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

Automatic Bayesian Density Analysis

Making sense of a dataset in an automatic and unsupervised fashion is a challenging problem in statistics and AI. Classical approaches for {exploratory data analysis} are usually not flexible enough to deal with the uncertainty inherent to real-world data: they are often restricted to fixed latent interaction models and homogeneous likelihoods; they are sensitive to missing, corrupt and anomalous data; moreover, their expressiveness generally comes at the price of intractable inference. As a result, supervision from statisticians is usually needed to find the right model for the data. However, since domain experts are not necessarily also experts in statistics, we propose Automatic Bayesian Density Analysis (ABDA) to make exploratory data analysis accessible at large. Specifically, ABDA allows for automatic and efficient missing value estimation, statistical data type and likelihood discovery, anomaly detection and dependency structure mining, on top of providing accurate density estimation. Extensive empirical evidence shows that ABDA is a suitable tool for automatic exploratory analysis of mixed continuous and discrete tabular data.Comment: In proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19

Parameter and Structure Learning Techniques for Sum Product Networks

Probabilistic graphical models (PGMs) provide a general and flexible framework for reasoning about complex dependencies in noisy domains with many variables. Among the various types of PGMs, sum-product networks (SPNs) have recently generated some interest because exact inference can always be done in linear time with respect to the size of the network. This is particularly attractive since it means that learning an SPN from data always yields a tractable model for inference. Learning the parameters and the structure for SPNs is being explored by various researchers, having algorithms that scale are essential in the era of big data. In this thesis, I present tractable parameter and structure learning techniques for SPNs. First, I propose a new Bayesian moment matching (BMM) algorithm to learn the parameters for SPNs generatively. BMM operates naturally in an online fashion and that can be easily distributed. I demonstrate the effectiveness and scalability of BMM in comparison to other online algorithms in the literature. Second, I present a discriminative learning algorithm for SPNs based on the Extended Baum-Welch (EBW) algorithm. The experiments show that this algorithm performs better than both generative Expectation-Maximization, and discriminative gradient descent on a wide variety of applications. I also demonstrate the robustness of the algorithm in the case of missing features by comparing its performance to Support Vector Machines and Neural Networks. Finally, I present the first online structure learning algorithm for recurrent SPNs. Recurrent SPNs were proposed by Mazen et. al to model sequential data. They also proposed a structure learning algorithm which is slow, and it only operates in batch mode. I present the first online algorithm to learn the structure of recurrent SPNs. I also show how the parameters can be learned simultaneously using a modified version of hard-EM algorithm. I compare the performance of the algorithm against different models on sequential data problems