Genetic models#

Bayesian alphabet framework#

**Summary of Effect Size Distributions for Polygenic Modeling**#
Models	Effect Size Distribution	Formula
BayesA	\(\beta_i \sim t(\nu, \sigma^2_{\alpha})\)	t Distribution
BayesB, BayesDπ	\(\beta_i \sim \pi t(0, \nu, \sigma^2_{\alpha}) + (1 - \pi) \delta_0\)	Point-t Distribution
BayesC	\(\beta_i \sim \pi t(0, \nu, \sigma^2_{\alpha}) + (1 - \pi) t(0, \nu, 0.01\sigma^2_{\alpha})\)	t Mixture
BayesCπ , BVSR	\(\beta_i \sim \pi N(0, \sigma^2_{\alpha}) + (1 - \pi) \delta_0\)	Point-Normal Distribution
Bayesian Lasso	\(\beta_i \sim DE(0, \theta)\)	Double Exponential Distribution
BayesR , SBayesR , SBayesRC	\[\beta_i \sim \pi_1 N(0, \sigma^2_{\alpha}) + \pi_2 N(0, 0.1\sigma^2_{\alpha}) + \pi_3 N(0, 0.01\sigma^2_{\alpha}) + (1 - \sum^{3}_{c = 1} \pi_c) \delta_0\]	Point-Normal Mixture
LMM, BLUP, Ridge Regression Ridge theory	\(\beta_i \sim N(0, \sigma^2_{\alpha})\)	Normal Distribution
NEG	\(\beta_i \sim NEG(0, \kappa, \theta)\)	Normal-Exponential-Gamma
BSLMM	\(\beta_i \sim \pi N(0, \sigma^2_{\alpha} + \sigma^2_{\beta}) + (1 - \pi) N(0, \sigma^2_{\beta})\)	Normal Mixture
SBayesS	\(\beta_i \sim \pi N(0,[1p_j(1-p_j)]^S \sigma^2_{\beta}) + \phi(1 - \pi)\)	Normal Mixture

**Bayesian Polygenic Models and Their Inference Methods**#
Models	Inference
BayesA	Gibbs sampling with Student-t prior on SNP effects. Variance components sampled via inverse-gamma.
BayesB	Gibbs sampling with a mixture prior (point mass at zero + Student-t). Requires latent variable updates.
BayesC	Gibbs sampling with a mixture prior (point mass at zero + Normal). SNP effects and variance components are sampled iteratively.
BayesCπ	Similar to BayesC, but includes estimation of π (the proportion of nonzero SNPs) within the Gibbs sampling framework.
BayesR	Gibbs sampling with a multi-component normal mixture prior (e.g., strong, moderate, weak effects + zero).
Bayesian Lasso	Uses Gibbs sampling with an auxiliary variable formulation for the Laplace prior.
NEG (Normal-Exponential-Gamma)	Hierarchical shrinkage prior estimated via Gibbs sampling.
BSLMM (Bayesian Sparse Linear Mixed Model)	Uses a mix of Gibbs sampling and variational inference, depending on implementation.
LMM (Linear Mixed Model)	Typically solved using REML (Restricted Maximum Likelihood) or Expectation-Maximization (EM), not Gibbs.

**LD score regression and its extension**#
Method	Formula	Key contribution
LDSC	\(E[\chi^2 \| \ell_j] = N h^2 \frac{\ell_j}{M} + N \alpha + 1\)	\(\chi^{2}_{j}=N\hat{\beta}^2_j,\quad\ell_j := \sum_{k=1}^{M} r_{jk}^2\)
HDL	\(\text{Cov}[\mathbf{z}] = \frac{N h^2}{M} \mathbf{L} + \mathbf{R}\)	\(\mathbf{L} := \mathbf{R}^\top \mathbf{R}\)
HDL-L	\(\text{Cov}[\mathbf{z}_1, \mathbf{z}_2] = \frac{\sqrt{N_1 N_2} h_{12}}{M} \mathbf{L}\)	\(\mathcal{R}(h_{12})\equiv \frac{\mathcal{L} \left( h_{12} \mid \mathbf{z}, \hat{h}_1^2, \hat{h}_2^2 \right)}{\mathcal{L} \left( \hat{h}_{12} \mid \mathbf{z}, \hat{h}_1^2, \hat{h}_2^2 \right)}\)
S-LDSC	\(E\left[\chi^2_j\right] = N \sum_{C} \tau_C \ell(j, C) + N \alpha + 1\)	\(\ell(j,C) = \sum_{k \in C} r_{jk}^2\)
MESC	\(E[\alpha_k^2 \mid \beta_{1k}, \dots, \beta_{Gk}] = E[\alpha^2] \sum_{i}^{G} \beta_{ik}^2 + E[\gamma^2]\)	\(\ell(j,C) = \sum_{k \in C} r_{jk}^2\)
FMR	\(\frac{d}{d t} E \left( e^{it z + \frac{1}{2} t^2 \hat{\sigma}^2} \right)\approx it N \sigma^2 \sum_K w_K \ell_K (t)\)	\(\ell_K (t) = \sum_j r_{j}^2 \phi_K (r_j t)\)

**LD score regression and its extension**#
Method	Formula	Key contribution
LDAK	\(\mathbb{E}[h_j^2] \sim [f_j(1 - f_j)]^{1 + \alpha} \times w_j \times r_j\)	\(\beta_j \sim \mathcal{N}\left(0, r_j w_j \frac{\sigma_g^2}{W} \right) \quad W = \sum_j r_j w_j \left[ 2 f_j (1 - f_j) \right]^{1 + \alpha}\)