Why Weight Variant-to-Gene Links?
Fine-mapping describes uncertainty about which variants in a locus are causal. Variant-to-gene mapping describes which genes those variants may affect. These are related but distinct questions: a variant can have a high probability of being causal while its target gene is uncertain, and a variant can map convincingly to a gene while having little probability of being causal. Posterior-weighted mapping keeps both uncertainties visible rather than treating every variant in a credible set as equally important.
The calculation is useful for summarizing support at the gene level, but its interpretation depends on what the mapping weights mean. A calibrated probability of a variant-to-gene link can support a probabilistic interpretation under stated assumptions. A heuristic score or binary annotation can still be combined with fine-mapping probabilities, but the result is a weighted score, not automatically a probability that the gene is causal.
Definitions and Interpretation
The sum runs over the variants being considered at the locus. Mapping weights may be based on position, regulatory evidence, or a prespecified combination of evidence. Their scale must be stated. In the probabilistic version, \(w_{ig}\) is the conditional probability that variant \(i\) maps to gene \(g\), given that the variant is causal and given the mapping evidence. If the candidate genes are mutually exclusive mapping destinations, the weights across those genes, together with any unassigned destination, sum to one for each variant. A deterministic link can be represented by a weight of one for a linked gene and zero otherwise.
If mapping weights are conditional probabilities, and the mapping evidence does not change the fine-mapping posterior, \(S_g\) is the expected number of causal variants assigned to gene \(g\). With exactly one causal variant at the locus and mapping destinations that are mutually exclusive, it is also the posterior probability that the causal variant maps to gene \(g\). If mapping weights are heuristic, the same sum is a weighted support score; it does not have a direct probability interpretation.
A variant may have evidence for more than one gene. If the weights represent separate, possibly overlapping links rather than mutually exclusive destinations, each gene’s score can still summarize weighted support, but scores across genes need not sum to one. In that setting, the score is not the probability that a causal variant maps exclusively to that gene.
Worked Example: One Causal Variant and Uncertain Gene Mapping
Consider a hypothetical locus with four fine-mapped variants. The fine-mapping model assigns posterior probabilities that sum to one, consistent here with a model in which exactly one variant is causal. A calibrated mapping procedure provides conditional probabilities for mapping each causal variant to gene A, gene B, or an unresolved destination.
| Variant | \(\mathrm{PIP}_i\) | Weight for A | Weight for B | Unresolved weight |
|---|---|---|---|---|
| 1 | 0.50 | 0.90 | 0.10 | 0.00 |
| 2 | 0.25 | 0.20 | 0.70 | 0.10 |
| 3 | 0.15 | 0.10 | 0.40 | 0.50 |
| 4 | 0.10 | 0.00 | 0.20 | 0.80 |
For each variant, the three mapping weights sum to one. The posterior-weighted support for gene A is $$ S_A=(0.50)(0.90)+(0.25)(0.20)+(0.15)(0.10)+(0.10)(0.00) =0.450+0.050+0.015+0=0.515. $$
For gene B, $$ S_B=(0.50)(0.10)+(0.25)(0.70)+(0.15)(0.40)+(0.10)(0.20) =0.050+0.175+0.060+0.020=0.305. $$
The unresolved contribution is \((0.50)(0.00)+(0.25)(0.10)+(0.15)(0.50)+(0.10)(0.80)=0.180\). The three results sum to \(1.000\), as expected under the assumptions in this example. Thus, the model assigns posterior mapping probabilities of 0.515 to gene A, 0.305 to gene B, and 0.180 to an unresolved destination. Gene A has the greatest support, but the unresolved mass shows why the result should not be reported as certainty about a target.
Why the Weighted Sum Works
The calculation follows from an expectation. Let \(C_i\) indicate that variant \(i\) is causal, and let \(A_{ig}\) indicate that it is assigned to gene \(g\) under the mapping model. When the mapping destinations are exclusive, define \(Y_{ig}=C_iA_{ig}\). This indicator equals one only when variant \(i\) is both causal and assigned to gene \(g\). If the mapping probability is \(w_{ig}=P(A_{ig}=1\mid C_i=1,D,M)\), where \(D\) denotes association data and \(M\) mapping evidence, and the mapping evidence does not alter the fine-mapping posterior, then:
The number of causal variants assigned to gene \(g\) is \(\sum_i Y_{ig}\). By additivity of expectation,
This derivation does not require the causal indicators for different variants to be independent: expectation of a sum is the sum of the expectations. It does require that the PIP and mapping weight represent the relevant conditional probabilities. If the mapping evidence also updates the probability that a variant is causal, the product must instead use \(P(C_i=1\mid D,M)\), or come from a joint model that represents both types of uncertainty. Combining a PIP conditional on \(D\) with a weight conditional on \(M\) is not automatically a valid joint posterior calculation.
Worked Example: Expected Support with Multiple Possible Causal Variants
Suppose a hypothetical fine-mapping analysis allows multiple causal variants. Three variants have PIPs of 0.60, 0.40, and 0.20. A calibrated mapping procedure assigns them probabilities of 0.80, 0.50, and 0.25, respectively, of mapping to gene X if causal. These weights need not sum to one across variants: they describe a separate mapping probability for each variant.
The support score for gene X is $$ S_X=(0.60)(0.80)+(0.40)(0.50)+(0.20)(0.25) =0.480+0.200+0.050=0.730. $$
Under the stated interpretation, 0.730 is the expected number of causal variants assigned to gene X, not a 73% probability that gene X is causal. An expected count can exceed one in other examples. To calculate the probability that at least one causal variant maps to X, one needs the joint distribution of causal status and mapping assignments across variants. The marginal products in this sum alone do not provide that joint distribution.
Using Positional and Functional Mapping Evidence
Position and functional annotations can inform \(w_{ig}\), but the weight should represent a clearly defined mapping rule. For example, a positional rule may link variants to genes based on genomic coordinates, while functional evidence may link a variant to a gene through a regulatory element. The earlier material on regulatory links and chromatin-contact evidence provides examples of the kinds of observations that can support such links. Here, the task is to combine the resulting variant-level mapping information with fine-mapping PIPs, not to treat an annotation alone as proof of the causal gene.
When both positional and functional information are available, use a prespecified joint mapping model or a calibrated weight based on the combined evidence. Do not multiply separate evidence scores merely because they are available: that multiplication would require a model justifying their scales and dependence. In particular, two annotations may reflect the same underlying biological signal. A combined weight should be interpreted as a probability only if its calibration and conditioning are appropriate for that interpretation.
Worked Example: Functional Mapping Changes Which Gene Leads
Consider a hypothetical locus with one causal variant under the fine-mapping model. A mapping procedure that uses positional and functional evidence jointly has produced the conditional mapping probabilities below. The weights for genes A and B sum to one for each variant, and the PIPs also sum to one.
| Variant | \(\mathrm{PIP}_i\) | Weight for A | Weight for B |
|---|---|---|---|
| 1 | 0.70 | 0.10 | 0.90 |
| 2 | 0.20 | 0.90 | 0.10 |
| 3 | 0.10 | 0.60 | 0.40 |
The PIP-weighted support for A is $$ S_A=(0.70)(0.10)+(0.20)(0.90)+(0.10)(0.60) =0.070+0.180+0.060=0.310. $$
For B, the calculation is $$ S_B=(0.70)(0.90)+(0.20)(0.10)+(0.10)(0.40) =0.630+0.020+0.040=0.690. $$
Thus, gene B receives posterior mapping probability 0.690 in this hypothetical single-causal-variant setting, compared with 0.310 for gene A. By contrast, averaging the mapping weights without the PIPs gives A an average weight of \((0.10+0.90+0.60)/3=0.533\) and B an average weight of \((0.90+0.10+0.40)/3=0.467\). That unweighted summary gives each variant equal influence, even though variant 1 carries 70% of the causal posterior probability. Posterior weighting gives more influence to the variant most likely to be causal.
Assumptions, Reporting, and Common Pitfalls
A defensible calculation starts by checking what the inputs represent. PIPs should come from a fine-mapping analysis appropriate to the locus and its association data. Mapping weights should refer to the same variant identifiers and genome coordinates, and their construction should be stated. If some variants cannot be mapped, retain an unresolved category when possible rather than silently assigning their weight to the most plausible available gene. The sum of PIPs itself need not equal one: it reflects the fine-mapping model’s assumptions about the number of causal variants and the variants included.
Also state whether mapping weights are probabilities, binary links, or heuristic scores. If they are calibrated conditional probabilities and the assumptions above hold, the weighted sum has the stated expectation interpretation. If they are binary, \(\sum_i\mathrm{PIP}_i w_{ig}\) is the expected number of causal variants among those linked to the gene. If they are arbitrary scores, it is a prioritization summary whose scale depends on the scoring rule. Scores from different mapping procedures should not be compared as though they were on a common probability scale unless that comparability has been established.
- Mistake: “The gene with the largest score is proven causal.” A score summarizes evidence under a mapping rule and fine-mapping model; it does not establish a molecular mechanism or causal gene.
- Mistake: “The sum is always the probability that the gene is causal.” It is a probability of a mapped causal variant in the single-causal, exclusive-mapping case. With multiple possible causal variants, it is generally an expected count; with heuristic weights, it is a score.
- Mistake: “Every variant in the credible set should count equally.” Equal weighting discards differences in PIP. Use each variant’s PIP when the goal is posterior-weighted support.
- Mistake: “Separate annotation scores can be multiplied without qualification.” That requires a justified joint model and appropriate handling of dependence. Prefer an explicitly defined combined mapping weight.
Check Your Understanding
Use the definitions and assumptions in this tutorial to answer the following questions.
- A variant has PIP 0.40 and a conditional mapping probability of 0.75 for gene Q. What is its contribution to the posterior-weighted support for Q?
- Why can a gene’s support score exceed one when multiple causal variants are possible?
- When can the sum of PIP-weighted mapping probabilities be interpreted as the probability that the single causal variant maps to a particular gene?
- If mapping weights are heuristic scores rather than calibrated probabilities, what can and cannot be claimed about the resulting weighted sum?
- Why is it not generally valid to calculate the probability that at least one variant maps causally to a gene using only the sum of the variant-level contributions?