  Defining the extent of epistasis - the non-independence of the effects of
mutations - is essential for understanding the relationship of genotype,
phenotype, and fitness in biological systems. The applications cover many areas
of biological research, including biochemistry, genomics, protein and systems
engineering, medicine, and evolutionary biology. However, the quantitative
definitions of epistasis vary among fields, and its analysis beyond just
pairwise effects remains obscure in general. Here, we show that different
definitions of epistasis are versions of a single mathematical formalism - the
weighted Walsh-Hadamard transform. We discuss that one of the definitions, the
backgound-averaged epistasis, is the most informative when the goal is to
uncover the general epistatic structure of a biological system, a description
that can be rather different from the local epistatic structure of specific
model systems. Key issues are the choice of effective ensembles for averaging
and to practically contend with the vast combinatorial complexity of mutations.
In this regard, we discuss possible approaches for optimally learning the
epistatic structure of biological systems.
