Fine-Mapping a Weak or Null Signal

Why Weak Signals Need an Explicit Null Model

Fine-mapping is often presented as a task of distributing posterior probability among candidate variants. That distribution can be misleading if the analysis presumes that at least one candidate is causal. When the association evidence is weak, the data may not support that presumption: a model in which no variant in the region is causal may remain plausible, or even receive most of the posterior probability.

The key is to keep two questions separate. First, do the data support a causal signal in the region rather than no signal? Second, conditional on a signal being present, which variant or configuration is most supported? A high probability for one variant among causal alternatives does not by itself answer the first question.

For clarity, the first calculations below use a model set containing \(M_0\) and one-causal-variant models \(M_1,\ldots,M_m\). These models are mutually exclusive and exhaustive within that simplified model set. The same calculation applies to a richer set of causal configurations, including configurations with multiple causal variants, provided the null configuration is included and the model probabilities are properly defined.

Posterior Probabilities With a Null Configuration

Write \(\pi_0=P(M_0)\) for the prior probability of no causal variant and \(\pi_j=P(M_j)\) for the prior probability of model \(M_j\). The prior probabilities sum to 1. Expressing every model's marginal likelihood relative to the null gives posterior model probabilities by Bayes' rule:

The denominator is the prior-weighted evidence across all models, with the null model's Bayes factor equal to 1 relative to itself. The unconditional posterior inclusion probability for variant \(j\), in this one-causal-variant model set, is \(P(M_j\mid D)\). The probability that any candidate is causal is \(1-P(M_0\mid D)\).

A different quantity is the probability that variant \(j\) is causal conditional on at least one candidate being causal. In this one-causal-variant setting it is:

This conditional probability renormalizes away the posterior mass on the null. It describes relative support among non-null models; it does not measure how strongly the data support any causal model over the null. In a multi-causal configuration model, the same principle applies: the unconditional inclusion probability for a variant sums posterior probabilities over all configurations containing that variant, while a conditional inclusion probability additionally conditions on the chosen non-null event.

A Bayes Factor for a Weak Summary Statistic

A simple normal approximation shows why weak association statistics often provide little evidence for a causal effect. Suppose an estimated effect \(b\), with known standard error \(s\), has sampling distribution \(b\mid\beta\sim N(\beta,s^2)\). Under the null, \(\beta=0\). Under the alternative, assign the effect prior \(\beta\sim N(0,W)\), where \(W>0\) is the prior variance. This is a model for one candidate variant; the effect scale and prior must be appropriate to the analysis.

Integrating over the effect prior gives \(b\mid M_1\sim N(0,s^2+W)\), whereas \(b\mid M_0\sim N(0,s^2)\). The ratio of these two normal densities is the Bayes factor for the alternative against the null. Defining \(z=b/s\) and \(r=W/(W+s^2)\), direct division of the densities yields:

This calculation assumes the stated normal sampling model, the specified normal effect prior, and a standard error that is treated as known. A value of \(BF_{10}\) near 1 means the observed estimate has similar marginal likelihood under the alternative and null. It is not a declaration that the effect is exactly zero.

Worked Comparison Across Candidate Variants

With multiple candidates, weak evidence can still produce a ranking among them. That ranking must be reported alongside the posterior mass assigned to the null. The next example uses Bayes factors directly, so it does not require estimating them from association statistics.

What Weak Evidence Can and Cannot Establish

A Bayes factor near 1 has a direct interpretation: the data do not materially change the prior odds between the compared models. For a single non-null model, posterior odds equal prior odds multiplied by the Bayes factor. For a collection of causal models, the total non-null evidence is the prior-weighted average of their Bayes factors, conditional on the non-null prior. Consequently, when these Bayes factors are all close to 1, the posterior probability of a signal may be driven mainly by the prior probability assigned to a signal.

This is not a defect in Bayes' rule, but it is a reason to state the prior and the model set. Fine-mapping results can appear decisive if the analysis assigns prior probability zero to the null: in that case the posterior must allocate all probability among causal alternatives, regardless of how weak their evidence is. Conversely, an explicit null with a high prior can retain high posterior probability unless the data provide evidence against it.

The null model is also relative to the analysis. “No causal variant” means no causal variant among the candidates and under the specified model and data assumptions. It does not rule out an unobserved causal variant, a variant outside the analyzed region, model misspecification, or a non-genetic explanation. The posterior answers a comparison among the models that were actually included.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting a weak-signal result, check that the comparison is internally consistent. Bayes factors must use the same null reference and the same data for the models being compared. The prior probabilities must sum to 1 across the model set, and the null must have positive prior probability if the analysis is intended to assess whether a signal exists. In a configuration analysis, verify that the null configuration is actually included rather than inferred from a set of causal configurations alone.

• Check the scale of the conclusion. Report posterior probability of any signal and unconditional variant inclusion probabilities, not only a ranking conditional on a signal.
• Inspect weak-evidence behavior. Bayes factors close to 1 should usually leave posterior odds near the prior odds. A dramatic posterior change calls for checking the arithmetic, priors, and model definitions.
• Check the candidate set and assumptions. The null is only as meaningful as the alternatives considered. Unmodeled variants, an unsuitable effect prior, or inaccurate LD and summary statistics can undermine the stated model comparison.
• Do not equate a weak association statistic with evidence of no effect. Weak evidence may reflect limited information, a small effect, or the chosen prior as well as a true null.

An alternative is to report fine-mapping probabilities only conditional on at least one causal variant. This is useful when the scientific question is explicitly “which variant, assuming a signal exists?” It is not an adequate substitute when the goal is to assess whether the region contains a signal. A thresholded association test answers a different question and does not, by itself, provide posterior probabilities over causal configurations. Neither approach should be presented as interchangeable with an explicit Bayesian comparison to the null.

Common mistake: “Variant 3 has the largest conditional posterior, so the data establish that it is causal.” This reasoning confuses a relative ranking among non-null alternatives with absolute evidence for a causal signal. The correction is to state both quantities: in the three-variant example, variant 3 has conditional probability \(0.4242\), but unconditional probability \(0.0915\), while the null has probability \(0.7843\).