Champ aléatoire de Markov (MRF)
A Champ aléatoire de Markov (MRF) is a type of modèle graphique probabiliste that captures the dependencies between a set of random variables. These variables are represented as nodes in a graph, where edges between nodes indicate direct relationships or dependencies. In an MRF, a variable is conditionally independent of its non-neighbors given its neighbors, which is a key property known as the propriété de Markov.
MRFs are particularly useful in scenarios where the data is structured in a way that allows for local interactions, such as in traitement d'image, spatial data analysis, and social network modeling. For example, in segmentation d'image, the pixels of an image can be treated as random variables, where the value of each pixel is influenced by its neighboring pixels.
Formellement, un MRF est défini par un graphe non orienté G = (V, E), where V is the set of vertices (random variables) and E is the set of edges (dependencies). The distribution de probabilité conjointe of the random variables is specified through potential functions associated with cliques (subsets of connected nodes) in the graph. These potential functions represent the compatibility of the variable configurations within the cliques.
To perform inference in MRFs—i.e., to compute the probability of certain variables given others—techniques such as Gibbs sampling and belief propagation are often employed. MRFs are widely utilized in various applications, including computer vision, traitement du langage naturel, and bioinformatics, due to their ability to model complex interactions in a flexible and interpretable manner.