  Williams and Beer (2010) proposed a nonnegative mutual information
decomposition, based on the construction of information gain lattices, which
allows separating the information that a set of variables contains about
another into components interpretable as the unique information of one
variable, or redundant and synergy components. In this work we extend the
framework of Williams and Beer (2010) focusing on the lattices that underpin
the decomposition. We generalize the type of constructible lattices and examine
the relations between the terms in different lattices, for example relating
bivariate and trivariate decompositions. We point out that, in information gain
lattices, redundancy components are invariant across decompositions, but unique
and synergy components are decomposition-dependent. Exploiting the connection
between different lattices we propose a procedure to construct, in the general
multivariate case, information decompositions from measures of synergy or
unique information. We introduce an alternative type of mutual information
decompositions based on information loss lattices, with the role and invariance
properties of redundancy and synergy components exchanged with respect to gain
lattices. We study the correspondence between information gain and information
loss lattices and we define dual decompositions that allow overcoming the
intrinsic asymmetry between invariant and decomposition-dependent components,
which hinders the consistent joint characterization of synergy and redundancy.
