both are alternating iterative procedures that successively converge on optimum parameter values. The initial steps to derive the respective procedures are also vaguely similar, both starting out with formulas for probability densities and both involving significant amounts of mathematical manipulations. However, there are a number of differences. Most important is what is being computed. EM computes point estimates of posterior distribution of those random variables that can be categorized as "parameters", but only estimates of the actual posterior distributions of the latent variables (at least in "soft EM", and often only when the latent variables are discrete). The point estimates computed are the modes of these parameters; no other information is available. VB, on the other hand, computes estimates of the actual posterior distribution of all variables, both parameters and latent variables. When point estimates need to be derived, generally the mean is used rather than the mode, as is normal in Bayesian inference. Concomitant with this, the parameters computed in VB do not have the same significance as those in EM. EM computes optimum values of the parameters of the Bayes network itself. VB computes optimum values of the parameters of the distributions used to approximate the parameters and latent variables of the Bayes network. For example, a typical Gaussian mixture model will have parameters for the mean and variance of each of the mixture components. EM would directly estimate optimum values for these parameters. VB, however, would first fit a distribution to these parameters — typically in the form of a prior distribution, e.g. a normal-scaled inverse gamma distribution — and would then compute values for the parameters of this prior distribution, i.e. essentially hyperparameters. In this case, VB would compute optimum estimates of the four parameters of the normal-scaled inverse gamma distribution that describes the joint distribution of the mean and variance of the component. A more complex example Imagine a Bayesian Gaussian mixture model described as follows: Note: SymDir() is the symmetric Dirichlet distribution of dimension , with the hyperparameter for each component set to . The Dirichlet distribution is the conjugate prior of the categorical distribution or multinomial distribution. is the Wishart distribution, which is the conjugate prior of the precision matrix (inverse covariance matrix) for a multivariate Gaussian distribution. Mult() is a multinomial distribution over a single observation (equivalent to a categorical distribution). The state space is a "one-of-K" representation, i.e. a -dimensional vector