Genotype Posteriors From Imputation

What a Genotype Posterior Represents

At a biallelic variant, a diploid individual can carry zero, one, or two copies of the alternate allele. Imputation does not always identify one of these genotypes with certainty. Instead, it can represent uncertainty with a probability for each possibility, conditional on the available data and the imputation model. These probabilities retain distinctions that would disappear if the result were immediately reduced to a single genotype call.

For example, a posterior probability of 0.90 for one genotype means that, under the fitted imputation model and given the information used for imputation, that genotype has posterior probability 0.90. It does not mean that 90% of the individual's genotype is that category, nor does it guarantee that the model is correctly specified or calibrated.

The genotype labels depend on which allele is designated alternate. If the reference allele is A and the alternate allele is G, then \(G=0\) means A/A, \(G=1\) means A/G, and \(G=2\) means G/G. If the allele designation is reversed, the labels for zero and two copies swap. The probability vector must therefore be read alongside the variant's allele coding.

For a valid posterior distribution over these three exhaustive, mutually exclusive possibilities, each probability is between zero and one, and the probabilities sum to one:

The three values are often called genotype probabilities or genotype posteriors. A file may store them in an order such as \(P(G=0\mid D),P(G=1\mid D),P(G=2\mid D)\), but the format's allele and genotype-order conventions must be checked rather than assumed.

Assumptions Behind the Interpretation

A posterior is conditional on a statistical model. The following conditions make the basic interpretation meaningful: the variant is treated as biallelic and diploid; the reference and alternate alleles are correctly identified and aligned; the three genotype categories exhaust the possible states being modeled; and the imputation procedure has produced probabilities from a model appropriate to the observed information. The probabilities also rely on the model's assumptions about the reference haplotypes and how the observed data relate to the unobserved genotype.

These conditions do not require that all three probabilities be positive. A posterior such as \((0.98,0.02,0)\) is valid. Nor do the conditions imply that a value such as 0.98 is empirically accurate in every application. It is a model-based probability. Its practical reliability depends on the quality of the model and input data, including whether relevant genetic patterns are represented.

From Genotype Weights to Posterior Probabilities

A useful way to understand the calculation is through Bayes' rule. Suppose the model assigns a prior probability \(P(G=g)\) to each genotype and a likelihood \(P(D\mid G=g)\) to the observed information under that genotype. The posterior probability is proportional to their product, normalized across the three possible genotypes:

The denominator is the total probability of the observed information across the modeled genotype possibilities. It ensures that the resulting probabilities sum to one. In an actual imputation procedure, likelihoods and prior weights arise from the model and data; the calculation below illustrates the normalization itself.

Summarizing a Posterior Distribution

The full three-probability vector is the most complete single-variant representation of genotype uncertainty. Two useful summaries are the largest probability and the entropy of the distribution. The largest probability identifies the most probable genotype and indicates how concentrated the posterior is in that category. It does not describe how the remaining probability is distributed between the other two genotypes.

For probabilities \(p_0,p_1,p_2\), the Shannon entropy in bits is

where a term with \(p_g=0\) is defined to be zero. Entropy is zero when one genotype has probability one, and is largest when all three genotypes have probability \(1/3\). It summarizes spread across categories, but it is not itself a probability that the imputation is wrong. The full probability vector remains important because different distributions can have similar entropy while placing probability on different genotypes.

Posterior Probabilities and Hard Calls

A hard call reports just one genotype, commonly the category with the greatest posterior probability. Formally, the maximum-posterior call is the genotype \(g\) that maximizes \(P(G=g\mid D)\). This is a decision rule that compresses three probabilities into one label. For example, the vector \((0.46,0.48,0.06)\) has a maximum-posterior call of \(G=1\), but that call has posterior probability only 0.48. Treating it as certain would hide substantial uncertainty.

A probability threshold can be used to flag calls as uncertain. For instance, a workflow might call the most probable genotype only when its posterior exceeds a prespecified threshold. The threshold is a decision choice, not a property of the posterior itself. Different applications may have different costs for an incorrect call, so one threshold is not universally appropriate.

Checking and Interpreting the Output

Several basic checks help prevent misinterpretation. Confirm that the three values correspond to the expected allele coding and genotype order. Check that each value lies between zero and one and that their sum is one, allowing for small differences due to rounding. If displayed values sum to 0.99 or 1.01, rounding may explain the discrepancy; a large discrepancy suggests a parsing, formatting, or data problem that should be investigated before interpretation.

The probabilities describe uncertainty for a particular individual at a particular variant. A high maximum probability means the model concentrates its probability on one genotype; it does not, by itself, establish that the model will be well calibrated across many individuals or variants. Calibration is an empirical property: among cases assigned a stated probability level, the corresponding genotype should occur at roughly that frequency when assessed against suitable known genotypes. Incorrect allele alignment, unrepresentative reference information, or model misspecification can undermine this interpretation even when the three probabilities sum perfectly to one.

Common Mistakes and Limitations

• Calling the largest probability certainty. A maximum of 0.55 identifies the favored genotype, not a nearly certain one. Report the other probabilities or an appropriate uncertainty summary as well.
• Interpreting the posterior as a frequency in the population. \(P(G=1\mid D)\) is an individual-level conditional probability under the model. It is not the proportion of the population with \(G=1\).
• Ignoring allele order. The genotype labels depend on which allele is alternate and on the output's ordering convention. A correct probability vector attached to the wrong allele labels yields a wrong interpretation.
• Assuming a normalized vector guarantees a good imputation. A sum of one is a mathematical consistency check, not evidence that the input, reference representation, or model is appropriate.
• Confusing a summary with the full posterior. The maximum probability and entropy each compress the vector. Neither alone shows exactly how uncertainty is allocated across the three genotypes.

Posterior probabilities matter because they make uncertainty visible at the genotype level. Retaining the full vector allows analysts to distinguish a sharply concentrated result from a weakly favored genotype, even when both would produce the same hard-call label. Their appropriate use still depends on correct allele coding, model assumptions, and the purpose of the analysis.