Propagating Uncertainty in Mixture-Based Frequency Estimates

Why Mixture Uncertainty Has More Than One Source

A mixture frequency depends on two kinds of information: the allele frequency in each source population and the fraction of the mixture contributed by each source. Even if the mixture formula is straightforward, its estimate can be uncertain because either input is uncertain. When the estimates are statistically related, their covariance also affects the uncertainty in the result.

This tutorial develops a first-order variance approximation for that setting. The goal is not to re-estimate source frequencies or mixture proportions, but to propagate their uncertainty through the mixture calculation. All numerical data below are hypothetical.

The estimated value \(\hat{q}_{mix}\) alone does not describe its precision. For example, the same estimated mixture frequency may be comparatively precise when source frequencies and proportions are known accurately, and much less precise when either input is noisy. Covariance matters as well: estimates that tend to increase together can contribute differently from estimates that tend to move in opposite directions.

Delta-Method Variance

Collect the estimated source frequencies and mixture proportions into the vector \(\hat{Z}=(\hat{p}_1,\ldots,\hat{p}_K,\hat{w}_1,\ldots,\hat{w}_K)\). Let \(\Sigma\) be the covariance matrix of this joint vector, or an estimate of that covariance matrix. The mixture calculation is a smooth function of these inputs. The delta method approximates its variance by evaluating the gradient of that function at the estimated inputs.

The quadratic form is compact, but its components show what uncertainty is being propagated. Write \(\widehat{\operatorname{Cov}}\) for an estimated covariance and \(\widehat{\operatorname{Var}}\) for an estimated variance. Expanding the quadratic form gives

The first line contains the marginal variances of source-frequency estimates and mixture-proportion estimates. The next two lines account for dependence among the source-frequency estimates and among the mixture-proportion estimates. The final line includes every cross-covariance between a source-frequency estimate and a mixture-proportion estimate. A positive covariance adds to the variance when its gradient coefficients have the same sign; a negative covariance can reduce it. The covariance matrix as a whole must be valid, so its resulting quadratic form cannot be negative.

Why the Approximation Has This Form

The key step is a first-order expansion of the mixture function around the estimated inputs. For small changes \(dp_i\) and \(dw_i\), its change is approximately

The coefficient of a source-frequency change is its mixture proportion; the coefficient of a proportion change is the corresponding source frequency. The variance of this linear combination is the gradient multiplied by the joint covariance matrix and then by the gradient transpose. Expanding that variance produces the marginal-variance and covariance terms above.

This is a first-order approximation, not an exact variance identity for arbitrary random inputs. It is most useful when the input estimates are reasonably concentrated around their true values and the mixture function is well approximated locally by its linear expansion. Strong uncertainty, estimates near boundaries, or substantial nonlinear effects can make the approximation less reliable. The calculation also requires a joint covariance matrix that represents the dependence in the actual estimation procedure.

Covariance Terms in a Two-Source Mixture

The two-source form makes the role of covariance especially clear. With \(q=wp_1+(1-w)p_2\), use the input vector \((p_1,p_2,w)\). The gradient is \((w,1-w,p_1-p_2)\), so the full first-order variance includes the covariance of the two frequency estimates and each frequency estimate with \(w\). In particular, uncertainty in \(w\) is multiplied by the squared frequency contrast: if the two source frequencies are similar, shifting the mixture proportion has less first-order effect on the resulting frequency.

Mixture Proportions Must Sum to One

Mixture proportions are constrained: they are nonnegative and sum to one. Consequently, their covariance matrix is often singular when all \(K\) proportions are included, because a change in one or more proportions must be balanced by changes in others. Singularity is not, by itself, a problem for the quadratic-form calculation. The covariance matrix must represent the constraint, and the full gradient and covariance matrix must use consistent coordinates.

An alternative is to express one proportion as the remainder. With three sources, set \(w_3=1-w_1-w_2\), so the mixture frequency becomes \(q=p_3+w_1(p_1-p_3)+w_2(p_2-p_3)\). The derivatives with respect to the two free proportions are \(p_1-p_3\) and \(p_2-p_3\). This representation automatically respects the sum-to-one constraint and can simplify variance calculations.

Assumptions, Interpretation, and a Common Pitfall

The delta-method result is an approximation based on local linearization. Its practical use requires estimates of the inputs and a joint covariance estimate for those same inputs. If source-frequency estimates and mixture-proportion estimates are dependent, that dependence belongs in the covariance matrix; assuming independence merely for convenience can misstate uncertainty. The standard error describes sampling variability under the specified estimation and covariance model. It does not account for sources of uncertainty that were not represented in that model.

A common mistake is to calculate only \(\sum_i\hat{w}_i^2\widehat{\operatorname{Var}}(\hat{p}_i)\), treating all mixture proportions as fixed. That reasoning is valid only when the proportions are known constants, or when their uncertainty is deliberately being excluded from the target calculation. If the proportions are estimated, their variance and relevant covariance terms must also be considered. A related mistake is to include the marginal variances but drop covariance terms without checking whether independence is justified. The correction is to define the full input vector, state the dependence assumptions, and use its joint covariance matrix.