On the Relationship between Sum-Product Networks and Bayesian Networks

In this paper, we establish some theoretical connections between Sum-Product Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be converted into a BN in linear time and space in terms of the network size. The key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent the local conditional probability distributions at each node in the resulting BN by exploiting context-specific independence (CSI). The generated BN has a simple directed bipartite graphical structure. We show that by applying the Variable Elimination algorithm (VE) to the generated BN with ADD representations, we can recover the original SPN where the SPN can be viewed as a history record or caching of the VE inference process. To help state the proof clearly, we introduce the notion of {\em normal} SPN and present a theoretical analysis of the consistency and decomposability properties. We conclude the paper with some discussion of the implications of the proof and establish a connection between the depth of an SPN and a lower bound of the tree-width of its corresponding BN.Comment: Full version of the same paper to appear at ICML-201

Directly Learning Tractable Models for Sequential Inference and DecisionMaking

Probabilistic graphical models such as Bayesian networks and Markov networks provide a general framework to represent multivariate distributions while exploiting conditional independence. Over the years, many approaches have been proposed to learn the structure of those networks. However, even if the resulting network is small, inference may be intractable (e.g., exponential in the size of the network) and practitioners must often resort to approximate inference techniques. Recent work has focused on the development of alternative graphical models such as arithmetic circuits (ACs) and sum-product networks (SPNs) for which inference is guaranteed to be tractable (e.g., linear in the size of the network for SPNs and ACs). This means that the networks learned from data can be directly used for inference without any further approximation. So far, previous work has focused on learning models with only random variables and for a fixed number of variables based on fixed-length data. In this thesis, I present two new probabilistic graphical models: Dynamic Sum-Product Networks (DynamicSPNs) and Decision Sum-Product-Max Networks (DecisionSPMNs), where the former is suitable for problems with sequence data of varying length and the latter is for problems with random, decision, and utility variables. Similar to SPNs and ACs, DynamicSPNs and DecisionSPMNs can be learned directly from data with guaranteed tractable exact inference and decision making in the resulting models. I also present a new online Bayesian discriminative learning algorithm for Selective Sum-Product Networks (SSPNs), which are a special class of SPNs with no latent variables. This new learning algorithm achieves tractability by utilizing a novel idea of mode matching, where the algorithm chooses a tractable distribution that matches the mode of the exact posterior after processing each training instance. This approach lends itself naturally to distributed learning since the data can be divided into subsets based on which partial posteriors are computed by different machines and combined into a single posterior

Decision Sum-Product-Max Networks

Sum-Product Networks (SPNs) were recently proposed as a new class of probabilistic graphical models that guarantee tractable inference, even on models with high-treewidth. In this paper, we propose a new extension to SPNs, called Decision Sum-Product-Max Networks (Decision-SPMNs), that makes SPNs suitable for discrete multi-stage decision problems. We present an algorithm that solves Decision-SPMNs in a time that is linear in the size of the network. We also present algorithms to learn the parameters of the network from data

Dynamic Sum Product Networks for Tractable Inference on Sequence Data

Abstract Sum-Product Networks (SPN) have recently emerged as a new class of tractable probabilistic models. Unlike Bayesian networks and Markov networks where inference may be exponential in the size of the network, inference in SPNs is in time linear in the size of the network. Since SPNs represent distributions over a fixed set of variables only, we propose dynamic sum product networks (DSPNs) as a generalization of SPNs for sequence data of varying length. A DSPN consists of a template network that is repeated as many times as needed to model data sequences of any length. We present a local search technique to learn the structure of the template network. In contrast to dynamic Bayesian networks for which inference is generally exponential in the number of variables per time slice, DSPNs inherit the linear inference complexity of SPNs. We demonstrate the advantages of DSPNs over DBNs and other models on several datasets of sequence data