BayesianGLASSO
Bayesian Graphical Lasso (BayesianGLASSO) is a Bayesian treatment of GLASSO that use a
double exponential prior and employs a block Gibbs sampler for exploring the posterior distribution.
GLASSO is still maintained. An efficient block Gibbs sampler is developed:1. For $i = 1, ..., p$
(a) Partition $\Omega, \hat{\Sigma}, T$ as following:
$$\Omega = \binom{\Omega_{11}\quad \omega_{12}}{\omega_{12}^{'}\quad \omega_{22}}, \quad S = \binom{S_{11}\quad s_{12}}{s_{12}^{'}\quad s_{22}}, \quad T = \binom{T_{11}\quad \tau_{12}}{\tau_{12}^{'}\quad \tau_{22}}.$$
(b) Sample $\gamma \sim Ga(n/2+1, (\hat{\sigma}_{22}+\lambda)/2$ and
$\beta \sim N(-C\hat{\sigma}_{21}, C)$, where
$C = \{(\hat{\sigma}_{22} + \lambda)\Omega^{-1}_{11} + D^{-1}_{\tau}\}$, $D_{\tau} = diag(\tau_{12})$.
(c) Update $\omega_{21} = \beta$, $\omega_{12} = \beta^T$, $\omega_{22} = \gamma + \beta^T\Omega^{-1}_{11}\beta$.
2. Sample $\mu_{ij} \sim Inv-Gau(\mu^{'}, \lambda^{'})$,
where $\mu^{'} = \sqrt{(\lambda^2/\omega^2_{ij})}, \lambda^{'} = \lambda^2$.
Update $\tau = 1/\mu_{ij}$.
3. Sample $\lambda \sim Ga(r + p(p+1)/2, s + ∥\omega∥_1/2)$.
In this form of the Bayesian graphical lasso, a single shrinkage parameter $\lambda$ is employed. The Bayesian adaptive graphical lasso, on the other hand, allows for different shrinkage parameters $\lambda_{ij}$ for different entries of the precision matrix $\Omega$. The model (data likelihood, prior, and hyperprior) is $$p(y_i|\Omega) = n(y_i|0, \Omega^{-1})$$ $$p(\Omega|\lambda) \propto \prod_{i \le j}{[\frac{\lambda_{ij}}{2}exp\{-\lambda_{ij}|\omega_{ij}|\}]}\prod^{p}_{i=1}[\frac{\lambda_{ii}}{2}exp\{-\frac{\lambda_{ii}}{2}\omega_{ii}\}]1_{\Omega \in M^+}$$ $$p(\{\lambda_{ij}\}_{i \le j}|\{\lambda_{ii}\}^{p}_{i=1}) \propto \prod_{i \le j}{\frac{s^r}{\Gamma(r)}\lambda^{r-1}_{ij}exp\{-\lambda_{ij}s_i\}}.$$ This allow the level of shrinkage to be automatically chosen based on the current value of $\omega_{ij}$.
Reference:
1. Wang, Hao. "Bayesian graphical lasso models and efficient posterior computation." Bayesian Analysis 7.4 (2012): 867-886.
Note:
1. BayesianGLASSO is time consuming. We require the input expression data to have no more than 50 genes (columns)
and no more than 100 observations (rows). (Otherwise, you won't be able to submit the job!)
2. Change the $\alpha$ level to control the sparsity of the network $(0 < \alpha < 1)$.
A small $\alpha$ will give you more estimated edges, but with lower confidence. If you don't know how to choose
a value, use the default one.