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Guan and Stephens (2011) examine the potential to apply Bayesian Variable Selection Regression to GWAS, which requires MCMC. They emphasize that, unlike penalized regression approaches, BVSR naturally produces easily-interpretable measures of confidence — specifically, posterior probabilities — that individual covariates have nonzero regression coefficients.
They define a binary indicator that indicate which element of \(\beta\) are nonzero, \(\gamma = (\gamma_1, \cdots, \gamma_p) \in \{0,1\}^p\). \[ y | \gamma, \mu, \beta, X, \tau \sim N_n( \mu + X_{\gamma}\beta_{\gamma}, \tau^{-1}I_n) \] \(X_\gamma\) denotes the design matrix \(X\) restricted to those columns \(j\) for which \(\gamma_j = 1\), \(\beta_\gamma\) denotes a corresponding vector of regression coefficients.
where \(|\gamma| = \sum_j \gamma_j\). Hyperparameters are \(\pi, \sigma_a, \lambda, \kappa, \sigma_\mu\). Rather than setting \(\pi\) and \(\sigma_a\) to prespecified values, they place priors on them, hence allowing their values to be informed by the data. The remaining hyperparameters are less critical, and they consider the posterior distributions for which \(\sigma_\mu \rightarrow \infty\) and \(\lambda, \kappa \rightarrow 0\), which has the attractive property that the resulting relative marginal likelihoods for \(\gamma\) are invariant to shifting or scaling of \(y\).
A uniform prior on \(\pi\) seems inappropriate, since this would inevitably place most of the prior mass on larger numbers of covariates (uniform on \(10^{-5}\) to \(10^{-3}\) puts about 90% probability on \(>10^{-4}\)). Instead, the prior on \(\log \pi\) is \[ \log \pi \sim U(\log(1/p), \log(M/p)) \]
Scaling the covariates to each have unit variance corresponds to a prior assumption that SNPs with less variable genotypes (lower MAF) have larger effect sizes.
Let \(V(\beta, \tau)\) denote the empirical variance of \(X\beta\) relative to the residual variance \(\tau^{-1}\). \[ V(\beta, \tau) = \frac{1}{n}\sum_{i=1}^{n} [(X\beta)_i]^2 \tau \] which assumes that the covariates have been centered (\(X\beta\) has mean 0). The PVE is \[ PVE(\beta, \tau) = V(\beta, \tau)/(1+ V(\beta, \tau)) \] The aim is to choose a prior on \(\beta\) given \(\tau\) so that the induced prior on \(PVE(\beta, \tau)\) is approximately uniform.
\[ \nu(\gamma, \sigma_a) = E\left[ V(\beta, \tau) | \gamma, \sigma_a, \tau \right] = \sigma_a^2 \sum_{j:\gamma_j = 1} s_j \] where \(s_j = \frac{1}{n}\sum_{i=1}^{n}x_{ij}\) is the variance of covariate \(j\). Define \[ h(\gamma, \sigma_a) = \frac{\nu(\gamma, \sigma_a)}{1+\nu(\gamma, \sigma_a)} \] which gives a rough guide to the expectation of PVE for a given \(\gamma\) and \(\sigma_a\). They specify a uniform prior on \(h\), independent of \(\gamma\), which induces a prior on \(\sigma_a\) given \(\gamma\) via \[ \sigma_a^2(h, \gamma) = \frac{h}{1-h}\frac{1}{\sum_{j:\gamma_j = 1} s_j}. \]
The posterior of \((h, \pi, \gamma)\) is \[ p(h, \pi, \gamma | y) \propto p(y | h, \gamma) p(h) p(\gamma|\pi)p(\pi) \]
Estimating PIP using Rao-Blackwellized estimate, \[ P(\gamma_j = 1|y) \approx \frac{1}{M} \sum_{i=1}^{M}P(\gamma_j = 1|y, \gamma_{-j}^{(i)}, \beta_{-j}^{(i)}, \tau^{(i)}, h^{(i)}, \pi^{(i)}) \] where \(\gamma^{(i)}, \beta^{(i)}, \tau^{(i)}, h^{(i)}, \pi^{(i)}\) denote the \(i\)th MCMC sample.
Guan, Yongtao, and Matthew Stephens. 2011. “Bayesian Variable Selection Regression for Genome-Wide Association Studies and Other Large-Scale Problems.” The Annals of Applied Statistics. JSTOR, 1780–1815.