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Markov Random Field
An undirected graphical model representing the joint distribution of a set of random variables where local dependencies are specified by potentials on the cliques of the graph.
Normalization constant
The normalization constant Z that ensures the sum of probabilities of a Markov random field equals 1, calculated as the sum of the products of potentials over all possible configurations.
Local potential
A non-negative function defined on a clique of the graph that quantifies the affinity between the variables in that clique, replacing the conditional probabilities of Bayesian models.
Maximal clique
A complete subgraph that cannot be extended by adding other vertices while remaining complete, serving as a support for defining potentials in a Markov random field.
Energy of a state
A scalar quantity associated with a particular configuration of the variables, calculated as the sum of local potentials, where low-energy states have a higher probability.
Gibbs distribution
An exponential probability distribution where the probability of a configuration is proportional to the negative exponential of its energy, forming the mathematical basis of Markov random fields.
Belief Propagation algorithm
An exact inference method on acyclic graphs and an approximate method on graphs with cycles, propagating messages between nodes to compute marginal beliefs.
Graph-cut
A combinatorial optimization technique used for inference in certain Markov random fields, transforming the problem into a minimum cut in a flow graph.
Ising Model
Simple binary random Markov field where each variable has only two states, originally developed in statistical physics to model ferromagnetism.
Potts Model
Multi-state generalization of the Ising model where variables can take more than two values, widely used in image segmentation and classification.
Conditional Random Field
Discriminative extension of random Markov fields that directly models the conditional probability P(y|x) for output structuring tasks such as sequence labeling.
Message Passing Inference
Inference paradigm where graph nodes communicate locally through messages to compute global marginals, including belief propagation and its variants.
Factor Graph
Bipartite representation of a random Markov field separating variables from potential factors, facilitating the implementation and analysis of inference algorithms.
Partition Function
Synonym of the normalization factor Z in statistical thermodynamics, representing the sum of weights of all possible system configurations.
Metropolis-Hastings Sampling
General MCMC algorithm that generates samples according to a target distribution by accepting or rejecting proposals based on a criterion derived from the probability ratio.