  Notwithstanding various attempts to construct a Partial Information
Decomposition (PID) for multiple variables by defining synergistic, redundant,
and unique information, there is no consensus on how one ought to precisely
define either of these quantities. One aim here is to illustrate how that
ambiguity -- or, more positively, freedom of choice -- may arise. Using the
basic idea that information equals the average reduction in uncertainty when
going from an initial to a final probability distribution, synergistic
information will likewise be defined as a difference between two entropies. One
term is uncontroversial and characterizes ``the whole'' information that source
variables carry jointly about a target variable $\varT$. The other term then is
meant to characterize the information carried by the ``sum of its parts.'' Here
we interpret that concept as needing a suitable probability distribution
aggregated (``pooled'') from multiple marginal distributions (the parts).
Ambiguity arises in the definition of the optimum way to pool two (or more)
probability distributions. Independent of the exact definition of optimum
pooling, the concept of pooling leads to a lattice that differs from the
often-used redundancy-based lattice. One can associate not just a number (an
average entropy) with each node of the lattice, but (pooled) probability
distributions. As an example,one simple and reasonable approach to pooling is
presented, which naturally gives rise to the overlap between different
probability distributions as being a crucial quantity that characterizes both
synergistic and unique information.
