Ancestral graphs are a prevalent mathematical tool to take into account latent (hidden) variables in a probabilistic graphical model. In ancestral graph representations, the nodes are only the observed (manifest) variables and the notion of m-separation fully characterizes the conditional independence relations among such variables, bypassing the need to explicitly consider latent variables. However, ancestral graph models do not necessarily represent the actual causal structure of the model, and do not contain information about, for example, the precise number and location of the hidden variables. Being able to detect the presence of latent variables while also inferring their precise location within the actual causal structure model is a more challenging task that provides more information about the actual causal relationships among all the model variables, including the latent ones. In this article, we develop an algorithm to exactly recover graphical models of random variables with underlying polytree structures when the latent nodes satisfy specific degree conditions. Therefore, this article proposes an approach for the full identification of hidden variables in a polytree. We also show that the algorithm is complete in the sense that when such degree conditions are not met, there exists another polytree with fewer number of latent nodes satisfying the degree conditions and entailing the same independence relations among the observed variables, making it indistinguishable from the actual polytree.

Learning causal structures with latent variables is a major challenge. This paper takes a shot at one of the simplest cases, polytree causal networks. While this is a limited special case, the ideas and methods may be useful more generally. It is interesting that the method needs only second and third order statistics of the observed variables. The paper would benefit from greater clarity in several areas.

After a discussion among reviewers, this submission has been considered a solid theoretical contribution to the specific problem of learning polytree networks (with latent variables). It is important to understand what can be learned. The work also proposes algorithms to recover the network under some conditions. On the downside, there is no good motivations and no evaluation of any kind. The latter is hard to be resolved quickly, but an effort could be put in improving the motivations and potential impact.

Ancestral graphs are a prevalent mathematical tool to take into account latent (hidden) variables in a probabilistic graphical model. In ancestral graph representations, the nodes are only the observed (manifest) variables and the notion of m-separation fully characterizes the conditional independence relations among such variables, bypassing the need to explicitly consider latent variables. However, ancestral graph models do not necessarily represent the actual causal structure of the model, and do not contain information about, for example, the precise number and location of the hidden variables. Being able to detect the presence of latent variables while also inferring their precise location within the actual causal structure model is a more challenging task that provides more information about the actual causal relationships among all the model variables, including the latent ones. In this article, we develop an algorithm to exactly recover graphical models of random variables with underlying polytree structures when the latent nodes satisfy specific degree conditions. Therefore, this article proposes an approach for the full identification of hidden variables in a polytree.

Ancestral graphs are a prevalent mathematical tool to take into account latent (hidden) variables in a probabilistic graphical model. In ancestral graph representations, the nodes are only the observed (manifest) variables and the notion of m-separation fully characterizes the conditional independence relations among such variables, bypassing the need to explicitly consider latent variables. However, ancestral graph models do not necessarily represent the actual causal structure of the model, and do not contain information about, for example, the precise number and location of the hidden variables. Being able to detect the presence of latent variables while also inferring their precise location within the actual causal structure model is a more challenging task that provides more information about the actual causal relationships among all the model variables, including the latent ones. In this article, we develop an algorithm to exactly recover graphical models of random variables with underlying polytree structures when the latent nodes satisfy specific degree conditions.

Unlike task-specific algorithms, Deep Learning is a part of Machine Learning family based on learning data representations. With massive amounts of computational power, machines can now recognize objects and translate speech in real time, enabling a smart Artificial intelligence in systems. The concept of a software simulating the neocortex's large array of neurons in an artificial neural network is decades old, and it has led to as many disappointments as breakthroughs. But because of improvements in mathematical formulas and increasingly powerful computers, today researchers & data scientists can model many more layers of virtual neurons than ever before. "Recent improvements in Deep Learning has reignited some of the grand challenges in Artificial Intelligence."

It's notoriously difficult to make sense of Quantum mechanics, and it's equally difficult to calculate the behavior of many quantum systems. That's due in part to the description of a quantum system called its wavefunction. The wavefunction for most single objects is pretty complicated on its own, and adding a second object makes predicting things even harder, since the wavefunction for the entire system becomes a mixture of the two individual ones. The more objects you add, the harder the calculations become. As a result, many-body calculations are usually done through methods that produce an approximation.

We propose Bayesian AutoEncoder (BAE) in order to construct a recognition system which uses feedback information. BAE constructs a generative model of input data as a Bayes Net. The network trained by BAE obtains its hidden variables as the features of given data. It can execute inference for each variable through belief propagation, using both feedforward and feedback information. We confirmed that BAE can construct small networks with one hidden layer and extract features as hidden variables from 3x3 and 5x5 pixel input data.