  The seminal work of Chow and Liu (1968) shows that approximation of a finite
probabilistic system by Markov trees can achieve the minimum information loss
with the topology of a maximum spanning tree. Our current paper generalizes the
result to Markov networks of tree width $\leq k$, for every fixed $k\geq 2$. In
particular, we prove that approximation of a finite probabilistic system with
such Markov networks has the minimum information loss when the network topology
is achieved with a maximum spanning $k$-tree. While constructing a maximum
spanning $k$-tree is intractable for even $k=2$, we show that polynomial
algorithms can be ensured by a sufficient condition accommodated by many
meaningful applications. In particular, we prove an efficient algorithm for
learning the optimal topology of higher order correlations among random
variables that belong to an underlying linear structure.
