58,071 research outputs found
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
On the Relationship between Sum-Product Networks and Bayesian Networks
Sum-Product Networks (SPNs), which are probabilistic inference machines, have attracted a lot of interests in recent years. They have a wide range of applications, including but not limited to activity modeling, language modeling and speech modeling. Despite their practical applications and popularity, little research has been done in understanding what is the connection and difference between Sum-Product Networks and traditional graphical models, including Bayesian Networks (BNs) and Markov Networks (MNs). In this thesis, I establish some theoretical connections between Sum-Product Networks and Bayesian Networks. First, I prove that every SPN can be converted into a BN in linear time and space in terms of the network size. Second, I show that by applying the Variable Elimination algorithm (VE) to the generated BN, I can recover the original SPN.
In the first direction, I 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. I establish the first connection between the depth of SPNs and the tree-width of the generated BNs, showing that the depth of SPNs is proportional to a lower bound of the tree-width of the BN.
In the other direction, I show that by applying the Variable Elimination algorithm (VE) to the generated BN with ADD representations, I 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, I introduce the notion of {\em normal} SPN and present a theoretical analysis of the consistency and decomposability properties. I provide constructive algorithms to transform any given SPN into its normal form in time and space quadratic in the size of the SPN. Combining the above two directions gives us a deep understanding about the modeling power of SPNs and their inner working mechanism
Bayesian Learning of Sum-Product Networks
Sum-product networks (SPNs) are flexible density estimators and have received
significant attention due to their attractive inference properties. While
parameter learning in SPNs is well developed, structure learning leaves
something to be desired: Even though there is a plethora of SPN structure
learners, most of them are somewhat ad-hoc and based on intuition rather than a
clear learning principle. In this paper, we introduce a well-principled
Bayesian framework for SPN structure learning. First, we decompose the problem
into i) laying out a computational graph, and ii) learning the so-called scope
function over the graph. The first is rather unproblematic and akin to neural
network architecture validation. The second represents the effective structure
of the SPN and needs to respect the usual structural constraints in SPN, i.e.
completeness and decomposability. While representing and learning the scope
function is somewhat involved in general, in this paper, we propose a natural
parametrisation for an important and widely used special case of SPNs. These
structural parameters are incorporated into a Bayesian model, such that
simultaneous structure and parameter learning is cast into monolithic Bayesian
posterior inference. In various experiments, our Bayesian SPNs often improve
test likelihoods over greedy SPN learners. Further, since the Bayesian
framework protects against overfitting, we can evaluate hyper-parameters
directly on the Bayesian model score, waiving the need for a separate
validation set, which is especially beneficial in low data regimes. Bayesian
SPNs can be applied to heterogeneous domains and can easily be extended to
nonparametric formulations. Moreover, our Bayesian approach is the first, which
consistently and robustly learns SPN structures under missing data.Comment: NeurIPS 2019; See conference page for supplemen
Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks
We discuss the computational complexity of approximating maximum a posteriori
inference in sum-product networks. We first show NP-hardness in trees of height
two by a reduction from maximum independent set; this implies
non-approximability within a sublinear factor. We show that this is a tight
bound, as we can find an approximation within a linear factor in networks of
height two. We then show that, in trees of height three, it is NP-hard to
approximate the problem within a factor for any sublinear function
of the size of the input . Again, this bound is tight, as we prove that
the usual max-product algorithm finds (in any network) approximations within
factor for some constant . Last, we present a simple
algorithm, and show that it provably produces solutions at least as good as,
and potentially much better than, the max-product algorithm. We empirically
analyze the proposed algorithm against max-product using synthetic and
realistic networks.Comment: 18 page
A Model of Consistent Node Types in Signed Directed Social Networks
Signed directed social networks, in which the relationships between users can
be either positive (indicating relations such as trust) or negative (indicating
relations such as distrust), are increasingly common. Thus the interplay
between positive and negative relationships in such networks has become an
important research topic. Most recent investigations focus upon edge sign
inference using structural balance theory or social status theory. Neither of
these two theories, however, can explain an observed edge sign well when the
two nodes connected by this edge do not share a common neighbor (e.g., common
friend). In this paper we develop a novel approach to handle this situation by
applying a new model for node types. Initially, we analyze the local node
structure in a fully observed signed directed network, inferring underlying
node types. The sign of an edge between two nodes must be consistent with their
types; this explains edge signs well even when there are no common neighbors.
We show, moreover, that our approach can be extended to incorporate directed
triads, when they exist, just as in models based upon structural balance or
social status theory. We compute Bayesian node types within empirical studies
based upon partially observed Wikipedia, Slashdot, and Epinions networks in
which the largest network (Epinions) has 119K nodes and 841K edges. Our
approach yields better performance than state-of-the-art approaches for these
three signed directed networks.Comment: To appear in the IEEE/ACM International Conference on Advances in
Social Network Analysis and Mining (ASONAM), 201
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