Flexible sampling of discrete data correlations without the marginal distributions

Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parametrization of joint distributions. Among other properties, copulas provide a recipe for combining flexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. Inference is typically done in a Bayesian framework with Gaussian copulas, and it is complicated by the fact this implies sampling within a space where the number of constraints increases quadratically with the number of data points. The result is slow mixing when using off-the-shelf Gibbs sampling. We present an efficient algorithm based on recent advances on constrained Hamiltonian Markov chain Monte Carlo that is simple to implement and does not require paying for a quadratic cost in sample size.

Ranking relations using analogies in biological and information networks

Analogical reasoning depends fundamentally on the ability to learn and generalize about relations between objects. We develop an approach to relational learning which, given a set of pairs of objects $\mathbf{S}=\{A^{(1)}:B^{(1)},A^{(2)}:B^{(2)},\ldots,A^{(N)}:B ^{(N)}\}$, measures how well other pairs A:B fit in with the set $\mathbf{S}$. Our work addresses the following question: is the relation between objects A and B analogous to those relations found in $\mathbf{S}$? Such questions are particularly relevant in information retrieval, where an investigator might want to search for analogous pairs of objects that match the query set of interest. There are many ways in which objects can be related, making the task of measuring analogies very challenging. Our approach combines a similarity measure on function spaces with Bayesian analysis to produce a ranking. It requires data containing features of the objects of interest and a link matrix specifying which relationships exist; no further attributes of such relationships are necessary. We illustrate the potential of our method on text analysis and information networks. An application on discovering functional interactions between pairs of proteins is discussed in detail, where we show that our approach can work in practice even if a small set of protein pairs is provided.

Learning Measurement Models for Unobserved Variables

Observed associations in a database may be due in whole or part to variations in unrecorded (latent) variables. Identifying such variables and their causal relationships with one another is a principal goal in many scientific and practical domains. Previous work shows that, given a partition of observed variables such that members of a class share only a single latent common cause, standard search algorithms for causal Bayes nets can infer structural relations between latent variables. We introduce an algorithm for discovering such partitions when they exist. Uniquely among available procedures, the algorithm is (asymptotically) correct under standard assumptions in causal Bayes net search algorithms, requires no prior knowledge of the number of latent variables, and does not depend on the mathematical form of the relationships among the latent variables. We evaluate the algorithm on a variety of simulated data sets.

Latent Composite Likelihood Learning for the Structured Canonical Correlation Model

Latent variable models are used to estimate variables of interest quantities which are observable only up to some measurement error. In many studies, such variables are known but not precisely quantifiable (such as "job satisfaction" in social sciences and marketing, "analytical ability" in educational testing, or "inflation" in economics). This leads to the development of measurement instruments to record noisy indirect evidence for such unobserved variables such as surveys, tests and price indexes. In such problems, there are postulated latent variables and a given measurement model. At the same time, other unantecipated latent variables can add further unmeasured confounding to the observed variables. The problem is how to deal with unantecipated latents variables. In this paper, we provide a method loosely inspired by canonical correlation that makes use of background information concerning the "known" latent variables. Given a partially specified structure, it provides a structure learning approach to detect "unknown unknowns," the confounding effect of potentially infinitely many other latent variables. This is done without explicitly modeling such extra latent factors. Because of the special structure of the problem, we are able to exploit a new variation of composite likelihood fitting to efficiently learn this structure. Validation is provided with experiments in synthetic data and the analysis of a large survey done with a sample of over 100,000 staff members of the National Health Service of the United Kingdom.

Bayesian Inference for Gaussian Mixed Graph Models

We introduce priors and algorithms to perform Bayesian inference in Gaussian models defined by acyclic directed mixed graphs. Such a class of graphs, composed of directed and bi-directed edges, is a representation of conditional independencies that is closed under marginalization and arises naturally from causal models which allow for unmeasured confounding. Monte Carlo methods and a variational approximation for such models are presented. Our algorithms for Bayesian inference allow the evaluation of posterior distributions for several quantities of interest, including causal effects that are not identifiable from data alone but could otherwise be inferred where informative prior knowledge about confounding is available.

Mixed Cumulative Distribution Networks

Directed acyclic graphs (DAGs) are a popular framework to express multivariate probability distributions. Acyclic directed mixed graphs (ADMGs) are generalizations of DAGs that can succinctly capture much richer sets of conditional independencies, and are especially useful in modeling the effects of latent variables implicitly. Unfortunately there are currently no good parameterizations of general ADMGs. In this paper, we apply recent work on cumulative distribution networks and copulas to propose one one general construction for ADMG models. We consider a simple parameter estimation approach, and report some encouraging experimental results.

Gaussian Process Structural Equation Models with Latent Variables

In a variety of disciplines such as social sciences, psychology, medicine and economics, the recorded data are considered to be noisy measurements of latent variables connected by some causal structure. This corresponds to a family of graphical models known as the structural equation model with latent variables. While linear non-Gaussian variants have been well-studied, inference in nonparametric structural equation models is still underdeveloped. We introduce a sparse Gaussian process parameterization that defines a non-linear structure connecting latent variables, unlike common formulations of Gaussian process latent variable models. The sparse parameterization is given a full Bayesian treatment without compromising Markov chain Monte Carlo efficiency. We compare the stability of the sampling procedure and the predictive ability of the model against the current practice.

Measuring Latent Causal Structure

Discovering latent representations of the observed world has become increasingly more relevant in data analysis. Much of the effort concentrates on building latent variables which can be used in prediction problems, such as classification and regression. A related goal of learning latent structure from data is that of identifying which hidden common causes generate the observations, such as in applications that require predicting the effect of policies. This will be the main problem tackled in our contribution: given a dataset of indicators assumed to be generated by unknown and unmeasured common causes, we wish to discover which hidden common causes are those, and how they generate our data. This is possible under the assumption that observed variables are linear functions of the latent causes with additive noise. Previous results in the literature present solutions for the case where each observed variable is a noisy function of a single latent variable. We show how to extend the existing results for some cases where observed variables measure more than one latent variable.