Cross-Gene Assumptions in Empirical Bayes

Why Cross-Gene Assumptions Matter

Empirical Bayes moderation uses information from many genes to stabilize the variance estimate for each gene. The crucial premise is not that genes have identical expression or identical biological roles. It is that their gene-wise residual variances can be described, perhaps after accounting for a trend or known groups, by a common distribution with estimable parameters. The number of genes can make that distribution informative, but only to the extent that the gene-wise information is sufficiently comparable and not so dependent that it amounts to far fewer independent pieces of information.

Two assumptions are often conflated. A shared marginal prior says that gene variances in a specified set follow the same distribution. Cross-gene independence says that information from different genes is independent, or is treated as independent when estimating that distribution. The first supports borrowing; the second is a convenient assumption for many prior-fitting likelihoods and their uncertainty calculations. Dependence does not automatically make a shared marginal prior false, but it can make the apparent amount of information in the gene collection much larger than the effective amount.

The Gene-Wise Model and the Shared Prior

For gene \(g\), a linear model represents its expression measurements across samples as \(y_g=X\beta_g+\varepsilon_g\). Here \(X\) is the design matrix, \(\beta_g\) is the vector of gene-specific coefficients, and the within-gene residual model is \(\varepsilon_g\mid\sigma_g^2\sim N(0,\sigma_g^2 I)\). If the model has \(df_g\) residual degrees of freedom, its residual variance estimate is \(s_g^2\), with \(df_gs_g^2/\sigma_g^2\sim\chi^2_{df_g}\) under the model. These are gene-wise statements; they do not by themselves specify the relationship between residuals or variances for different genes.

The shared-prior premise is about a distribution of variances, not one common variance value. Under this prior and the gene-wise sampling model, \(s_g^2/s_0^2\) has an \(F\) distribution with \(df_g\) and \(df_0\) degrees of freedom. This marginal distribution is the basis for estimating the prior from observed gene-wise variances. Conditional on the fitted prior, the moderated variance estimate is a weighted average:

The weight \(df_0\) represents the strength of the prior information, while \(df_g\) represents the gene's own residual information. Thus the prior must be plausible for the gene set being moderated. If prior parameters are estimated from many genes, their precision also depends on how much independent information those genes provide.

What Dependence Changes

Genes measured in the same samples need not be independent. Shared sample quality, preprocessing, normalization, and biological co-expression can induce relationships across gene-level measurements. Such relationships may also make gene-wise variance estimates dependent. However, correlation of expression levels across genes does not, by itself, prove that their residual variance estimates have a particular correlation: the dependence relevant to prior fitting concerns the gene-level quantities used to estimate the variance distribution.

If the marginal variance distribution is correctly specified, dependence need not make each gene's marginal prior false. Its clearest consequence is that treating all genes as independent can overstate the precision of estimated prior parameters. With positive dependence, groups of similar genes may contribute less new information than the same number of independent genes. Under weak dependence, a pooled prior estimate may still be useful, but uncertainty assessments based on independence can be too optimistic. Strong or pervasive dependence can make the prior estimate unstable even when the gene count is large.

A short calculation illustrates the loss of information. Let \(Z_g\) be a gene-level summary used to learn a prior parameter, with variance \(\tau^2\) and common pairwise correlation \(\rho\) across \(G\) genes. Then

If this variance is expressed as \(\tau^2/G_{\mathrm{eff}}\), the corresponding effective number of independent summaries is \(G_{\mathrm{eff}}=G/[1+(G-1)\rho]\). This is an illustration under equal variances and equicorrelation, not a universal adjustment formula for empirical Bayes fitting. Real dependence can be uneven, clustered, or more complicated than one common correlation.

When One Shared Prior Is Not Enough

A common prior is most defensible when the genes being pooled have comparable marginal variance distributions after any relevant modeling of systematic trends. If variability changes with average expression intensity, a single distribution across all genes can be a poor description. A similar problem arises if identifiable gene groups have substantially different variance distributions. In such cases, moderation toward one pooled prior can pull low-variance genes upward and high-variance genes downward toward an inappropriate center.

This is distinct from cross-gene dependence. Dependence concerns how much independent information the observed genes supply; heterogeneity concerns whether the genes share the same marginal prior in the first place. A variance trend can address systematic changes in the prior with a covariate such as average intensity, while a robust prior fit can reduce the influence of extreme genes. Neither automatically removes dependence among the gene-level summaries.

Diagnostics, Alternatives, and Limits

No single diagnostic proves cross-gene independence or validates a prior. Useful checks include inspecting gene-wise residual variance summaries against average expression and other design-relevant covariates; comparing variance distributions across pre-specified gene groups; and examining whether residual patterns or variance summaries cluster by pathway, sample quality, or technical features. Strong structure in these checks suggests that an independent, identically distributed prior for all genes may be too simple. A smooth marginal histogram alone cannot rule out dependence: correlated observations can have the same marginal distribution as independent ones.

The main alternatives answer different problems. Unmoderated gene-wise variances avoid borrowing across genes, but can be unstable when \(df_g\) is small. A trend prior allows the prior variance distribution to vary with a measured covariate. Group-specific priors allow distinct distributions for defensible groups but can be imprecise if groups are small. Robust fitting limits the influence of unusual genes when estimating prior parameters, but does not make all genes independent. These approaches can address some forms of heterogeneity or outlier influence; none should be described as a general correction for arbitrary cross-gene dependence.

Dependence among genes also matters when interpreting a collection of gene-level results: discoveries may occur in related groups rather than representing independent biological signals. The variance prior is not a model of the dependence structure among gene effects or test statistics. Moderation stabilizes gene-wise variance estimates under its assumptions; it does not establish that genes are independent, that a pathway is enriched, or that a gene-level association is causal.

Common Mistake

Wrong reasoning: “There are thousands of genes, so the prior parameters must be estimated very precisely, and the genes can be treated as independent.”

Correction: The gene count alone does not establish either claim. Precision depends on the dependence among the quantities used for prior fitting, and the shared-prior assumption depends on whether their marginal distributions are sufficiently comparable. Check for trends and groups, consider the effect of dependence on uncertainty, and describe the fitted prior as an empirical summary rather than a known biological constant.