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 $k=1,\dots ,K$ 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 $g_k(\theta )$) $$g_{\mathrm{\setminus }k}(\theta )\propto f_k(\theta )g_{-k}(\theta )$$ The algorithm updates the site approximation $g_k(\theta )$ so that the new global approximation approximates the tilted distribution $$g_k(\theta {)}^{\text{new}}{g}_{-k}(\theta )\approx f_k(\theta )g_{-k}(\theta )$$