Markov Networks | Roberto Santana

Markov networks (also called Markov random fields or undirected graphical models) represent joint probability distributions through potential functions defined on the cliques of an undirected graph. Unlike Bayesian networks, Markov networks naturally represent symmetric statistical dependencies between variables, making them particularly suitable for problems where the direction of influence is unclear or genuinely symmetric. My research on Markov networks spans their use in estimation of distribution algorithms, energy-based generative modeling, and applications to image denoising, classification, and brain connectivity.

Foundations of Markov Networks

Markov Random Fields

A Markov random field (MRF) is a set of random variables having the Markov property with respect to an undirected graph: each variable is conditionally independent of all other variables given its neighbors in the graph. The joint distribution is expressed as a product of potential functions (factors) over the cliques of the graph.

Key properties of MRFs include the global Markov property (separation in the graph implies conditional independence), the local Markov property (a variable is independent of non-neighbors given neighbors), and the pairwise Markov property. Research on learning and inference in MRFs has been a central topic in probabilistic graphical models.

Comparison with Bayesian Networks

Markov networks and Bayesian networks are two complementary frameworks for probabilistic graphical models. Bayesian networks use directed acyclic graphs to represent conditional independence and have a natural causal interpretation. Markov networks use undirected graphs and are more natural for representing symmetric dependencies such as those found in image models, Ising models, and spatial statistics. Research on when each representation is more appropriate and on methods for converting between the two formalisms.

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Restricted Boltzmann Machines

Boltzmann Machines as Markov Networks

Boltzmann machines are a type of Markov network where the potential functions have an energy-based form. The joint distribution of visible (observed) and hidden (latent) variables is defined through an energy function, and the model is trained by maximizing the likelihood of the observed data under the model. Restricted Boltzmann machines (RBMs) are a special case with a bipartite graph connecting visible and hidden units, enabling efficient learning through contrastive divergence.

Structural Restricted Boltzmann Machines

Research on structural RBMs that go beyond the standard bipartite connectivity pattern. By adding structure within the visible and/or hidden layers, structural RBMs can capture more complex dependency patterns in the data. Applications to image denoising (exploiting local spatial correlations) and classification (exploiting label structure).

Deep Boltzmann Machines

Deep Boltzmann machines (DBMs) stack multiple layers of RBMs to create deep hierarchical generative models. Research on training methods for DBMs, including layer-wise pre-training and joint fine-tuning, and on the relationship between DBMs and deep belief networks (which are hybrid directed/undirected models).

Markov Network-Based EDAs

Markov Network EDAs

Development of estimation of distribution algorithms (EDAs) that use Markov networks as the probabilistic model. Markov network EDAs learn an undirected graphical model from the selected population and sample new candidate solutions from this model. The symmetric nature of Markov networks makes them particularly appropriate for problems where the dependencies between variables are symmetric.

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Learning Markov Network Structures for EDAs

Research on algorithms for learning the structure of Markov networks from data within the EDA loop. Structure learning algorithms for Markov networks must balance the quality of the learned structure (how well it represents the dependencies in the data) against computational cost (Markov network structure learning is generally more computationally demanding than Bayesian network structure learning).

Structure Learning for Markov Networks

Score-Based Markov Network Learning

Development of score-based algorithms for learning the structure of Markov networks from data. These algorithms search over possible edge sets to find the structure that maximizes a penalized log-likelihood score. Key challenges include the intractability of the partition function (which prevents direct likelihood computation) and the combinatorial nature of the search space.

Constraint-Based Learning

Constraint-based methods for Markov network structure learning use statistical tests of conditional independence to identify the edges of the Markov network. Research on how the choice of conditional independence test, the sample size, and the significance level affect the quality of the learned structure. Development of algorithms that scale to high-dimensional problems.

Applications

Markov Networks for Image Modeling

Application of Markov networks to image modeling and denoising. Image models based on Markov random fields exploit local spatial correlations in images, with each pixel depending primarily on its spatial neighbors. Research on how to learn effective image priors from data and use them for denoising, segmentation, and super-resolution tasks.

Markov Networks for Brain Connectivity

Application of Markov networks to model functional and structural connectivity in the brain. By learning a Markov network from neuroimaging data (fMRI, MEG, EEG), it is possible to identify which brain regions interact with each other and to study how connectivity patterns change across different cognitive states or populations.

See Neuroscience page →

Selected Publications