some well known EDAs of different levels of complexity. It is always assumed a population at the generation , a selection operator , a model-building operator and a sampling operator . Univariate factorizations The most simple EDAs assume that decision variables are independent, i.e. . Therefore, univariate EDAs rely only on univariate statistics and multivariate distributions must be factorized as the product of univariate probability distributions, Such factorizations are used in many different EDAs, next we describe some of them. Univariate marginal distribution algorithm (UMDA) The UMDA is a simple EDA that uses an operator to estimate marginal probabilities from a selected population . By assuming contain elements, produces probabilities: Every UMDA step can be described as follows Population-based incremental learning (PBIL) The PBIL, represents the population implicitly by its model, from which it samples new solutions and updates the model. At each generation, individuals are sampled and are selected. Such individuals are then used to update the model as follows where is a parameter defining the learning rate, a small value determines that the previous model should be only slightly modified by the new solutions sampled. PBIL can be described as Compact genetic algorithm (cGA) The CGA, also relies on the implicit populations defined by univariate distributions. At each generation , two individuals are sampled, . The population is then sorted in decreasing order of fitness, , with being the best and being the worst solution. The CGA estimates univariate probabilities as follows where, is a constant defining the learning rate, usually set to . The CGA can be defined as Bivariate factorizations Although univariate models can be computed efficiently, in many cases they are not representative enough to provide better performance than GAs. In order to overcome such a drawback, the use of bivariate factorizations was proposed in the EDA community, in which dependencies between pairs of variables could be modeled. A bivariate factorization can be defined as follows, where contains a possible variable dependent to , i.e. . Bivariate and multivariate distributions are usually represented as probabilistic graphical models (graphs), in which edges denote statistical dependencies (or conditional probabilities) and vertices denote variables. To learn the structure of a PGM from data linkage-learning is employed. Mutual information maximizing input clustering (MIMIC) The MIMIC factorizes the joint probability distribution in a chain-like model representing successive dependencies between variables. It finds a permutation of the decision variables, , such