Algebraic statistics

Graphical models are a class of statistical models that represent independencies among random variables by a graph. They include hidden Markov models, Bayesian networks, and Markov random fields. Graphical models are very popular in computational biology. They are used for sequence alignment, gene finding, phylogeny reconstruction, gene regulatory network inference, and many other tasks. We do not only use graphical models in several biomedical applications, but we are also interested in the mathematical properties of these models. Among other methods, we use algebraic tools for studying graphical models, an approach termed algebraic statistics.

Algebraic statistics offers a conceptual and computational framework for parametric inference in hidden Markov models, i.e., for finding all maximum a posteriori estimates of the hidden variables for all choices of the model parameters. Parametric inference facilitates the analysis of robustness of annotations and is therefore of general importance in computational biology. For example, we use parametric inference for detecting HIV recombinant genomes and for quantifying the confidence in model predictions.

We have also used techniques from algebraic combinatorics for the analysis of fitness landscapes. A fitness landscape is a simple model of differential reproductive success of individuals in a population. We have introduced a geometric classification of fitness landscapes which is based on triangulations of polytopes. The shape of a fitness landscape generalizes the notion of epistasis (the interaction between two genes).

Selected references

• Beerenwinkel N, Pachter L, Sturmfels B (2007). Epistasis and shapes of fitness landscapes. Statistica Sinica 17:1317-1342.
• Beerenwinkel N, Eriksson N, Sturmfels B (2007). Conjunctive Bayesian networks. Bernoulli 13:893–909.
• Beerenwinkel N, Dewey CN, Woods KM (2005). Parametric inference of recombination in HIV genomes. arXiv:q-bio/0512019.