	0.050
\(c_B\)	0.40	0.060	0.100
\(c_{AB}\)	0.20	0.010	0.010

Using \(R_1\), the unnormalized weights are \(0.40(0.120)=0.048\), \(0.40(0.060)=0.024\), and \(0.20(0.010)=0.002\). Their total is \(0.074\), so the posterior probabilities are \(0.048/0.074=0.6486\), \(0.024/0.074=0.3243\), and \(0.002/0.074=0.0270\), respectively.

Using \(R_2\), the weights are \(0.40(0.050)=0.020\), \(0.40(0.100)=0.040\), and \(0.20(0.010)=0.002\). Their total is \(0.062\), giving posterior probabilities \(0.020/0.062=0.3226\), \(0.040/0.062=0.6452\), and \(0.002/0.062=0.0323\), respectively.

The most probable configuration changes from \(\{A\}\) under \(R_1\) to \(\{B\}\) under \(R_2\). This is possible because LD affects how a configuration predicts the observed association pattern under the model. The example does not establish that either LD input is appropriate; it shows why the LD information must be reported as part of the conditioning context.

Worked Example: Probability of a Group of Configurations

Suppose a hypothetical model allows four configurations for variants \(A\) and \(B\): \(\varnothing\), \(\{A\}\), \(\{B\}\), and \(\{A,B\}\). Assume the prior probabilities are \(0.20, 0.30, 0.30,\) and \(0.20\), respectively, and the marginal likelihoods under the same data, LD input, and model are \(0.050, 0.120, 0.100,\) and \(0.040\). The unnormalized weights are:

0.20(0.050)=0.010,\quad 0.30(0.120)=0.036,\quad 0.30(0.100)=0.030,\quad 0.20(0.040)=0.008.

The weights sum to \(0.084\). Dividing each by \(0.084\), the posterior probabilities are \(0.1190\), \(0.4286\), \(0.3571\), and \(0.0952\), in the order listed. The posterior probability that exactly one candidate variant is causal is the sum of the probabilities for \(\{A\}\) and \(\{B\}\):

P(C\in\{\{A\},\{B\}\}\mid D,R,M) = 0.4286+0.3571 = 0.7857.

This is a probability for a specified group of configurations. It is obtained by adding their posterior probabilities because the configurations are mutually exclusive. The exact-configuration probabilities remain important: the total 0.7857 alone does not distinguish whether \(\{A\}\) or \(\{B\}\) is better supported.

What Determines the Posterior?

The posterior combines three ingredients. First, the configuration prior assigns probability before considering the observed association data. Second, the data likelihood measures how compatible the observed data are with each configuration under the model. Third, the normalization compares each configuration’s prior-weighted likelihood with the corresponding quantities for all other allowed configurations.

The likelihood depends on how the model connects causal variants to observed data. With individual-level data, this connection is represented through the model for outcomes and genotypes. With association summaries, LD may be needed to represent the dependence among variant-level results. The same association data can therefore lead to different configuration posteriors if the LD input or statistical assumptions change.

A posterior is not defined by association data alone. A stated analysis must identify the candidate variants, the allowed configurations, the LD information, and the statistical model, including how unknown parameters and configuration probabilities are treated. If the model rules out a configuration, no amount of data can give that excluded configuration posterior probability within that model.

Common Pitfall: Treating the Posterior as Model-Free

Incorrect reasoning: “The variant with the strongest association has a 90% chance of being causal.” A strong association statistic by itself does not define a posterior probability for a causal configuration.
Correction: A posterior probability follows from a specified data likelihood, LD input, configuration prior, and statistical model. State these conditions when interpreting the result.
Incorrect reasoning: “The top configuration is proven to be the true causal set.” The configuration with the largest posterior probability is the most probable among the allowed alternatives under the specified analysis, but uncertainty remains.
Correction: Report the posterior distribution, or a clearly defined summary of it, and retain its conditioning context. Do not turn model-conditional support into a claim of certainty or causation established by association alone.

Key takeaway: Fine-mapping targets a posterior distribution over causal configurations. Each probability refers to an exact set of variants and is conditional on the association data, LD information, candidate set, prior, and statistical model used to construct it.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

If a locus has three candidate variants and the model permits any subset, how many possible configurations are there, including the empty configuration?
In the configuration posterior formula, why must the denominator include every configuration allowed by the model?
What does \(P(C=c\mid D,R,M)\) condition on, and why is it not a model-free probability of causation?
If two mutually exclusive configurations have posterior probabilities 0.25 and 0.35, what is the posterior probability that the true configuration is one of those two, under the model?
In the LD example, why could changing \(R\) change which configuration has the largest posterior probability even though the data and prior were held fixed?