We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable, meaning that every latent variable has four children that do not have any other parents in common. We show that the existence of such a quartet allows us to uniquely identify each latent variable and to learn all parameters involving that latent variable. Underlying our algorithm are two new techniques for structure learning: a quartet test to determine whether a set of binary variables are singly coupled, and a conditional mutual information test that we use to learn parameters. We also show how to subtract already learned latent variables from the model to create new singly-coupled quartets, which substantially expands the class of structures that we can learn. Finally, we give a proof of the polynomial sample complexity of our learning algorithm, and experimentally compare it to variational EM.

The presented ideas are based on a recent paper [Halpern & Sontag, 2013], extending to a more general class of networks. The performance of the proposed algorithm is illustrated on synthetic examples. The exposition of paper is generally logical but the writing needs a lot of improvement. The paper contains potentially interesting algorithms but it is at times hard to tell which ideas are new and which ones are borrowed. A summary of the main contributions and their significance would be useful.

We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable. For each latent variable, the existence of a singly-coupled quartet allows us to uniquely identify and learn all parameters involving that latent variable. We give a proof of the polynomial sample complexity of our learning algorithm, and experimentally compare it to variational EM.

An important class of problems can be cast as inference in noisy(cid:173) OR Bayesian networks, where the binary state of each variable is a logical OR of noisy versions of the states of the variable's par(cid:173) ents. For example, in medical diagnosis, the presence of a symptom can be expressed as a noisy-OR of the diseases that may cause the symptom - on some occasions, a disease may fail to activate the symptom. Inference in richly-connected noisy-OR networks is in(cid:173) tractable, but approximate methods (e .g., variational techniques) are showing increasing promise as practical solutions. One prob(cid:173) lem with most approximations is that they tend to concentrate on a relatively small number of modes in the true posterior, ig(cid:173) noring other plausible configurations of the hidden variables. We introduce a new sequential variational method for bipartite noisy(cid:173) OR networks, that favors including all modes of the true posterior and models the posterior distribution as a tree.

We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable, meaning that every latent variable has four children that do not have any other parents in common. We show that the existence of such a quartet allows us to uniquely identify each latent variable and to learn all parameters involving that latent variable. Underlying our algorithm are two new techniques for structure learning: a quartet test to determine whether a set of binary variables are singly coupled, and a conditional mutual information test that we use to learn parameters.

One of the most surprising and exciting discoveries in supervising learning was the benefit of overparametrization (i.e. training a very large model) to improving the optimization landscape of a problem, with minimal effect on statistical performance (i.e. generalization). In contrast, unsupervised settings have been under-explored, despite the fact that it has been observed that overparameterization can be helpful as early as Dasgupta & Schulman (2007). In this paper, we perform an exhaustive study of different aspects of overparameterization in unsupervised learning via synthetic and semi-synthetic experiments. We discuss benefits to different metrics of success (held-out log-likelihood, recovering the parameters of the ground-truth model), sensitivity to variations of the training algorithm, and behavior as the amount of overparameterization increases. We find that, when learning using methods such as variational inference, larger models can significantly increase the number of ground truth latent variables recovered.

We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable, meaning that every latent variable has four children that do not have any other parents in common. We show that the existence of such a quartet allows us to uniquely identify each latent variable and to learn all parameters involving that latent variable. Underlying our algorithm are two new techniques for structure learning: a quartet test to determine whether a set of binary variables are singly coupled, and a conditional mutual information test that we use to learn parameters. We also show how to subtract already learned latent variables from the model to create new singly-coupled quartets, which substantially expands the class of structures that we can learn. Finally, we give a proof of the polynomial sample complexity of our learning algorithm, and experimentally compare it to variational EM.

These techniques become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the bounds by numerical experiments.

In the literature on graphical models, there has been increased attention paid to the problems of learning hidden structure (see Heckerman [H96] for survey) and causal mechanisms from sample data [H96, P88, S93, P95, F98]. In most settings we should expect the former to be difficult, and the latter potentially impossible without experimental intervention. In this work, we examine some restricted settings in which perfectly reconstruct the hidden structure solely on the basis of observed sample data.