Graphs represent an intuitive way to visualize and understand complex relationships among different random variables. The nodes of the graph represent random variables, while missing connections represent conditional independencies among variables. Rules that translate graph properties into statements of conditional independence among variables are called Markov properties. Using separation criteria we can define three different types of Markov properties: global, local, and pairwise Markov property. Conditional independencies defined by Markov properties impose limitations on the graphical model. A probabilistic graphical model on an undirected graph is called a Markov network, and a Bayesian network on a directed graph. Discrete Markov networks can be parameterized with standard log-linear parameterization for undirected graphical models, derived through the application of Möbius inversion formula. Markov and Bayesian networks have diverse applications in practice, such as image processing or making informative decisions in managing financial portfolios.
|
