Polynomial-time derivation of optimal k-tree topology from Markov networks

Characterization of joint probability distribution for large networks of random variables remains a challenging task in data science. Probabilistic graph approximation with simple topologies has practically been resorted to; typically the tree topology makes joint probability computation much simpler and can be effective for statistical inference on insufficient data. However, to characterize network components where multiple variables cooperate closely to influence others, model topologies beyond a tree are needed, which unfortunately are infeasible to acquire. In particular, our previous work has related optimal approximation of Markov networks of tree-width k >=2 closely to the graph-theoretic problem of finding maximum spanning k-tree (MSkT), which is a provably intractable task. This paper investigates optimal approximation of Markov networks with k-tree topology that retains some designated underlying subgraph. Such a subgraph may encode certain background information that arises in scientific applications, for example, about a known significant pathway in gene networks or the indispensable backbone connectivity in the residue interaction graphs for a biomolecule 3D structure. In particular, it is proved that the \beta-retaining MSkT problem, for a number of classes \beta of graphs, admit O(n^{k+1})-time algorithms for every fixed k>= 1. These \beta-retaining MSkT algorithms offer efficient solutions for approximation of Markov networks with k-tree topology in the situation where certain persistent information needs to be retained.

α-HMM: A Graphical Model for RNA Folding

RNA secondary structure is modeled with the novel arbitrary-order hidden Markov model ({\alpha}-HMM). The {\alpha}-HMM extends over the traditional HMM with capability to model stochastic events that may be in influenced by historically distant ones, making it suitable to account for long-range canonical base pairings between nucleotides, which constitute the RNA secondary structure. Unlike previous heavy-weight extensions over HMM, the {\alpha}-HMM has the flexibility to apply restrictions on how one event may influence another in stochastic processes, enabling efficient prediction of RNA secondary structure including pseudoknots.

Efficient Learning of Optimal Markov Network Topology with k-Tree Modeling

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.