Taken from Vehtari et al. (2014). Assume that you can factorize your target distribution \[ f(\theta) \propto \prod_{k=0}^K f_k(\theta) \] Approximate target distribution using a distribution with a similar factorization, called the global approximation \[ g(\theta) \propto \prod_{k=0}^K g_k(\theta) \] At each iteration of the algorithm, and for every factor \(k=1, \ldots, K\) form the cavity distribution by removing \(k^{\text{th}}\) term from the global approximation \[ g_{-k} \propto \frac{g(\theta)}{g_k(\theta)} \] also form the tilted distribution by including \(f_k(\theta)\) to the cavity distribution (in place of \(g_k(\theta)\)) \[ g_{\backslash k}(\theta) \propto f_k(\theta) g_{-k}(\theta) \] The algorithm updates the site approximation \(g_k(\theta)\) so that the new global approximation approximates the tilted distribution \[ g_k(\theta)^{\text{new}}g_{-k}(\theta) \approx f_k(\theta)g_{-k}(\theta) \]