  An information-theoretic development is given for the problem of compound
Poisson approximation, which parallels earlier treatments for Gaussian and
Poisson approximation. Let $P_{S_n}$ be the distribution of a sum $S_n=\Sumn
Y_i$ of independent integer-valued random variables $Y_i$. Nonasymptotic bounds
are derived for the distance between $P_{S_n}$ and an appropriately chosen
compound Poisson law. In the case where all $Y_i$ have the same conditional
distribution given $\{Y_i\neq 0\}$, a bound on the relative entropy distance
between $P_{S_n}$ and the compound Poisson distribution is derived, based on
the data-processing property of relative entropy and earlier Poisson
approximation results. When the $Y_i$ have arbitrary distributions,
corresponding bounds are derived in terms of the total variation distance. The
main technical ingredient is the introduction of two "information functionals,"
and the analysis of their properties. These information functionals play a role
analogous to that of the classical Fisher information in normal approximation.
Detailed comparisons are made between the resulting inequalities and related
bounds.
