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Carbonetto, Stephens, and others (2012) introduce varbvs, which uses variational Bayes to do variable selection. The method scales linearly with the number of variables. They assume each regression coefficient \(\beta_i\) is independent of the other regression coefficients a posteriori. Variational estimates of posterior distributions can be inaccurate. For instance, they are often too concentrated. They focus on assessing the accuracy of the variational approximation. In situations where the conditional independence assumption (\(X_i\) and \(X_j\) are independent) is violated, we would not expect accurate approximations of the posterior inclusion probabilities.

The model is \[ Y = \sum_{k=1}^{p} X_k \beta_k + \epsilon, \epsilon \sim N(0, \sigma^2) \] The binary variables \(\gamma = (\gamma_1, \cdots, \gamma_p)^\intercal\) to indicate whether or not each variable is included in the model. They assign priors to the indicator variables \(\gamma\) and coefficients \(\beta\), then compute posterior probabilities by averaging over choices of \(\beta\) and \(\gamma\), and any additional model parameters, such as \(\sigma^2\). Priors: \[\begin{align*} p(\gamma_k = 1) &= \pi \\ \beta_k | \gamma_k=1 &\sim N(0, \sigma^2 \sigma_{\beta}^2). \end{align*}\]

The hyperparameters \(\theta = (\sigma^2, \sigma_{\beta}^2, \pi)\).

The posterior probability that variable \(X_k\) is included in the linear model of \(Y\) is \[ PIP(k) = p(\gamma_k = 1 | X, y) = \frac{\sum_{\gamma_{-k}} \int \int p(y, \beta, \gamma_k = 1, \gamma_{-k} | X, \theta) d \beta d \theta}{\sum_{\gamma} \int \int p(y, \beta, \gamma|X, \theta) d \beta d \theta}. \] The joint probability of \(y\), \(\beta\) and \(\gamma\) given \(X\) and \(\theta\) is \[ p(y, \beta, \gamma|X, \theta) = p(y | X, \beta, \sigma^2) \prod_{k=1}^{p} p(\beta_k | \gamma_k, \sigma^2, \sigma_{\beta}^2) \prod_{k=1}^{p} p(\gamma_k | \pi). \] MCMC methods approximate the intractable sums and integrals by implementing Metropolis-Hastings moves that explore models with strong support under the posterior.

\[\begin{align*} q(\beta, \gamma; \phi) &= \prod_{k=1}^{p} q(\beta_k, \gamma_k; \phi_k) \\ q(\beta_k, \gamma_k; \phi_k) &= \begin{cases} \alpha_k N(\beta_k | \mu_k, s_k^2) \quad &\text{if } \gamma_k = 1 \\ (1-\alpha_k) \delta_0(\beta_k) \quad & \text{ow} \end{cases} \end{align*}\]

\(\phi_k = (\alpha_k, \mu_k, s_k^2)\).

\[\begin{align*} Var(\beta_k|\gamma_k = 1) &= s_k^2 = \frac{\sigma^2}{(X^\intercal X)_{kk} + 1/\sigma_{\beta}^2} \\ E(\beta_k | \gamma_k = 1) &= \mu_k = \frac{s_k^2}{\sigma^2} \left( (X^\intercal y)_k - \sum_{j \neq k}(X^\intercal X)_{jk} \alpha_j \mu_j \right) \\ \text{Posterior odds = Bayes factor × prior odds } \quad \frac{p(\gamma_k = 1|X, y, \theta)}{p(\gamma_k = 0|X, y, \theta)} &= \frac{\alpha_k}{1-\alpha_k} = \frac{\pi}{1-\pi} \times \frac{s_k}{\sigma_{\beta} \sigma} \times e^{SSR_k/2} \\ SSR_k &= \mu_{k}^2/s_k^2 \end{align*}\]

They use importance sampling to integrate over the hyperparameters.

Carbonetto, Peter, Matthew Stephens, and others. 2012. “Scalable Variational Inference for Bayesian Variable Selection in Regression, and Its Accuracy in Genetic Association Studies.” Bayesian Analysis 7 (1). International Society for Bayesian Analysis: 73–108.