On a Class of Tensor Markov Fields

Hernández‐Lemus, Enrique

doi:10.3390/e22040451

Cited by 4 publications

(4 citation statements)

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“…However, MI distribution changes for every type of omics paired complicating MI comparisons. Perhaps a better alternative will be to resort to tensor representations of probabilistic multilayer networks (Hernández-Lemus, 2020 ).…”

Section: Discussionmentioning

confidence: 99%

An Information Theoretical Multilayer Network Approach to Breast Cancer Transcriptional Regulation

2021

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Breast cancer is a complex, highly heterogeneous disease at multiple levels ranging from its genetic origins and molecular processes to clinical manifestations. This heterogeneity has given rise to the so-called intrinsic or molecular breast cancer subtypes. Aside from classification, these subtypes have set a basis for differential prognosis and treatment. Multiple regulatory mechanisms—involving a variety of biomolecular entities—suffer from alterations leading to the diseased phenotypes. Information theoretical approaches have been found to be useful in the description of these complex regulatory programs. In this work, we identified the interactions occurring between three main mechanisms of regulation of the gene expression program: transcription factor regulation, regulation via noncoding RNA, and epigenetic regulation through DNA methylation. Using data from The Cancer Genome Atlas, we inferred probabilistic multilayer networks, identifying key regulatory circuits able to (partially) explain the alterations that lead from a healthy phenotype to different manifestations of breast cancer, as captured by its molecular subtype classification. We also found some general trends in the topology of the multi-omic regulatory networks: Tumor subtype networks present longer shortest paths than their normal tissue counterpart; epigenomic regulation has frequently focused on genes enriched for certain biological processes; CpG methylation and miRNA interactions are often part of a regulatory core of conserved interactions. The use of probabilistic measures to infer information regarding theoretical-derived multilayer networks based on multi-omic high-throughput data is hence presented as a useful methodological approach to capture some of the molecular heterogeneity behind regulatory phenomena in breast cancer, and potentially other diseases.

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Section: Discussionmentioning

confidence: 99%

An Information Theoretical Multilayer Network Approach to Breast Cancer Transcriptional Regulation

2021

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“…Hence HRFs are theoretically founded, but developed thinking in applications. The scope of applicability of MRFs has also become broader by expanding its applicability to model tensor valued quantities [35], giving rise to the so-called multilayer graphical models, also called multilayer networks [36][37][38][39].…”

Section: Mrfs In Statistical Mechanics and Mathematical Physicsmentioning

confidence: 99%

Random Fields in Physics, Biology and Data Science

Hernández‐Lemus

2021

Front. Phys.

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A random field is the representation of the joint probability distribution for a set of random variables. Markov fields, in particular, have a long standing tradition as the theoretical foundation of many applications in statistical physics and probability. For strictly positive probability densities, a Markov random field is also a Gibbs field, i.e., a random field supplemented with a measure that implies the existence of a regular conditional distribution. Markov random fields have been used in statistical physics, dating back as far as the Ehrenfests. However, their measure theoretical foundations were developed much later by Dobruschin, Lanford and Ruelle, as well as by Hammersley and Clifford. Aside from its enormous theoretical relevance, due to its generality and simplicity, Markov random fields have been used in a broad range of applications in equilibrium and non-equilibrium statistical physics, in non-linear dynamics and ergodic theory. Also in computational molecular biology, ecology, structural biology, computer vision, control theory, complex networks and data science, to name but a few. Often these applications have been inspired by the original statistical physics approaches. Here, we will briefly present a modern introduction to the theory of random fields, later we will explore and discuss some of the recent applications of random fields in physics, biology and data science. Our aim is to highlight the relevance of this powerful theoretical aspect of statistical physics and its relation to the broad success of its many interdisciplinary applications.

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“…In terms of the stochastic process, ref. [ 3 ] considers the fundamental properties of Tensor Markov Fields. Under a Bayesian paradigm of Statistical Inference, ref.…”

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confidence: 99%

“…Ref. [ 3 ] presented the definitions and properties of Tensor Markov Fields (random Markov fields over tensor spaces). The author shows that tensor Markov fields are indeed Gibbs fields whenever strictly positive probability measures are considered.…”

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confidence: 99%