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Latent tree analysis seeks to model the correlations among a set of random variables using a tree of latent variables. It was proposed as an improvement to latent class analysis — a method widely used in social sciences and medicine to identify homogeneous subgroups in a population. It provides new and fruitful perspectives on a number of machine learning areas, including cluster analysis, topic detection, and deep probabilistic modeling. This paper gives an overview of the research on latent tree analysis and various ways it is used in practice. Much of machine learning is about modeling and utilizing correlations among variables. In classification, the task is to establish relationships between attributes and class variables so that unseen data can be classified accurately. In Bayesian networks, dependencies among variables are represented as directed acyclic graphs and the graphs are used to facilitate efficient probabilistic inference. In topic models, word cooccurrences are accounted for by assuming that all words are generated probabilistically from the same set of topics, and the generation process is reverted via statistical inference to determine the topics. In deep belief networks, correlations among observed units are modeled using multiple levels of hidden units, and the top-level hidden units are used as a representation of the data for further analysis. Latent tree analysis (LTA) seeks to model the correlations among a set of observed variables using a tree model, called latent tree model (LTM), where the leaf nodes represent observed variables and the internal nodes represent latent variables.The dependence between two observed variables is explained by the path between them. Despite their simplicity, LTMs subsume two classes of models widely used in academic research. The first one is latent class models (LCMs) (Lazarsfeld and Henry, 1968; Knott and Bartholomew, 1999), which are LTMs with a single latent variable. They are used for categorical data clustering in social sciences and medicine. The second class is probabilistic phylogenetic trees (Durbin et al., 1998), which are a tool for determining the evolution history of a set of species. Phylogenetic trees are special LTMs where the model structures are binary (bifurcating) trees and all the Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. variables have the same number of possible states. LTA also provides new and fruitful perspectives on a number of machine learning areas. One area is cluster analysis. Here finite mixture models such as LCMs are commonly used. A finite mixture model has one latent variable and consequently it gives one soft partition of data. An LTM typically has multiple latent variables and hence LTA yields multiple soft partitions of data simultaneously. In other words, LTA performs multidimensional clustering (Chen et al., 2012; Liu et al., 2013). It is interesting because complex data usually have multiple facets and can be meaningfully clustered in multiple ways. Another area is topic detection. Applying LTA to text data, we can partition a collection of documents in multiple ways. The document clusters in the partitions can be interpreted as topics. Furthermore, it is possible to learn hierarchical LTMs where the latent variables are organized into multiple layers. This leads to an alternative method for hierarchical topic detection (Liu, Zhang, and Chen, 2014; Chen et al., 2016a), which has been shown to find more meaningful topics and topic hierarchies than the state-of-the-art method based on latent Dirichlet allocation (Paisley et al., 2015). The third area is deep probabilistic modeling. Hierarchical LTM and deep belief network (DBN) (Hinton, Osindero, and Teh, 2006) are similar in that they both consist of multiple layers of variables, with an observed layer at the bottom and multiple layers of hidden units on top of it. One difference is that, in DBN, units from adjacent layers are fully connected, while HLTM is tree-structured. It would be interesting to explore the middle ground between the two extreme and develop algorithms for learning what might be called sparse DBNs. Learning structures for deep models is an interesting open problem. Extension of LTA might offer one solution (Chen et al., 2016b). The concept of latent tree models was introduced in (Zhang, 2002, 2004), where they were referred to as hierarchical latent class models. The term “latent tree models” first appeared in (Zhang et al., 2008; Wang, Zhang, and Chen, 2008). Mourad et al. (2013) surveyed the research on latent tree models as of 2012 in details. This paper provides a concise overview of the methodology. The exposition are more conceptual and less technical than (Mourad et al., 2013). Developments after 2012 are also included. ar X iv :1 61 0. 00 08 5v 1 [ cs .L G ] 1 O ct 2 01 6
Latent Tree Analysis

In the mixture models problem it is assumed that there are K distributions θ1, . . . , θK and one gets to observe a sample from a mixture of these distributions with unknown coefficients. The goal is to associate instances with their generating distributions, or to identify the parameters of the hidden distributions. In this work we make the assumption that we have access to several samples drawn from the same K underlying distributions, but with different mixing weights. As with topic modeling, having multiple samples is often a reasonable assumption. Instead of pooling the data into one sample, we prove that it is possible to use the differences between the samples to better recover the underlying structure. We present algorithms that recover the underlying structure under milder assumptions than the current state of art when either the dimensionality or the separation is high. The methods, when applied to topic modeling, allow generalization to words not present in the training data.
Using multiple samples to learn mixture models

Fundamental discrepancy between first order logic and statistical inference (global versus local properties of universe) is shown to be the obstacle for integration of logic and probability in L.p. logic of Bacchus. To overcome the counterintuitiveness of L.p. behaviour, a 3-valued logic is proposed.

Beliefs and Probability in Bacchus’ l.p. Logic: A 3-Valued Logic Solution to Apparent Counter-intuition