Expectation Propagation performs a smoothed gradient descent

Abstract: Bayesian inference is a popular method to build learning algorithms but it is hampered by the fact that its key object, the posterior probability distribution, is often uncomputable. Expectation Propagation (EP) (Minka (2001)) is a popular algorithm that solves this issue by computing a parametric approximation (e.g: Gaussian) to the density of the posterior. However, while it is known empirically to quickly compute fine approximations, EP is extremely poorly understood which prevents it from being adopted by a larger fraction of the community. The object of the present article is to shed intuitive light on EP, by relating it to other better understood methods. More precisely, we link it to using gradient descent to compute the Laplace approximation of a target probability distribution. We show that EP is exactly equivalent to performing gradient descent on a smoothed energy landscape: i.e: the original energy landscape convoluted with some smoothing kernel. This also relates EP to algorithms that compute the Gaussian approximation which minimizes the reverse KL divergence to the target distribution, a link that has been conjectured before but has not been proved rigorously yet. These results can help practitioners to get a better feel for how EP works, as well as lead to other new results on this important method.