Markov networks and other probabilistic graphical modes have recently received an upsurge in attention from Evolutionary computation community, particularly in the area of Estimation of distribution algorithms (EDAs). EDAs have arisen as one of the most successful experiences in the application of machine learning methods in optimization, mainly due to their efficiency to solve complex real-world optimization problems and their suitability for theoretical analysis.

This book focuses on the different steps involved in the conception, implementation and application of EDAs that use Markov networks, and undirected models in general. It can serve as a general introduction to EDAs but covers also an important current void in the study of these algorithms by explaining the specificities and benefits of modeling optimization problems by means of undirected probabilistic models.

All major developments to date in the progressive introduction of Markov networks based EDAs are reviewed in the book. Hot current research trends and future perspectives in the enhancement and applicability of EDAs are also covered. The contributions included in the book address topics as relevant as the application of probabilistic-based fitness models, the use of belief propagation algorithms in EDAs and the application of Markov network based EDAs to real-world optimization problems. The book should be of interest to researchers and practitioners from areas such as optimization, evolutionary computation, and machine learning.

The Robert Gordon University, Aberdeen, UK (April 2006) 60. Shakya, S., McCall, J.: Optimisation by Estimation of Distribution with DEUM framework based on Markov Random Fields. International Journal of Automation and Computing 4, 262–272 (2007) 61. Shakya, S., McCall, J., Brown, D.: Updating the probability vector using MRF technique for a univariate EDA. In: Onaindia, E., Staab, S. (eds.) Proceedings of the Second Starting AI Researchers’ Symposium. Frontiers in Artificial Intelligence and

between two nodes should be seen as a neighbourhood relationship, rather than a parenthood relationship in Bayesian Networks. We use N = {N1 , N2 , ..., Nn } to define a neighbourhood system on G, where each Ni is a set of nodes neighbouring to node Xi . Figure 4.1 shows an example of a Markov network structure on 6 random variables. Here, variable X1 has 2 neighbours, N1 = {X2 , X3 }. Similarly, variable X2 has 4 neighbours N2 = {X1 , X3 , X4 , X5 }. X1 X2 X4 X3 X5 X6 Fig. 4.1 A Markov

previous section, this is achieved by solving the system of linear equations (4.15). The p(xi ) is then calculated from using α (as shown in figure 4.3) and sampled to generate the child population. The child then replaces the parent, P, and this process continues until termination criteria are satisfied. DEUMd Workflow 1. Generate a population, P, consisting of M solutions 2. Select a set D from P consisting of N best solutions, where N ≤ M. 3. For each solution, x, in D, build a linear equation

Multivariate Gaussian Distribution Multivariate Gaussian distribution (MGD) is the most frequently used probability distributions for continuous optimization problems in estimation of distribution algorithms [2, 4, 28]. A multivariate Gaussian distribution N (µ , Σ ) over a vector of n random variables X = (X1 , . . . , Xn ) is defined with two parameters: µ is the n-dimensional vector of mean values for each variable, and Σ is a positive-definite and symmetric n × n covariance matrix. A square

Markov network codes information about conditional independencies. In this case, x7 is conditionally independent of x5 , x9 , given x6 , x10 . So we can separate the set in two parts: x5 , x6 , x9 , x10 and x6 , x7 , x10 . So we got rid of the pentavariate set and just have two sets of four and three variables, thus less degrees of freedom. For the 2-D grid, the gain is not very large; there are other structures where the difference is larger. E. g. on an analogous 3-D grid, the method reduces