  Bayesian networks and their accompanying graphical models are widely used for
prediction and analysis across many disciplines. We will reformulate these in
terms of linear maps. This reformulation will suggest a natural extension,
which we will show is equivalent to standard textbook quantum mechanics.
Therefore, this extension will be termed "quantum". However, the term "quantum"
should not be taken to imply this extension is necessarily only of utility in
situations traditionally thought of as in the domain of quantum mechanics. In
principle, it may be employed in any modeling situation, say forecasting the
weather or the stock market--it is up to experiment to determine if this
extension is useful in practice. Even restricting to the domain of quantum
mechanics, with this new formulation the advantages of Bayesian networks can be
maintained for models incorporating quantum and mixed classical-quantum
behavior. The use of these will be illustrated by various basic examples.
  Parrondo's paradox refers to the situation where two, multi-round games with
a fixed winning criteria, both with probability greater than one-half for one
player to win, are combined. Using a possibly biased coin to determine the rule
to employ for each round, paradoxically, the previously losing player now wins
the combined game with probability greater than one-half. Using the extended
Bayesian networks, we will formulate and analyze classical observed, classical
hidden, and quantum versions of a game that displays this paradox, finding
bounds for the discrepancy from naive expectations for the occurrence of the
paradox. A quantum paradox inspired by Parrondo's paradox will also be
analyzed. We will prove a bound for the discrepancy from naive expectations for
this paradox as well. Games involving quantum walks that achieve this bound
will be presented.
