Posterior Mean Scores and Uncertainty

What a Score Mean Does—and Does Not—Tell You

A polygenic score is often reported as one number for each individual. That number is useful, but it can hide uncertainty in the estimated variant effects used to construct it. Two individuals can have the same calculated score even when the posterior uncertainty in their scores differs, for example because their genotypes place different weights on uncertain effect estimates.

This tutorial separates two quantities. The posterior mean score is the score calculated using posterior mean effect weights. The posterior uncertainty in the score describes how the score would vary across plausible effect-weight vectors under the posterior distribution, holding the individual’s genotype fixed. This is uncertainty about the genetic contribution represented by the score; it is not, by itself, uncertainty about the individual’s phenotype.

Definition: Let \(g_i=(g_{i1},\ldots,g_{ip})^\mathsf{T}\) be the fixed vector of genotype dosages for individual \(i\) at \(p\) variants, using the same allele coding and variant order as the effect estimates. Let \(\beta=(\beta_1,\ldots,\beta_p)^\mathsf{T}\) be the vector of variant effects, with posterior mean \(m=E(\beta\mid D)\) and posterior covariance matrix \(V=\operatorname{Cov}(\beta\mid D)\), conditional on discovery data \(D\) and the specified model. The score is \(S_i=g_i^\mathsf{T}\beta\). Its posterior mean is \(E(S_i\mid D)=g_i^\mathsf{T}m\), and its posterior variance is \(\operatorname{Var}(S_i\mid D)=g_i^\mathsf{T}Vg_i\).

The effect scale determines the score units. If \(\beta_j\) is measured in standard deviations of a quantitative trait per allele, then \(S_i\) is in those trait-standard-deviation units. If effects are on the log-odds scale, the score is a log-odds contribution relative to the chosen genotype reference and model intercept. The dosage vector must use compatible allele coding; reversing an allele without changing the corresponding effect sign changes the score incorrectly.

Why the Full Covariance Matrix Matters

A tempting shortcut is to add the posterior variances of the individual effects, perhaps multiplying each by the squared genotype dosage. That shortcut is valid only when posterior covariances between distinct effects are zero. In general, the score is a weighted sum of effects, and the uncertainty of a sum includes covariance terms.

\operatorname{Var}(S_i\mid D) = \sum_{j=1}^{p} g_{ij}^{2}V_{jj} + 2\sum_{j<k}g_{ij}g_{ik}V_{jk}.

Here \(V_{jj}\) is the posterior variance of effect \(j\), and \(V_{jk}\) is the posterior covariance between effects \(j\) and \(k\). The covariance may be positive or negative. Consequently, dependence can increase or reduce score uncertainty. Linkage disequilibrium can contribute to posterior dependence because correlated genotypes provide overlapping information about effects; the relevant quantity in this calculation is the covariance in the fitted posterior, not an LD correlation inserted as a substitute.

The result follows directly from the definition of covariance. Since \(g_i\) is fixed conditional on the individual’s genotype, \(S_i-E(S_i\mid D)=g_i^\mathsf{T}(\beta-m)\). Therefore,

\begin{aligned} \operatorname{Var}(S_i\mid D) &=E\left[\{g_i^\mathsf{T}(\beta-m)\}^2\mid D\right]\\ &=E\left[g_i^\mathsf{T}(\beta-m)(\beta-m)^\mathsf{T}g_i\mid D\right]\\ &=g_i^\mathsf{T}E\left[(\beta-m)(\beta-m)^\mathsf{T}\mid D\right]g_i\\ &=g_i^\mathsf{T}Vg_i. \end{aligned}

This identity does not require normally distributed effects. If the posterior effect vector is multivariate normal, the score is normally distributed as a linear combination of its components. In that case, the posterior mean and variance fully specify the score’s posterior distribution. With a non-normal posterior, they still give its mean and variance, but may not describe its shape or tail probabilities well.

Worked Example: A Two-Variant Score With Correlated Effects

Suppose an individual has dosages \(g_i=(2,1)^\mathsf{T}\). For this hypothetical example, effects are measured in log-odds units per allele. Suppose the posterior mean and covariance matrix are

m= \begin{pmatrix} 0.08\\ -0.03 \end{pmatrix}, \qquad V= \begin{pmatrix} 0.0009&0.0003\\ 0.0003&0.0016 \end{pmatrix}.

The posterior mean score is

g_i^\mathsf{T}m=(2)(0.08)+(1)(-0.03)=0.16-0.03=0.13.

Thus the estimated genetic contribution is \(0.1300\) log-odds units. The posterior variance includes both individual variances and the covariance term:

\begin{aligned} \operatorname{Var}(S_i\mid D) &=(2^2)(0.0009)+(1^2)(0.0016)+2(2)(1)(0.0003)\\ &=0.0036+0.0016+0.0012\\ &=0.0064. \end{aligned}

The posterior standard deviation is \(\sqrt{0.0064}=0.0800\) log-odds units. If the score posterior is normal, an approximate 95% equal-tail credible interval is \(0.13\pm1.96(0.08)\), or \((-0.0268,\ 0.2868)\) log-odds units. Exponentiating the endpoints gives an interval of approximately \((0.974,\ 1.332)\) for the odds multiplier associated with this genetic contribution. This interval concerns the contribution represented by the score, not an individual’s full odds or outcome.

The covariance matrix is admissible: its diagonal entries are positive and its determinant is \(0.0009(0.0016)-(0.0003)^2=0.00000135>0\). This check is useful because a covariance matrix must be positive semidefinite, and an invalid matrix could produce a negative variance for some genotype vector.

Reading Score Uncertainty in Context

A posterior standard deviation summarizes dispersion around the posterior mean, in the same units as the score. A credible interval gives a range containing a specified posterior probability under the model and data, when calculated from the posterior distribution. For example, a 95% posterior credible interval has 95% posterior probability under that specified posterior distribution. It is not a statement that there is a 95% probability that a fixed parameter lies in an interval after the data have been observed.

For a normal posterior score, an interval can be calculated from its mean and variance. Otherwise, posterior draws are generally preferable for intervals and tail probabilities. In either case, the uncertainty reported is conditional on the chosen model, prior, discovery data, and effect scale. It does not automatically account for every source of uncertainty in applying a score, such as imperfect genotype measurement, uncertainty in phenotype definition, or a mismatch between the discovery and target populations.

Worked Example: Why Ignoring Covariance Can Misstate Uncertainty

Consider a hypothetical individual with \(g_i=(1,1)^\mathsf{T}\). Suppose the posterior mean effects, in quantitative-trait standard-deviation units per allele, are \(0.20\) and \(0.10\), and the posterior covariance matrix is

V= \begin{pmatrix} 0.04&0.03\\ 0.03&0.04 \end{pmatrix}.

The posterior mean score is \(1(0.20)+1(0.10)=0.30\) trait-standard-deviation units. Using the full covariance matrix, the variance is

\operatorname{Var}(S_i\mid D) =(1^2)(0.04)+(1^2)(0.04)+2(1)(1)(0.03)=0.14,

so the posterior standard deviation is \(\sqrt{0.14}=0.3742\) trait-standard-deviation units. If one incorrectly treated the effects as independent, one would calculate variance \(0.04+0.04=0.08\), giving standard deviation \(0.2828\). That understates uncertainty because the covariance is positive. The matrix is valid: its determinant is \(0.04(0.04)-0.03^2=0.0007>0\), and its diagonal entries are positive.

Had the covariance been negative, the covariance term would instead reduce the variance for this genotype vector. It is therefore not generally conservative to omit covariance; the direction of the error depends on the covariance structure and the individual’s dosages.

Posterior Draws, Diagnostics, and Alternatives

Many shrinkage methods provide posterior draws or approximations to posterior effect distributions rather than a simple multivariate normal posterior. If \(\beta^{(1)},\ldots,\beta^{(M)}\) are draws from the posterior, calculate one score for each draw:

S_i^{(r)}=g_i^\mathsf{T}\beta^{(r)},\qquad r=1,\ldots,M.

The average of these scores estimates the posterior mean, and the average squared deviation from that average estimates the posterior variance. Quantiles of the score draws estimate posterior credible-interval endpoints; the fraction of draws exceeding a specified threshold estimates the posterior probability of that event. When the draws are autocorrelated, effective sample size and mixing affect the precision of these Monte Carlo estimates.

Worked Example: Score Uncertainty From Posterior Draws

Suppose one hypothetical individual has \(g_i=(1,2)^\mathsf{T}\), with dosages fixed. Five equally weighted posterior draws of the two effects, in trait-standard-deviation units per allele, are \((0.02,0)\), \((0,0.04)\), \((0.10,0)\), \((0.04,0.04)\), and \((0,0.14)\). The corresponding scores are

0.02,\quad 0.08,\quad 0.10,\quad 0.12,\quad 0.28.

Their mean is \((0.02+0.08+0.10+0.12+0.28)/5=0.12\) trait-standard-deviation units. The empirical posterior variance, using the average squared deviation, is

\frac{(0.02-0.12)^2+(0.08-0.12)^2+(0.10-0.12)^2+(0.12-0.12)^2+(0.28-0.12)^2}{5} =\frac{0.0376}{5}=0.00752.

The resulting posterior standard deviation is \(\sqrt{0.00752}=0.08672\) trait-standard-deviation units. One of the five equally weighted draws exceeds \(0.15\), so the estimated posterior probability that the score exceeds \(0.15\) is \(1/5=0.20\). With only five draws this probability estimate is very coarse; a practical analysis would use many more draws and assess their sampling quality. These draws are intentionally uneven in their score values, illustrating why a normal approximation should not be assumed merely because a mean and variance have been calculated.

The posterior mean score is a useful point summary, but it is not the only possible summary. A maximum a posteriori effect vector, if available, produces a score at the posterior mode rather than the posterior mean. It can differ substantially in skewed or multimodal posteriors. Posterior-draw summaries retain information about shape and tail probabilities; a normal approximation is simpler but can be misleading when the score distribution is asymmetric or heavy-tailed. The best summary depends on whether the aim is a point score, a measure of uncertainty, or a probability about a score threshold.

A further distinction is essential when the scientific target is an outcome. If a model writes an outcome as a genetic contribution plus residual variation, the posterior variance \(g_i^\mathsf{T}Vg_i\) quantifies uncertainty in the genetic contribution from uncertain weights. Predictive uncertainty for the outcome also includes residual variation and may include uncertainty in other model components. It is therefore incorrect to label the score’s posterior standard deviation as the standard deviation of the individual’s future phenotype.

Assumptions and Failure Modes

The calculation \(g_i^\mathsf{T}Vg_i\) is exact for a fixed genotype vector and a posterior distribution with finite second moments. Its interpretation depends on several conditions:

Compatible variants and coding. The dosages and effects refer to the same variants, alleles, order, and effect scale. Allele flips or genome-build mismatches can corrupt both the mean and uncertainty calculations.
Appropriate posterior covariance. The matrix \(V\) represents joint posterior uncertainty under the fitted model. Using only marginal variances discards dependence; using covariance from a different model or variant set may also be inappropriate.
Fixed, correctly measured genotype. The formula conditions on \(g_i\). If genotype is uncertain or imputed with substantial uncertainty, this calculation alone does not incorporate that source of uncertainty.
Model-conditional interpretation. A narrow posterior can reflect strong assumptions or a restrictive prior, not necessarily a universally precise estimate. The result is conditional on the prior, likelihood, data, and modeling choices.

Useful diagnostics include checking that \(V\) is symmetric and positive semidefinite, verifying that the computed variance is nonnegative up to numerical tolerance, and comparing the matrix calculation with the variance of scores computed from posterior draws. Large disagreement can signal an indexing or allele-alignment error, a covariance matrix that does not correspond to the reported effects, or a mistake in the draw calculation. For draw-based summaries, inspect convergence and effective sample size rather than treating repeated, highly correlated draws as independent information.

Common mistake: “The effects are uncertain, so I can add their posterior variances and ignore their covariances.” This is correct only when the relevant posterior covariances are zero. For a general score, use \(g_i^\mathsf{T}Vg_i\), which includes every covariance contribution. A second common mistake is to interpret a score interval as an interval for the individual’s outcome; the score interval covers uncertainty in the genetic contribution under the specified posterior, not all sources of phenotype variation.

Key takeaway: The posterior mean effect vector gives the posterior mean score, while the full posterior covariance matrix determines uncertainty in that score. Keep genotype fixed for this calculation, retain dependence among effects, and distinguish uncertainty in the genetic contribution from uncertainty in the phenotype.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

An individual has dosage vector \(g=(1,2)^\mathsf{T}\) and posterior mean effects \(m=(0.10,0.03)^\mathsf{T}\). What is the posterior mean score?
For the first question, suppose \(V_{11}=0.01\), \(V_{22}=0.0025\), and \(V_{12}=V_{21}=0.002\). Calculate the posterior score variance and standard deviation.
Why is a posterior score variance not generally the same as predictive variance for an individual’s phenotype?
If posterior draws show a strongly right-skewed score distribution, what is a better approach than reporting a normal interval based only on the mean and variance?

Solutions. (1) \(g^\mathsf{T}m=1(0.10)+2(0.03)=0.16\), in the effect scale’s score units. (2) The variance is \(1^2(0.01)+2^2(0.0025)+2(1)(2)(0.002)=0.01+0.01+0.008=0.028\). The standard deviation is \(\sqrt{0.028}=0.1673\) score units. (3) Score variance describes uncertainty in the genetic contribution due to uncertain effects, conditional on genotype and model. Phenotype prediction also includes residual variation and may include uncertainty in other model components. (4) Calculate score quantiles directly from posterior draws and report a draw-based credible interval, while checking that the draws adequately represent the posterior distribution.