  There exists a dichotomy between classical probabilistic graphical models,
such as Bayesian networks (BNs), and modern tractable models, such as
sum-product networks (SPNs). The former generally have intractable inference,
but provide a high level of interpretability, while the latter admits a wide
range of tractable inference routines, but are typically harder to interpret.
Due to this dichotomy, tools to convert between BNs and SPNs are desirable.
While one direction -- compiling BNs into SPNs -- is well discussed in
Darwiche's seminal work on arithmetic circuit compilation, the converse
direction -- decompiling SPNs into BNs -- has received surprisingly little
attention.
  In this paper, we fill this gap by proposing SPN2BN, an algorithm that
decompiles an SPN into a BN. SPN2BN has several salient features when compared
to the only other two works decompiling SPNs. Most significantly, the BNs
returned by SPN2BN are minimal independence-maps that are more parsimonious
with respect to the introduction of latent variables. Secondly, the output BN
produced by SPN2BN can be precisely characterized with respect to a compiled
BN. More specifically, a certain set of directed edges will be added to the
input BN, giving what we will call the moral-closure. Lastly, it is established
that our compilation-decompilation process is idempotent. This has practical
significance as it limits the size of the decompiled SPN.
