  We generalize the exponential family of probability distributions. In our
approach, the exponential function is replaced by a $\varphi$-function,
resulting in a $\varphi$-family of probability distributions. We show how
$\varphi$-families are constructed. In a $\varphi$-family, the analogue of the
cumulant-generating function is a normalizing function. We define the
$\varphi$-divergence as the Bregman divergence associated to the normalizing
function, providing a generalization of the Kullback-Leibler divergence. A
formula for the $\varphi$-divergence where the $\varphi$-function is the
Kaniadakis' $\kappa$-exponential function is derived.
