With the Naive Bayes model, we do not take only a small set of positive and negative words into account, but all words the NB Classifier was trained with, i.e. all words presents in the training set. If a word has not appeared in the training set, we have no data available and apply Laplacian smoothing (use 1 instead of the conditional probability of the word). The probability a document belongs to a class C is given by the class probability P(C) multiplied by the products of the conditional probabilities of each word for that class. Here count(d_i, C) is the number of occurences of word d_i in class C, V_C is the total number of words in class C and n is the number of words in the document we are currently classifying. In theory we want a training set as large as possible, since that will increase the accuracy.

Real-time systems differ in the way they perform analytics. Specifically, Real-time systems perform analytics on short time windows for Data Streams. Hence, the scope of Real Time analytics is a'window' which typically comprises of the last few time slots. Making Predictions on Real Time Data streams involves building an Offline model and applying it to a stream. Models incorporate one or more machine learning algorithms which are trained using the training Data.

Quantum annealing (QA) is a hardware-based heuristic optimization and sampling method applicable to discrete undirected graphical models. While similar to simulated annealing, QA relies on quantum, rather than thermal, effects to explore complex search spaces. For many classes of problems, QA is known to offer computational advantages over simulated annealing. Here we report on the ability of recent QA hardware to accelerate training of fully visible Boltzmann machines. We characterize the sampling distribution of QA hardware, and show that in many cases, the quantum distributions differ significantly from classical Boltzmann distributions. In spite of this difference, training (which seeks to match data and model statistics) using standard classical gradient updates is still effective. We investigate the use of QA for seeding Markov chains as an alternative to contrastive divergence (CD) and persistent contrastive divergence (PCD). Using $k=50$ Gibbs steps, we show that for problems with high-energy barriers between modes, QA-based seeds can improve upon chains with CD and PCD initializations. For these hard problems, QA gradient estimates are more accurate, and allow for faster learning. Furthermore, and interestingly, even the case of raw QA samples (that is, $k=0$) achieved similar improvements. We argue that this relates to the fact that we are training a quantum rather than classical Boltzmann distribution in this case. The learned parameters give rise to hardware QA distributions closely approximating classical Boltzmann distributions that are hard to train with CD/PCD.

We consider the problem of identifying the causal direction between two discrete random variables using observational data. Unlike previous work, we keep the most general functional model but make an assumption on the unobserved exogenous variable: Inspired by Occam's razor, we assume that the exogenous variable is simple in the true causal direction. We quantify simplicity using R\'enyi entropy. Our main result is that, under natural assumptions, if the exogenous variable has low $H_0$ entropy (cardinality) in the true direction, it must have high $H_0$ entropy in the wrong direction. We establish several algorithmic hardness results about estimating the minimum entropy exogenous variable. We show that the problem of finding the exogenous variable with minimum entropy is equivalent to the problem of finding minimum joint entropy given $n$ marginal distributions, also known as minimum entropy coupling problem. We propose an efficient greedy algorithm for the minimum entropy coupling problem, that for $n=2$ provably finds a local optimum. This gives a greedy algorithm for finding the exogenous variable with minimum $H_1$ (Shannon Entropy). Our greedy entropy-based causal inference algorithm has similar performance to the state of the art additive noise models in real datasets. One advantage of our approach is that we make no use of the values of random variables but only their distributions. Our method can therefore be used for causal inference for both ordinal and also categorical data, unlike additive noise models.

How can we efficiently propagate uncertainty in a latent state representation with recurrent neural networks? This paper introduces stochastic recurrent neural networks which glue a deterministic recurrent neural network and a state space model together to form a stochastic and sequential neural generative model. The clear separation of deterministic and stochastic layers allows a structured variational inference network to track the factorization of the model's posterior distribution. By retaining both the nonlinear recursive structure of a recurrent neural network and averaging over the uncertainty in a latent path, like a state space model, we improve the state of the art results on the Blizzard and TIMIT speech modeling data sets by a large margin, while achieving comparable performances to competing methods on polyphonic music modeling.

A BSTRACT Restricted Boltzmann Machine (RBM) is a bipartite graphical model that is used as the building block in energy-based deep generative models. Due to numerical stability and quantifiability of the likelihood, RBM is commonly used with Bernoulli units. Here, we consider an alternative member of exponential family RBM with leaky rectified linear units - called leaky RBM. We first study the joint and marginal distributions of leaky RBM under different leakiness, which provides us important insights by connecting the leaky RBM model and truncated Gaussian distributions. The connection leads us to a simple yet efficient method for sampling from this model, where the basic idea is to anneal the leakiness rather than the energy; - i.e., start from a fully Gaussian/Linear unit and gradually decrease the leakiness over iterations. This serves as an alternative to the annealing of the temperature parameter and enables numerical estimation of the likelihood that are more efficient and more accurate than the commonly used annealed importance sampling (AIS). We further demonstrate that the proposed sampling algorithm enjoys faster mixing property than contrastive divergence algorithm, which benefits the training without any additional computational cost. 1 I NTRODUCTION In this paper, we are interested in deep generative models. There is a family of directed deep generative models which can be trained by back-propagation (e.g., Kingma & Welling, 2013; Goodfellow et al., 2014). The other family is the deep energy-based models, including deep belief network (Hinton et al., 2006) and deep Boltzmann machine (Salakhutdinov & Hinton, 2009).

Recent sequential pattern mining methods have used the minimum description length (MDL) principle to define an encoding scheme which describes an algorithm for mining the most compressing patterns in a database. We present a novel subsequence interleaving model based on a probabilistic model of the sequence database, which allows us to search for the most compressing set of patterns without designing a specific encoding scheme. Our proposed algorithm is able to efficiently mine the most relevant sequential patterns and rank them using an associated measure of interestingness. The efficient inference in our model is a direct result of our use of a structural expectation-maximization framework, in which the expectation-step takes the form of a submodular optimization problem subject to a coverage constraint. We show on both synthetic and real world datasets that our model mines a set of sequential patterns with low spuriousness and redundancy, high interpretability and usefulness in real-world applications. Furthermore, we demonstrate that the quality of the patterns from our approach is comparable to, if not better than, existing state of the art sequential pattern mining algorithms.

We develop a supervised machine learning model that detects anomalies in systems in real time. Our model processes unbounded streams of data into time series which then form the basis of a low-latency anomaly detection model. Moreover, we extend our preliminary goal of just anomaly detection to simultaneous anomaly prediction. We approach this very challenging problem by developing a Bayesian Network framework that captures the information about the parameters of the lagged regressors calibrated in the first part of our approach and use this structure to learn local conditional probability distributions.

Probabilistic Temporal Tensor Factorization (PTTF) is an effective algorithm to model the temporal tensor data. It leverages a time constraint to capture the evolving properties of tensor data. Nowadays the exploding dataset demands a large scale PTTF analysis, and a parallel solution is critical to accommodate the trend. Whereas, the parallelization of PTTF still remains unexplored. In this paper, we propose a simple yet efficient Parallel Probabilistic Temporal Tensor Factorization, referred to as P$^2$T$^2$F, to provide a scalable PTTF solution. P$^2$T$^2$F is fundamentally disparate from existing parallel tensor factorizations by considering the probabilistic decomposition and the temporal effects of tensor data. It adopts a new tensor data split strategy to subdivide a large tensor into independent sub-tensors, the computation of which is inherently parallel. We train P$^2$T$^2$F with an efficient algorithm of stochastic Alternating Direction Method of Multipliers, and show that the convergence is guaranteed. Experiments on several real-word tensor datasets demonstrate that P$^2$T$^2$F is a highly effective and efficiently scalable algorithm dedicated for large scale probabilistic temporal tensor analysis.

We consider several estimation and learning problems that networked agents face when making decisions given their uncertainty about an unknown variable. Our methods are designed to efficiently deal with heterogeneity in both size and quality of the observed data, as well as heterogeneity over time (intermittence). The goal of the studied aggregation schemes is to efficiently combine the observed data that is spread over time and across several network nodes, accounting for all the network heterogeneities. Moreover, we require no form of coordination beyond the local neighborhood of every network agent or sensor node. The three problems that we consider are (i) maximum likelihood estimation of the unknown given initial data sets, (ii) learning the true model parameter from streams of data that the agents receive intermittently over time, and (iii) minimum variance estimation of a complete sufficient statistic from several data points that the networked agents collect over time. In each case we rely on an aggregation scheme to combine the observations of all agents; moreover, when the agents receive streams of data over time, we modify the update rules to accommodate the most recent observations. In every case, we demonstrate the efficiency of our algorithms by proving convergence to the globally efficient estimators given the observations of all agents. We supplement these results by investigating the rate of convergence and providing finite-time performance guarantees.