Empirical Bayes

Unlike rigid thresholding methods, Empirical Bayes adaptively shrinks variant effect estimates based on the overall signal distribution. Variants with noisy, outsized effects are pulled toward zero, while robust signals are preserved. This reduces the Winner's Curse bias inherent in standard GWAS discovery, producing more realistic effect sizes for PRS construction.

The core innovation is bypassing subjective prior specification. The framework treats the prior distribution of SNP effects as an unknown parameter to be estimated directly from the GWAS summary statistics. Common assumptions include:

• A point-normal mixture: a fraction of null variants and a fraction of causal variants with effects drawn from a normal distribution.
• A continuous shrinkage prior, such as the horseshoe or Strawderman-Berger, allowing for a flexible spectrum of effect sizes.

LDpred2 is a widely adopted Empirical Bayes method for PRS. It models genetic architecture using a point-normal mixture prior and employs a Gibbs sampler to compute posterior mean effect sizes. The algorithm accounts for linkage disequilibrium (LD) by using a reference panel, effectively disentangling the causal signal from correlated neighboring variants without relying on arbitrary clumping thresholds.

PRS-CS applies continuous shrinkage priors with a global-local scale parameterization. This allows each variant to have its own shrinkage factor, placing a high density near zero for null effects while maintaining heavy tails to accommodate large causal effects. The method operates directly on GWAS summary statistics and an external LD reference panel, making it computationally efficient for biobank-scale data.

Standard C+T methods use hard p-value cutoffs and LD-based pruning, which discards information and creates discontinuous risk profiles. Empirical Bayes provides a probabilistic alternative:

• All variants are retained in the model.
• Effect sizes are weighted by their posterior probability of being causal.
• This often yields higher variance explained (R²) and better cross-ancestry portability than heuristic selection methods.

Empirical Bayes flexibly accommodates diverse genetic architectures. The inferred prior hyperparameters—such as the polygenicity (fraction of causal variants) and heritability—provide direct insight into the trait's underlying biology. This contrasts with methods that assume a uniform, infinitesimal architecture, and allows the model to adapt whether a trait is highly polygenic or driven by a few large-effect loci.

The foundational discovery analysis that scans millions of genetic variants across the genome to identify genotype-phenotype associations. Empirical Bayes methods directly consume GWAS summary statistics—including effect sizes, standard errors, and p-values—as input data to estimate the prior distribution of genetic effects.

• Provides the raw marginal effect size estimates that Empirical Bayes shrinks
• Quality of GWAS data directly determines the reliability of the estimated prior
• Requires large sample sizes to detect the small effect sizes typical of polygenic traits

The non-random correlation structure between nearby genetic variants on a chromosome. Empirical Bayes methods must explicitly model LD patterns using an external reference panel to distinguish true causal signals from tagging associations.

• Variants in high LD are co-inherited, creating correlated test statistics
• LD reference panels (e.g., 1000 Genomes) provide the correlation matrix for prior estimation
• Failure to account for LD leads to double-counting of genetic effects and inflated predictions

A Bayesian PRS method that uses a point-normal mixture prior on variant effect sizes. It models the genetic architecture by assuming a fraction of variants have zero effect while the remainder follow a normal distribution, with the prior parameters estimated directly from GWAS data.

• Uses a Gibbs sampler to infer posterior mean effect sizes
• Automatically adapts the shrinkage to match the trait's polygenicity
• Represents a direct implementation of the Empirical Bayes framework for PRS

A polygenic prediction method applying continuous shrinkage priors on SNP effect sizes. Unlike discrete mixture models, PRS-CS places a global-local scale parameter on each variant, allowing flexible, data-adaptive shrinkage informed by the marginal GWAS signal.

• Uses summary statistics and an external LD reference panel
• Employs a gamma-gamma hierarchical prior for automatic relevance determination
• Computationally efficient compared to MCMC-based alternatives

A statistical adjustment for the overestimation bias inherent in GWAS discovery. Variants reaching genome-wide significance tend to have inflated effect sizes due to random sampling error. Empirical Bayes shrinkage naturally addresses this by regressing extreme observed effects toward the estimated prior mean.

• Discovery bias is most severe for underpowered GWAS
• Empirical Bayes provides a principled alternative to ad-hoc correction factors
• Critical for ensuring PRS weights are not systematically over-optimistic

The comprehensive characterization of the number, frequency, and effect size distribution of causal variants underlying a complex trait. Empirical Bayes methods estimate this architecture directly from the data to determine the optimal degree of shrinkage.

• Highly polygenic traits (many small effects) require different priors than oligogenic traits
• Parameters include polygenicity (proportion of causal variants) and SNP heritability
• Misspecifying the genetic architecture leads to suboptimal prediction accuracy