One of the fundamental problems in machine learning is the estimation of a probability distribution from data. Many techniques have been proposed to study the structure of data, most often building around the assumption that observations lie on a lower-dimensional manifold of high probability. It has been more difficult, however, to exploit this insight to build explicit, tractable density models for high-dimensional data. In this paper, we introduce the deep density model (DDM), a new approach to density estimation. We exploit insights from deep learning to construct a bijective map to a representation space, under which the transformation of the distribution of the data is approximately factorized and has identical and known marginal densities. The simplicity of the latent distribution under the model allows us to feasibly explore it, and the invertibility of the map to characterize contraction of measure across it. This enables us to compute normalized densities for out-of-sample data. This combination of tractability and flexibility allows us to tackle a variety of probabilistic tasks on high-dimensional datasets, including: rapid computation of normalized densities at test-time without evaluating a partition function; generation of samples without MCMC; and characterization of the joint entropy of the data.

When modeling a probability distribution with a Bayesian network, we are faced with the problem of how to handle continuous variables. Most previous work has either solved the problem by discretizing, or assumed that the data are generated by a single Gaussian. In this paper we abandon the normality assumption and instead use statistical methods for nonparametric density estimation. For a naive Bayesian classifier, we present experimental results on a variety of natural and artificial domains, comparing two methods of density estimation: assuming normality and modeling each conditional distribution with a single Gaussian; and using nonparametric kernel density estimation. We observe large reductions in error on several natural and artificial data sets, which suggests that kernel estimation is a useful tool for learning Bayesian models.

This paper considers a classic problem in machine learning and theoretical computer science, namely graph clustering, i.e., given an undirected unweighted graph, partition the nodes into disjoint clusters, so that the density of edges within one cluster is higher than those across clusters. Graph clustering arises naturally in many application across science and engineering. Some prominent examples include community detection in social network Mishra et al. [2007], submarket identification in E-commerce and sponsored search Yahoo!-Inc [2009], and co-authorship analysis in analyzing document database Ester et al. [1995], among others. From a purely binary classification theoretical point of view, the edges of the graph are (noisy) labels of similarity or affinity between pairs of objects, and the concept class consists of clusterings of the objects (encoded graphically by identifying clusters with cliques). Many theoretical results in graph clustering [e.g., Boppana, 1987, Chen et al., 2012, McSherry, 2001] consider the planted partition model, in which the edges are generated randomly; see Section 1.1 for more details. While numerous different methods have been proposed, their performance guarantees all share the following manner - under certain condition of the density of edges (within clusters and across clusters), the proposed method succeeds to recover the correct clusters exactly if all clusters are larger than a threshold size, typically Ω( n).

Dawid, Kjaerulff and Lauritzen (1994) provided a preliminary description of a hybrid between Monte-Carlo sampling methods and exact local computations in junction trees. Utilizing the strengths of both methods, such hybrid inference methods has the potential of expanding the class of problems which can be solved under bounded resources as well as solving problems which otherwise resist exact solutions. The paper provides a detailed description of a particular instance of such a hybrid scheme; namely, combination of exact inference and Gibbs sampling in discrete Bayesian networks. We argue that this combination calls for an extension of the usual message passing scheme of ordinary junction trees.

Stochastic simulation algorithms such as likelihood weighting often give fast, accurate approximations to posterior probabilities in probabilistic networks, and are the methods of choice for very large networks. Unfortunately, the special characteristics of dynamic probabilistic networks (DPNs), which are used to represent stochastic temporal processes, mean that standard simulation algorithms perform very poorly. In essence, the simulation trials diverge further and further from reality as the process is observed over time. In this paper, we present simulation algorithms that use the evidence observed at each time step to push the set of trials back towards reality. The first algorithm, "evidence reversal" (ER) restructures each time slice of the DPN so that the evidence nodes for the slice become ancestors of the state variables. The second algorithm, called "survival of the fittest" sampling (SOF), "repopulates" the set of trials at each time step using a stochastic reproduction rate weighted by the likelihood of the evidence according to each trial. We compare the performance of each algorithm with likelihood weighting on the original network, and also investigate the benefits of combining the ER and SOF methods. The ER/SOF combination appears to maintain bounded error independent of the number of time steps in the simulation.

Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives of the density functions have received only relatively scant attention. Kernel estimators of density functions are widely used as they exhibit excellent theoretical and practical properties, though their generalization to density derivatives has progressed more slowly due to the mathematical intractabilities encountered in the crucial problem of bandwidth (or smoothing parameter) selection. This paper presents the first fully automatic, data-based bandwidth selectors for multivariate kernel density derivative estimators. This is achieved by synthesizing recent advances in matrix analytic theory which allow mathematically and computationally tractable representations of higher order derivatives of multivariate vector valued functions. The theoretical asymptotic properties as well as the finite sample behaviour of the proposed selectors are studied. {In addition, we explore in detail the applications of the new data-driven methods for two other statistical problems: clustering and bump hunting. The introduced techniques are combined with the mean shift algorithm to develop novel automatic, nonparametric clustering procedures which are shown to outperform mixture-model cluster analysis and other recent nonparametric approaches in practice. Furthermore, the advantage of the use of smoothing parameters designed for density derivative estimation for feature significance analysis for bump hunting is illustrated with a real data example.

Our study is broadly motivated by questions in high-dimensional learning. As is well known, learning in high dimensions is feasible only if the data distribution satisfies suitable prior assumptions. One such assumption is that the data distribution lies on, or is close to, a low-dimensional set embedded in a high dimensional space, for instance a low dimensional manifold. This latter assumption has proved to be useful in practice, as well as amenable to theoretical analysis, and it has led to a significant amount of recent work. Starting from [29, 40, 7], this set of ideas, broadly referred to as manifold learning, has been applied to a variety of problems from supervised [42] and semi-supervised learning [8], to clustering [45] and dimensionality reduction [7], to name a few. Interestingly, the problem of learning the manifold itself has received less attention: given samples from a d-manifold M embedded in some ambient space X, the problem is to learn a set that approximates M in a suitable sense. This problem has been considered in computational geometry, but in a setting in which typically the manifold is a hyper-surface in a low-dimensional space (e.g.

In many practical applications of clustering, the objects to be clustered evolve over time, and a clustering result is desired at each time step. In such applications, evolutionary clustering typically outperforms traditional static clustering by producing clustering results that reflect long-term trends while being robust to short-term variations. Several evolutionary clustering algorithms have recently been proposed, often by adding a temporal smoothness penalty to the cost function of a static clustering method. In this paper, we introduce a different approach to evolutionary clustering by accurately tracking the time-varying proximities between objects followed by static clustering. We present an evolutionary clustering framework that adaptively estimates the optimal smoothing parameter using shrinkage estimation, a statistical approach that improves a naive estimate using additional information. The proposed framework can be used to extend a variety of static clustering algorithms, including hierarchical, k-means, and spectral clustering, into evolutionary clustering algorithms. Experiments on synthetic and real data sets indicate that the proposed framework outperforms static clustering and existing evolutionary clustering algorithms in many scenarios.

In this letter, we derive the optimal discriminant functions for modulation classification based on the sampled distribution distance. The proposed method classifies various candidate constellations using a low complexity approach based on the distribution distance at specific testpoints along the cumulative distribution function. This method, based on the Bayesian decision criteria, asymptotically provides the minimum classification error possible given a set of testpoints. Testpoint locations are also optimized to improve classification performance. The method provides significant gains over existing approaches that also use the distribution of the signal features.

Unawareness refers to the lack of conception rather than the lack of information. There is a fundamental difference between not knowing about which events obtain and the inability to conceive of some events. Unawareness is an interdisciplinary topic that fascinates economists, computer scientists, logicians, and philosophers alike. Traditionally, computer scientists, logicians and philosophers are interested in epistemic models while most economists are mainly interested in the behavioral implications. In the literature, unawareness has been defined epistemically using syntactic and semantic approaches.