Algebraic Statistics in Practice: Applications to Networks

Abstract

Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra and computational algebra), geometry and combinatorics to provide insight into knotty problems in mathematical statistics. In this survey we illustrate this on three problems related to networks, namely network models for relational data, causal structure discovery and phylogenetics. For each problem we give an overview of recent results in algebraic statistics with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines. 1 arXiv:1906.09537v1 [math.ST] 23 Jun 2019 Casanellas, Petrović, Uhler /Algebraic Statistics in Practice 2the nodes of an unknown underlying causal network), and phylogenetics (a particular network structure discovery problem where the underlying network is a tree with latent variables).Section 2 focuses on statistical models for relational data, typical uses of which arise in the social and biological sciences. In these applications, nodes in the network may represent individuals, organizations, proteins, neurons, or brain regions, while links represent observed relationships between the nodes, such as personal or organizational affinities, social/financial relationships, binding between proteins or physical links between brain regions. A key problem in this area is to test whether a proposed statistical model fits the data at hand; such a test typically involves generating a sufficiently large and generic sample of networks from the model and comparing it to the observed network. Perhaps somewhat surprisingly, algorithms for sampling networks with given network statistics for goodness-of-fit testing are often efficiently encoded by algebraic constraints. In Section 2, we outline how techniques from commutative algebra and combinatorics are applied to this problem for several families of network models for which a formal test is otherwise unavailable.In Section 3, we turn to applications where the network structure cannot directly be observed and we only have access to observations on the nodes of the network. Such applications range from data on consumer behavior to click statistics for ads or websites, DNA sequences of related species, gene expression data, etc. The use of such data to gain insight into complex phenomena requires characterizing the relationships among the observed variables. Probabilistic graphical models explicitly capture the statistical relationships between the variables as a network. A good representation of a complex system should not only enable predicting the state of one component given others, but also the effect that local operations have on the global system. This requires causal modeling and making use of interventional data. In Section 3, we discuss the role that algebraic and discrete geometry play in analyzing prominent algorithms for causal structure discovery and in developing the first provably consistent algorithms for causal inference from a mix of observational ...

The model-specific Markov embedding problem for symmetric group-based models

“…Biologically, the subspace dd of diagonally dominant matrices consists of matrices with probability of not mutating at least as large as the probability of mutating. If a diagonally dominant matrix is embeddable, it has an identifiable rate matrix (Cuthbert 1972(Cuthbert , 1973, namely a unique Markov generator, which is crucial for proving the consistency of many phylogenetic reconstruction methods, such as those based on maximum likelihood methods (Casanellas et al 2020c;Chang 1996). What is more, the set of Markov matrices with positive eigenvalues + includes the multiplicative closure of the transition matrices in the continuous-time version of the model (Sumner et al 2012).…”