Posterior Probabilities for Causal Configurations

From Configuration Priors to Posterior Probabilities

In a multi-causal fine-mapping model, the unknown is not always just which single variant is causal. Several variants may be causal together, so the model assigns probability to possible configurations: sets of variants included as causal. The posterior probability of a configuration combines its prior probability with how well that configuration explains the observed data. Variant inclusion probabilities then follow by adding the posterior probabilities of all configurations that contain the variant.

This calculation depends on comparing configurations on the same data and under a specified model. In particular, the likelihood for a configuration must account for the variants jointly; evidence calculated for each variant separately does not generally determine the joint evidence.

Definition: Let \(D\) denote the observed genetic data and let \(\gamma\) be a configuration, represented by a subset of the \(p\) candidate variants. Its prior probability is \(\Pr(\gamma)\). Its marginal likelihood is \(m(D\mid\gamma)=\int f(D\mid\theta_\gamma,\gamma)\,\pi(\theta_\gamma\mid\gamma)\,d\theta_\gamma\), where \(\theta_\gamma\) denotes the model parameters under that configuration, \(f\) is the sampling density, and \(\pi(\theta_\gamma\mid\gamma)\) is the parameter prior. The configuration posterior probability is \(\Pr(\gamma\mid D)\), the probability assigned to that configuration after observing \(D\).

The configuration specifies which variants are included in the causal model. It does not by itself specify their effect sizes; those are parameters whose uncertainty is integrated over in the marginal likelihood. Other nuisance parameters, if present, must also be handled consistently across configurations. A maximized likelihood, which uses only the best-fitting parameter value, is not the marginal likelihood in this definition.

Assume that the set of configurations under consideration is mutually exclusive and exhaustive: exactly one of them describes the included causal variants, and their prior probabilities sum to one. Also assume a coherent probability model for the observed data and proper parameter priors where integration is required. Bayes’ rule then gives

\Pr(\gamma\mid D) = \frac{\Pr(\gamma)m(D\mid\gamma)} {\displaystyle\sum_{\gamma'\in\Gamma}\Pr(\gamma')m(D\mid\gamma')}, \qquad \sum_{\gamma\in\Gamma}\Pr(\gamma\mid D)=1.

Here \(\Gamma\) is the set of allowed configurations, and the denominator is the marginal probability of the data under the configuration mixture. It is the same for every configuration in the comparison, so it normalizes the prior-weighted evidence into probabilities.

Often it is convenient to divide every marginal likelihood by the likelihood under the empty, or null, configuration \(\varnothing\). The resulting Bayes factor is \(BF_\gamma=m(D\mid\gamma)/m(D\mid\varnothing)\), with \(BF_\varnothing=1\). Since this divides every likelihood by the same positive quantity, posterior probabilities can be calculated as

\Pr(\gamma\mid D) = \frac{\Pr(\gamma)BF_\gamma} {\displaystyle\sum_{\gamma'\in\Gamma}\Pr(\gamma')BF_{\gamma'}}.

Worked Configuration Calculations

Worked Example: Normalizing Joint Evidence for Two Variants

Suppose a hypothetical analysis considers two candidate variants, \(A\) and \(B\), and all four configurations: \(\varnothing\), \(\{A\}\), \(\{B\}\), and \(\{A,B\}\). The prior probabilities are \(0.70, 0.10, 0.15,\) and \(0.05\), respectively. Suppose their joint Bayes factors relative to the null configuration are \(1, 4, 2,\) and \(12\). The prior probabilities sum to \(1\), so they form a valid prior distribution over these configurations.

Multiply each prior probability by its configuration Bayes factor to obtain unnormalized posterior weights:

\begin{aligned} w_{\varnothing}&=0.70(1)=0.70, & w_{\{A\}}&=0.10(4)=0.40,\\ w_{\{B\}}&=0.15(2)=0.30, & w_{\{A,B\}}&=0.05(12)=0.60. \end{aligned}

The normalizing constant is \(Z=0.70+0.40+0.30+0.60=2.00\). Dividing each weight by \(Z\) gives posterior probabilities \(0.35, 0.20, 0.15,\) and \(0.30\), in the same configuration order. These sum to \(1.00\).

Variant \(A\) is included in \(\{A\}\) and \(\{A,B\}\), so its posterior inclusion probability is \(0.20+0.30=0.50\). Variant \(B\) is included in \(\{B\}\) and \(\{A,B\}\), giving \(0.15+0.30=0.45\). The posterior probability of at least one causal variant is \(1-0.35=0.65\). These are probabilities under the specified model and priors, not statements that either variant causes an outcome with certainty.

Worked Example: Independent Bernoulli Configuration Priors

Consider three hypothetical variants \(1,2,3\). Suppose each has prior inclusion probability \(0.10\), independently of the others. This assumption defines a prior over all eight configurations: a configuration with \(k\) variants has prior probability \(0.10^k(0.90)^{3-k}\). Thus the null prior is \(0.90^3=0.729\), each singleton prior is \(0.10(0.90)^2=0.081\), each two-variant prior is \(0.10^2(0.90)=0.009\), and the three-variant prior is \(0.10^3=0.001\). These eight priors sum to \(0.729+3(0.081)+3(0.009)+0.001=1\).

Suppose the hypothetical joint Bayes factors are \(1\) for the null, \(5,2,1\) for \(\{1\},\{2\},\{3\}\), \(8,3,4\) for \(\{1,2\},\{1,3\},\{2,3\}\), and \(10\) for \(\{1,2,3\}\). The prior-weighted evidence values, in that order, are \(0.729, 0.405, 0.162, 0.081, 0.072, 0.027, 0.036,\) and \(0.010\). The normalization constant is

$$ Z=0.729+0.405+0.162+0.081+0.072+0.027+0.036+0.010=1.522. $$

For variant \(1\), sum the weights for \(\{1\},\{1,2\},\{1,3\},\{1,2,3\}\), then divide by \(Z\): \(PIP_1=(0.405+0.072+0.027+0.010)/1.522=0.514/1.522\approx0.3377\). For variants \(2\) and \(3\), the corresponding calculations are \(PIP_2=(0.162+0.072+0.036+0.010)/1.522=0.280/1.522\approx0.1840\) and \(PIP_3=(0.081+0.027+0.036+0.010)/1.522=0.154/1.522\approx0.1012\).

The null configuration posterior is \(0.729/1.522\approx0.4790\), so the posterior probability of at least one included variant is \(1-0.4790\approx0.5210\). The three inclusion probabilities do not add to this probability: configurations containing multiple variants contribute once to each relevant inclusion probability.

Worked Example: Using Marginal Likelihoods Directly

Suppose a hypothetical model allows only three configurations: the null configuration, variant \(A\) alone, or variant \(B\) alone. Their prior probabilities are \(0.50, 0.30,\) and \(0.20\), and their marginal likelihoods on a common scale are \(0.02, 0.12,\) and \(0.04\). The priors sum to \(1\), and each likelihood uses the same data and the specified parameter priors.

The prior-weighted likelihoods are \(0.50(0.02)=0.010\), \(0.30(0.12)=0.036\), and \(0.20(0.04)=0.008\). Their sum is \(0.054\). Therefore the posterior probabilities are \(0.010/0.054\approx0.1852\) for the null, \(0.036/0.054=0.6667\) for \(A\), and \(0.008/0.054\approx0.1481\) for \(B\). Since this model permits only singleton causal configurations, \(PIP_A=0.6667\) and \(PIP_B=0.1481\). The probability of at least one included variant is \(1-0.1852=0.8148\).

The same result follows from Bayes factors relative to the null: \(BF_A=0.12/0.02=6\) and \(BF_B=0.04/0.02=2\). The null-relative weights are \(0.50, 0.30(6)=1.80,\) and \(0.20(2)=0.40\), which sum to \(2.70\). Dividing each by \(2.70\) gives the same posterior probabilities. This confirms that using a common rescaling of all marginal likelihoods leaves the posterior unchanged.

Variant Inclusion Probabilities and Their Meaning

Definition: The posterior inclusion probability for variant \(j\) is the posterior probability that \(j\) belongs to the causal configuration: \(PIP_j=\Pr(j\in\gamma\mid D)=\sum_{\gamma\in\Gamma:\,j\in\gamma}\Pr(\gamma\mid D)\).

This is a marginal probability obtained by summing over all configurations that include the variant, regardless of which other variants are included. It is not the posterior probability of the single-variant configuration \(\{j\}\), which excludes every other variant. Consequently, PIPs can sum to more than \(1\) when configurations with multiple variants are allowed.

A useful consistency identity is that the sum of all variant PIPs equals the posterior expected number of included variants. If \(|\gamma|\) denotes the number of variants in configuration \(\gamma\), then

\sum_{j=1}^{p}PIP_j = \sum_{j=1}^{p}\sum_{\gamma:\,j\in\gamma}\Pr(\gamma\mid D) = \sum_{\gamma\in\Gamma}|\gamma|\Pr(\gamma\mid D) = E(|\gamma|\mid D).

The middle equality follows by switching the order of two finite sums: each configuration contributes its posterior probability once for each variant it contains. This gives a practical check on calculations. In the three-variant example, the PIPs sum to approximately \(0.6229\), which is the posterior expected number of included variants; it is distinct from the probability \(0.5210\) of at least one inclusion.

Why the Joint Likelihood Matters

The marginal likelihood \(m(D\mid\gamma)\) evaluates the observed data under the entire configuration. It can reflect dependence among variant genotypes, shared uncertainty in effect estimates, and the model’s effect-size priors. Even if evidence has been calculated separately for each variant, multiplying those single-variant Bayes factors is justified only under additional assumptions that make the relevant likelihood and prior contributions factorize. Such factorization is not automatic in genetic data, where variants can be correlated through linkage disequilibrium.

Configuration posterior probabilities are also conditional on the set \(\Gamma\), the configuration priors, the likelihood model, and the parameter priors used to form each marginal likelihood. A PIP is therefore not a prior-free property of a variant. A different prior distribution over configurations can alter the posterior even when the data likelihoods are unchanged.

Calculation Checks and Common Failure Modes

Check the configuration space.
List the configurations allowed by the model and verify that they are mutually exclusive and cover the possibilities the model intends to represent. Confirm that their prior probabilities are nonnegative and sum to \(1\).

Compute comparable evidence.
For every configuration, use the same observed data and a coherent definition of its marginal likelihood. If using Bayes factors, divide by the same reference marginal likelihood.

Normalize before summing.
Multiply each prior by its marginal likelihood or Bayes factor, sum the weights across all configurations, and divide each weight by that common total. Then verify the posterior probabilities sum to \(1\).

Marginalize for each variant.
For variant \(j\), add the posterior probabilities of every configuration containing \(j\). Check that the sum of the PIPs equals the posterior expected configuration size.

A common mistake is to call the posterior probability of \(\{j\}\) the PIP for variant \(j\). That omits configurations such as \(\{j,k\}\), even though they also include \(j\). The correction is to sum over all configurations containing \(j\), as in the PIP definition.

Other warning signs include posterior configuration probabilities that do not sum to \(1\), prior weights that omit some allowed configurations, or PIPs inconsistent with the posterior expected number of included variants. These can indicate a missing configuration, an arithmetic error, or confusion between a prior and a posterior. If the number of candidate variants is large, exhaustive enumeration may also be computationally impractical because there are \(2^p\) possible subsets before restrictions. Any computational shortcut must still target the same posterior quantity if it is to be interpreted as the configuration posterior defined here.

Quantity	Calculation	Interpretation
Configuration posterior	Normalize \(\Pr(\gamma)m(D\mid\gamma)\) across configurations	Posterior probability of that entire included-variant set
Variant inclusion probability	Sum configuration posteriors over \(\gamma\) containing \(j\)	Posterior probability that variant \(j\) is included
Probability of any inclusion	Sum posteriors for non-null configurations	Posterior probability that at least one variant is included

Check Your Understanding

Try each calculation before reading the solutions below.

Two configurations have prior probabilities \(0.8\) and \(0.2\), with Bayes factors \(1\) and \(5\). What are their posterior probabilities?
Three configurations have posterior probabilities \(0.2\) for the null, \(0.5\) for \(\{A\}\), and \(0.3\) for \(\{A,B\}\). Find \(PIP_A\), \(PIP_B\), and the probability of at least one included variant.
In a model allowing only the null and \(\{A\}\), the posterior null probability is \(0.35\). What is \(PIP_A\)? Why would the same subtraction not generally give the PIP of \(A\) in a model that also permits other variants?
Explain why multiplying single-variant Bayes factors does not generally give a joint Bayes factor for a multi-variant configuration.

Solutions. (1) The unnormalized weights are \(0.8(1)=0.8\) and \(0.2(5)=1.0\), totaling \(1.8\). The posterior probabilities are \(0.8/1.8\approx0.4444\) and \(1.0/1.8\approx0.5556\). (2) \(PIP_A=0.5+0.3=0.8\), \(PIP_B=0.3\), and the probability of at least one inclusion is \(1-0.2=0.8\). (3) In this restricted model, every non-null configuration contains \(A\), so \(PIP_A=1-0.35=0.65\). If other variants or configurations are allowed, some non-null configurations may not contain \(A\), so their posterior probabilities must not be counted toward \(PIP_A\). (4) A joint Bayes factor evaluates the full data under the joint configuration relative to the reference model. Dependence among variants or non-factorizing likelihood and prior terms can prevent the joint evidence from equaling the product of separate single-variant Bayes factors.