  Understanding how different information sources together transmit information
is crucial in many domains. For example, understanding the neural code requires
characterizing how different neurons contribute unique, redundant, or
synergistic pieces of information about sensory or behavioral variables.
Williams and Beer (2010) proposed a partial information decomposition (PID)
which separates the mutual information that a set of sources contains about a
set of targets into nonnegative terms interpretable as these pieces.
Quantifying redundancy requires assigning an identity to different information
pieces, to assess when information is common across sources. Harder et al.
(2013) proposed an identity axiom stating that there cannot be redundancy
between two independent sources about a copy of themselves. However,
Bertschinger et al. (2012) showed that with a deterministically related
sources-target copy this axiom is incompatible with ensuring PID nonnegativity.
Here we study systematically the effect of deterministic target-sources
dependencies. We introduce two synergy stochasticity axioms that generalize the
identity axiom, and we derive general expressions separating stochastic and
deterministic PID components. Our analysis identifies how negative terms can
originate from deterministic dependencies and shows how different assumptions
on information identity, implicit in the stochasticity and identity axioms,
determine the PID structure. The implications for studying neural coding are
discussed.
