We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of such Gaussian graphical models are also developed. We apply these rules to compare conditional dependencies on Gaussian trees. In particular for trees, we show that such dependence can be completely characterized by the length of the paths joining the dependent vertices to each other and to the vertices conditioned on. We also apply our results to postulate rules for model selection for polytree models. Our rules apply to mutual information of Gaussian random vectors as well.

We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions belong. Our formulation provides a unified framework so that the classical notions, such as covariance standardization, Mahalanobis distance, SIR and linear discriminant analysis, can be naturally and smoothly carried out in our enlarged space. This enlarged dimension reduction space also links to the linear discriminant space of Gaussian measures on a separable Hilbert space.

This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete distribution. The probability of this event is referred to as the "missing mass". The impossibility result can then be stated as: the missing mass is not distribution-free PAC-learnable in relative error. The proof is semi-constructive and relies on a coupling argument using a dithered geometric distribution. This result formalizes the folklore that in order to predict rare events, one necessarily needs distributions with "heavy tails".

Community detection is a fundamental problem in network analysis which is made more challenging by overlaps between communities which often occur in practice. Here we propose a general, flexible, and interpretable generative model for overlapping communities, which can be thought of as a generalization of the degree-corrected stochastic block model. We develop an efficient spectral algorithm for estimating the community memberships, which deals with the overlaps by employing the K-medians algorithm rather than the usual K-means for clustering in the spectral domain. We show that the algorithm is asymptotically consistent when networks are not too sparse and the overlaps between communities not too large. Numerical experiments on both simulated networks and many real social networks demonstrate that our method performs very well compared to a number of benchmark methods for overlapping community detection.

Conditional restricted Boltzmann machines are undirected stochastic neural networks with a layer of input and output units connected bipartitely to a layer of hidden units. These networks define models of conditional probability distributions on the states of the output units given the states of the input units, parametrized by interaction weights and biases. We address the representational power of these models, proving results their ability to represent conditional Markov random fields and conditional distributions with restricted supports, the minimal size of universal approximators, the maximal model approximation errors, and on the dimension of the set of representable conditional distributions. We contribute new tools for investigating conditional probability models, which allow us to improve the results that can be derived from existing work on restricted Boltzmann machine probability models.

Graph matching is an increasingly important problem in inferential graph statistics, with applications across a broad spectrum of fields including computer vision ([38], [10]), shape matching and object recognition ([4], [7]), and biology and neuroscience ([22], [34], [37]), to name a few. The graph matching problem (GMP) seeks to find an alignment between the vertex sets of two graphs that best preserves common structure across graphs. Unfortunately, the GMP is inherently combinatorial, and no efficient exact graph matching algorithms are known. Indeed, even the simpler problem of determining if two graphs are isomorphic is famously of unknown complexity ([19], 1 [30]), and if the graphs are allowed to be loopy, weighted and directed, then the simplest version of GMP is equivalent to the NPhard quadratic assignment problem. Due to its wide applicability, there exist a vast number of approximating algorithms for GMP; see the paper "30 Years of Graph Matching in Pattern Recognition" ([11]) for an excellent survey of the existing literature.

Remember those charming smartphone ads a few years ago where people held conversations with Siri, Apple iOS' enchanting virtual assistant? The concept was innovative for operating the phone hands-free but not so helpful in searching the web on your behalf. Oftentimes, Siri comes up with entirely wrong answers, too many options or services only tangentially relevant to your inquiry. Asking for a modestly-priced tailor or an upscale sushi bar known for uni is too ambitious. She hears us, but she's not listening.

ABSTRACT Several efforts to predict student failure rate (SFR) at school accurately still remains a core problem area faced by many in the educational sector. The procedure for forecasting SFR are rigid and most often times require data scaling or conversion into binary form such as is the case of the logistic model which may lead to lose of information and effect size attenuation. Currently the application of Genetic Programming (GP) holds great promises and has produced tremendous positive results in different sectors. In this regard, this study developed GPSFARPS, a software application to provide a robust solution to the prediction of SFR using an evolutionary algorithm known as multi-gene genetic programming. The approach is validated by feeding a testing data set to the evolved GP models. Result obtained from GPSFARPS simulations show its unique ability to evolve a suitable failure rate expression with a fast convergence at 30 generations from a maximum specified generation of 500. The multigene system was also able to minimize the evolved model expression and accurately predict student failure rate using a subset of the original expression. Keywords: Genetic Programming, Student Failure Rate, Multi-Gene GP 1. INTRODUCTION SFR has always being and will continue to be a major concern to stakeholders in the educational sector.

Geotagged data can be used to describe regions in the world and discover local themes. However, not all data produced within a region is necessarily specifically descriptive of that area. To surface the content that is characteristic for a region, we present the geographical hierarchy model (GHM), a probabilistic model based on the assumption that data observed in a region is a random mixture of content that pertains to different levels of a hierarchy. We apply the GHM to a dataset of 8 million Flickr photos in order to discriminate between content (i.e., tags) that specifically characterizes a region (e.g., neighborhood) and content that characterizes surrounding areas or more general themes. Knowledge of the discriminative and non-discriminative terms used throughout the hierarchy enables us to quantify the uniqueness of a given region and to compare similar but distant regions. Our evaluation demonstrates that our model improves upon traditional Naive Bayes classification by 47% and hierarchical TF-IDF by 27%. We further highlight the differences and commonalities with human reasoning about what is locally characteristic for a neighborhood, distilled from ten interviews and a survey that covered themes such as time, events, and prior regional knowledge.

Distributed estimation, detection, and learning theory in networks have attracted much attention over the past decades [1], [2], [3], [4], with applications that range from sensor and robotic networks [5], [6], [7], [8], [9] to social and economic networks [10], [11], [12]. In these scenarios, agents in a network need to learn the value of a parameter that they may not be able to infer on their own, but the global spread of information in the network provides them with adequate data to learn the truth collectively. As a result, agents iteratively exchange information with their neighbors. For instance, in distributed sensor and robotic networks, agents use local diffusion to augment their imperfect observations with information from their neighbors and achieve consensus and coordination [13], [14]. Similarly, agents exchange beliefs in social networks to benefit from each other's observations and private information and learn the unknown state of the world [15], [16]. Existing literature on distributed learning focuses mostly on environments where individuals communicate at every round. Of particular relevance to our discussion are a host of algorithms that follow the non-Bayesian learning scheme in Jadbabaie et.