Towards an information theory of quantitative genetics

Abstract

Quantitative genetics has evolved dramatically in the past century, and the proliferation of genetic data enables the characterization of complex interactions beyond the scope of its theoretical foundations. In this paper, we lay the foundations of an alternative formulation of quantitative genetics based in information theory. Information theory can provide sensitive measures of statistical dependencies among variables, and provides a natural mathematical language for an alternative view of quantitative genetics. In previous work we examined the information content of discrete functions and applied this formalism to the analysis of genetic data. We present here a set of relationships that both unifies the information measures for the set of discrete functions, and uses them to express key quantitative genetic relationships. Information theory measures of variable interdependency are used to identify significant interactions, and a general approach is described for inferring functional relationships within genotype and phenotype data. We present information-based measures of the genetic quantities: penetrance, heritability and degrees of statistical epistasis. Our scope includes the consideration of three variable dependencies and independently segregating variants, which captures two locus effects, genetic interactions, and two phenotype pleiotropy. However, this formalism and general theory naturally applies to multi-variable interactions and higherorder complex dependencies, and can be adapted to account for population structure, linkage and nonrandomly segregating markers. This paper therefore lays the groundwork for a full formulation of quantitative genetics based in information theory. *Communication

Higher-order in-and-outeractions reveal synergy and logical dependence beyond Shannon-information

“…On a powerset ordered by inclusion, the Möbius function takes the simple form µ(x, y) = (−1) |x|−|y| [29,27]. is definition allows us to write the mutual information among a set of variables τ as [5,11]:…”

“…(0 0) 21), (12), or (11) gives us the other categorical interactions. e transitivity property of the interactions, i.e.…”

“…where ∆ is known as the differential mutual information [10]. It describes the change in mutual information when leaving out X, and has been used to describe information structures in genetics [11]. On the Boolean algebra of three variables {X, Y, Z}, the dual mutual information of X can be wri en out as:…”

Tutorial - Machine Learning and Information Theoretic Methods for Molecular Biology and Medicine

“…Information theory and information theoretic concepts contribute to many statistical and machine learning tasks also in medicine and molecular biology and provide a natural paradigm to process information [4,5]. For example, the information theoretic perspective on medical diagnostic inference is considered in [23] and its relevance for clinical diagnostic testing is investigated in [10], whereas their applications in computational biology and quantitative genetics are discussed in [18] and [26], respectively. Considered broadly, one can distinguish foundational information theoretic concepts for data proximity evaluation (potentially leading to distance metrics) with statistical inference on the one hand and information theory based learning concepts in machine learning applied for bio-molecular and medical data analysis tasks on the other hand.…”

Information Theory in Computational Biology: Where We Stand Today

“…Information theoretic methods dealing with such quantitative phenotypes and environmental variables can be found in [ 116 , 117 , 131 ] that use differential or cross-entropy based generalizations of multivariate test statistics. In a recent review, Galas et al [ 132 ] presented a detailed discussion on the information theoretic formalism for gene association with quantitative phenotypes.…”