Coordinate Updates in SuSiE

Why Update One Component at a Time?

A sum of single effects model represents a genetic association as the sum of several components, each of which can select at most one variant. The components make a complicated multi-signal model computationally manageable: updating one component reduces to a single-effect calculation, even when the other components are present. The central detail is that the update must account for the effects currently attributed to those other components.

The introductory material on sum of single effects models establishes this component representation. Here the focus is the coordinate update: how to construct its residualized response, calculate the evidence for each candidate variant, and turn that evidence into component probabilities. These are iterative updates, not a claim that the components have been identified without uncertainty.

Model and Assumptions

Let \(y\) be a centered \(n\)-vector of quantitative outcomes and let \(X\) be an \(n\times p\) matrix of centered variant predictors. Write \(x_j\) for column \(j\) of \(X\). With \(L\) effects, the Gaussian model is

Here \(b_l\) is a \(p\)-vector for component \(l\). Its prior assigns probability \(\pi_{l0}\) to a null component and probability \(\pi_{lj}\) to selecting variant \(j\), with \(\pi_{l0}+\sum_{j=1}^{p}\pi_{lj}=1\). If variant \(j\) is selected, its component effect has prior \(b_{lj}\sim N(0,\tau_l)\), and all other entries of \(b_l\) are zero. The null component has every entry zero. The variance \(\sigma^2\), effect prior variance \(\tau_l\), and prior probabilities are treated as fixed during the coordinate update described here.

The Gaussian likelihood assumes the outcome is adequately modeled by the stated linear predictor and independent, equal-variance Gaussian errors. In an application, measured covariates can be included and accounted for before this calculation, provided that the resulting likelihood and predictors are handled consistently. The update also assumes the specified single-effect prior: a component cannot select two variants at once. It does not assume that the full model has only one causal variant; that is the purpose of having multiple components.

Residualizing the Other Components

For each component \(k\), let \(a_{kj}\) be its current posterior probability of selecting variant \(j\), and let \(\mu_{kj}\) be the posterior mean effect conditional on that selection. Its expected effect vector has entries \(a_{kj}\mu_{kj}\). To update component \(l\), subtract the expected fitted effects of all other components:

The vector \(r_l\) is the response remaining for component \(l\) after accounting for the current expected contributions of the other components. For a candidate variant \(j\), define \(S_j=x_j^\mathsf{T}x_j\) and \(z_{lj}=x_j^\mathsf{T}r_l\). The likelihood for a proposed effect \(b\) on that variant is proportional to \(\exp\{(2bz_{lj}-b^2S_j)/(2\sigma^2)\}\). This expression supplies the data-dependent part of the update.

The use of expected effects matters. The update does not subtract a definite, known causal effect for each other component; it subtracts the current posterior mean fitted value. Uncertainty and dependence among components are not fully retained by this mean-field coordinate step.

Bayes Factors and Component Probabilities

Compare the model in which component \(l\) selects variant \(j\), integrating over its \(N(0,\tau_l)\) effect prior, with the null-component model. Completing the square in the Gaussian integral gives the Bayes factor

Conditional on selecting \(j\), the posterior distribution of its effect is normal, with variance \(V_{lj}\) and mean \(\mu_{lj}\):

The posterior probability for each component choice follows by multiplying its prior probability by its Bayes factor and normalizing. For the null choice, the Bayes factor is 1:

Thus \(a_{l0}+\sum_j a_{lj}=1\). The posterior mean effect for variant \(j\) in this component is \(a_{lj}\mu_{lj}\), not just the conditional mean \(\mu_{lj}\). The probability that the component selects the variant accounts for the possibility that it selects another variant or is null.

Worked Updates

Why the Coordinate Formula Holds

The update can be justified directly from the Gaussian likelihood. For component \(l\), write the fitted contribution of all other components as \(X\sum_{k\ne l}\widehat b_k\). Subtracting it from \(y\) leaves \(r_l\). If component \(l\) selects \(j\) with effect \(b\), its residual likelihood is proportional to

Expanding the quadratic and dropping terms that do not involve \(b\) gives \(\exp\{(2bz_{lj}-b^2S_j)/(2\sigma^2)\}\). Multiplication by the prior density proportional to \(\exp\{-b^2/(2\tau_l)\}\) yields a quadratic in \(b\). Completing the square gives posterior precision \(S_j/\sigma^2+1/\tau_l\), hence posterior variance \(V_{lj}\) and mean \(\mu_{lj}\) as stated above. Integrating this normal likelihood times the prior over \(b\), then dividing by the null likelihood, gives \(BF_{lj0}\). Prior weighting and normalization then give the component probabilities.

In the variational interpretation, this is a coordinate optimization: the distribution for one component is updated while the other component distributions are held fixed. Taking their expected fitted contributions produces the residual above. Repeating this operation across components produces an iterative procedure; one update alone is not the final model fit.

Diagnostics, Alternatives, and Limitations

Useful checks include verifying that each component’s probabilities are nonnegative and sum to 1, that the expected effect vector uses \(a_{lj}\mu_{lj}\), and that the residual score reflects the current expected contributions of every other component. With fixed hyperparameters, monitor whether the variational objective improves and whether component probabilities and expected effects stabilize across iterations. Large changes between iterations, persistent cycling, or components repeatedly exchanging similar signals warrant investigation. Different starting values can help reveal sensitivity to initialization.

Highly correlated predictors can make their individual evidence difficult to distinguish: several variants may receive similar component probabilities, and small changes in residuals can shift probability among them. A large component probability is conditional on the model, priors, and current coordinate approximation; it is not proof that a variant is causal. Check that the predictors, outcome scale, variance assumptions, and prior effect scale are sensible. Numerical implementation should also evaluate very large or small Bayes factors on a log scale to avoid overflow or underflow.

Coordinate updates are computationally attractive compared with enumerating all joint assignments of variants to components, whose number can grow rapidly with \(p\) and \(L\). The trade-off is that the factorized approximation does not represent all posterior dependence among components. Joint configuration calculations can retain that dependence in small problems but may be infeasible at genomic scale. Coordinate updating is therefore an efficient approximation, not an exact substitute for the full joint posterior.

A common mistake is to treat the residualized response as \(y\) minus the other components’ most likely selected variants, or to update from the unadjusted \(y\) each time. The correction is to use the current expected effect vector of each other component and compute \(r_l=y-\sum_{k\ne l}X\widehat b_k\). Another mistake is to interpret \(\mu_{lj}\) as the component’s overall expected effect. It is conditional on selecting \(j\); the overall expected entry is \(a_{lj}\mu_{lj}\).