Discussion of: Brownian distance covariance

Discussion on "Brownian distance covariance" by G\'abor J. Sz\'ekely and Maria L. Rizzo [arXiv:1010.0297]Comment: Published in at http://dx.doi.org/10.1214/09-AOAS312F the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Infinite dimensional exponential families by reproducing kernel Hilbert spaces

The purpose of this paper is to propose a method of constructing exponential families of Hilbert manifold, on which estimation theory can be built. Although there have been works on infinite dimensional exponential families of Banach manifolds (Pistone and Sempi, 1995; Gibilisco and Pistone, 1998; Pistone and Rogantin, 1999), they are not appropriate to discuss statistical estimation with finite number of samples; the likelihood function with finite samples is not continuous on the manifold. In this paper we use a reproducing kernel Hilbert space as a functional space for constructing an exponential manifold. A reproducing kernel Hilbert space is dened as a Hilbert space of functions such that evaluation of a function at an arbitrary point is a continuous functional on the Hilbert space. Since we can discuss the value of a function with this space, it is very natural to use a manifold associated with a reproducing kernel Hilbert space as a basis of estimation theory. We focus on the maximum likelihood estimation (MLE) with the exponential manifold of a reproducing kernel Hilbert space. As in many non-parametric estimation methods, straightforward extension of MLE to an infinite dimensional exponential manifold suffers the problem of ill-posedness caused by the fact that the estimator should be chosen from the infinite dimensional space with only finite number of constraints given by the data. To solve this problem, a pseudo-maximum likelihood method is proposed by restricting the infinite dimensional manifold to a series of finite dimensional submanifolds, which enlarge as the number of samples increases. Some asymptotic results in the limit of infinite samples are shown, including the consistency of the pseudo-MLE

Discussion of: Brownian distance covariance

Discussion on "Brownian distance covariance" by G\'{a}bor J. Sz\'{e}kely and Maria L. Rizzo [arXiv:1010.0297]Comment: Published in at http://dx.doi.org/10.1214/09-AOAS312E the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Computing Functions of Random Variables via Reproducing Kernel Hilbert Space Representations

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as {\em kernel probabilistic programming}. We illustrate it on synthetic data, and show how it can be used for nonparametric structural equation models, with an application to causal inference

On-Line Learning Theory of Soft Committee Machines with Correlated Hidden Units - Steepest Gradient Descent and Natural Gradient Descent -

The permutation symmetry of the hidden units in multilayer perceptrons causes the saddle structure and plateaus of the learning dynamics in gradient learning methods. The correlation of the weight vectors of hidden units in a teacher network is thought to affect this saddle structure, resulting in a prolonged learning time, but this mechanism is still unclear. In this paper, we discuss it with regard to soft committee machines and on-line learning using statistical mechanics. Conventional gradient descent needs more time to break the symmetry as the correlation of the teacher weight vectors rises. On the other hand, no plateaus occur with natural gradient descent regardless of the correlation for the limit of a low learning rate. Analytical results support these dynamics around the saddle point.Comment: 7 pages, 6 figure

Model-based kernel sum rule: kernel Bayesian inference with probabilistic model

Kernel Bayesian inference is a principled approach to nonparametric inference in probabilistic graphical models, where probabilistic relationships between variables are learned from data in a nonparametric manner. Various algorithms of kernel Bayesian inference have been developed by combining kernelized basic probabilistic operations such as the kernel sum rule and kernel Bayes’ rule. However, the current framework is fully nonparametric, and it does not allow a user to flexibly combine nonparametric and model-based inferences. This is inefficient when there are good probabilistic models (or simulation models) available for some parts of a graphical model; this is in particular true in scientific fields where “models” are the central topic of study. Our contribution in this paper is to introduce a novel approach, termed the model-based kernel sum rule (Mb-KSR), to combine a probabilistic model and kernel Bayesian inference. By combining the Mb-KSR with the existing kernelized probabilistic rules, one can develop various algorithms for hybrid (i.e., nonparametric and model-based) inferences. As an illustrative example, we consider Bayesian filtering in a state space model, where typically there exists an accurate probabilistic model for the state transition process. We propose a novel filtering method that combines model-based inference for the state transition process and data-driven, nonparametric inference for the observation generating process. We empirically validate our approach with synthetic and real-data experiments, the latter being the problem of vision-based mobile robot localization in robotics, which illustrates the effectiveness of the proposed hybrid approach

A Kernel-Based Causal Learning Algorithm

We describe a causal learning method, which employs measuring the strength of statistical dependences in terms of the Hilbert-Schmidt norm of kernel-based cross-covariance operators. Following the line of the common faithfulness assumption of constraint-based causal learning, our approach assumes that a variable Z is likely to be a common effect of X and Y, if conditioning on Z increases the dependence between X and Y. Based on this assumption, we collect "votes" for hypothetical causal directions and orient the edges by the majority principle. In most experiments with known causal structures, our method provided plausible results and outperformed the conventional constraint-based PC algorithm

Hilbert Space Representations of Probability Distributions

Many problems in unsupervised learning require the analysis of features of probability distributions. At the most fundamental level, we might wish to determine whether two distributions are the same, based on samples from each - this is known as the two-sample or homogeneity problem. We use kernel methods to address this problem, by mapping probability distributions to elements in a reproducing kernel Hilbert space (RKHS). Given a sufficiently rich RKHS, these representations are unique: thus comparing feature space representations allows us to compare distributions without ambiguity. Applications include testing whether cancer subtypes are distinguishable on the basis of DNA microarray data, and whether low frequency oscillations measured at an electrode in the cortex have a different distribution during a neural spike. A more difficult problem is to discover whether two random variables drawn from a joint distribution are independent. It turns out that any dependence between pairs of random variables can be encoded in a cross-covariance operator between appropriate RKHS representations of the variables, and we may test independence by looking at a norm of the operator. We demonstrate this independence test by establishing dependence between an English text and its French translation, as opposed to French text on the same topic but otherwise unrelated. Finally, we show that this operator norm is itself a difference in feature means