Algorithm 1.

The Exact EM Algorithm

Initial values: Obtain an initial values ${\hat{\mathrm{\Theta }}}^{(0)}$ based on existing group ICA software.

repeat

E-step:

1. Evaluate the conditional distribution of the latent variables $p(\mathit{L}(v)|\mathbf{y}(v),{\hat{\mathrm{\Theta }}}^{(k)})$ using the proposed three-step approach:

1.a Evaluate the multivariate Gaussian $p[\mathit{L}(v)|\mathbf{y}(v),\mathit{z}(v),{\hat{\mathrm{\Theta }}}^{(k)}]$;

1.b Evaluate $p[\mathit{z}(v)|\mathbf{y}(v);{\hat{\mathrm{\Theta }}}^{(k)}]$ via Bayes’ Theorem

1.c integrate out the latent states z(v) $$p(\mathit{L}(v)|\mathbf{y}(v),{\hat{\mathrm{\Theta }}}^{(k)})=\sum _{\mathit{z}(v)\in \mathcal{R}}p(\mathit{L}(v)|\mathbf{y}(v),\mathit{z}(v),{\hat{\mathrm{\Theta }}}^{(k)})p[\mathit{z}(v)|\mathbf{y}(v);{\hat{\mathrm{\Theta }}}^{(k)}]$$

2. Estimate conditional expectation $Q(\mathrm{\Theta }|{\hat{\mathrm{\Theta }}}^{(k)})$ based on $p(\mathit{L}(v)|\mathbf{y}(v),{\hat{\mathrm{\Theta }}}^{(k)})$.

M-step:

Update parameters estimates $${\hat{\mathrm{\Theta }}}^{(k+1)}=\underset{\mathrm{\Theta }}{\mathrm{argmax}}Q(\mathrm{\Theta }|{\hat{\mathrm{\Theta }}}^{(k)}).$$

until convergence, i.e. $\frac{\Vert {\hat{\mathrm{\Theta }}}^{(k+1)}-{\hat{\mathrm{\Theta }}}^{(k)}\Vert }{\Vert {\hat{\mathrm{\Theta }}}^{(k)}\Vert }<ϵ$