We learn the structure of a Markov Network between two groups of random variables from joint observations. Since modelling and learning the full MN structure may be hard, learning the links between two groups directly may be a preferable option. We introduce a novel concept called the \emph{partitioned ratio} whose factorization directly associates with the Markovian properties of random variables across two groups. A simple one-shot convex optimization procedure is proposed for learning the \emph{sparse} factorizations of the partitioned ratio and it is theoretically guaranteed to recover the correct inter-group structure under mild conditions. The performance of the proposed method is experimentally compared with the state of the art MN structure learning methods using ROC curves. Real applications on analyzing bipartisanship in US congress and pairwise DNA/time-series alignments are also reported.

Recently, there has been considerable progress on designing algorithms with provable guarantees -- typically using linear algebraic methods -- for parameter learning in latent variable models. But designing provable algorithms for inference has proven to be more challenging. Here we take a first step towards provable inference in topic models. We leverage a property of topic models that enables us to construct simple linear estimators for the unknown topic proportions that have small variance, and consequently can work with short documents. Our estimators also correspond to finding an estimate around which the posterior is well-concentrated. We show lower bounds that for shorter documents it can be information theoretically impossible to find the hidden topics. Finally, we give empirical results that demonstrate that our algorithm works on realistic topic models. It yields good solutions on synthetic data and runs in time comparable to a {\em single} iteration of Gibbs sampling.

Topic models have emerged as fundamental tools in unsupervised machine learning. Most modern topic modeling algorithms take a probabilistic view and derive inference algorithms based on Latent Dirichlet Allocation (LDA) or its variants. In contrast, we study topic modeling as a combinatorial optimization problem, and propose a new objective function derived from LDA by passing to the small-variance limit. We minimize the derived objective by using ideas from combinatorial optimization, which results in a new, fast, and high-quality topic modeling algorithm. In particular, we show that our results are competitive with popular LDA-based topic modeling approaches, and also discuss the (dis)similarities between our approach and its probabilistic counterparts.

The Naive Bayes classifier is a simple classifier that is often used as a baseline for comparison with more complex classifiers. We will use the famous MNIST data set (pre-processed via PCA and normalized [TODO]) for this tutorial, so our class labels are {0, 1, …, 9}. If you're like me, you may have found this notation a little confusing at first. We can read the left side P(C X) as "the probability that the class is C given the data X". We can read the right side P(X C) as "the probability that the data X belongs to the class C". (this is called the "likelihood") And we can compute the probability that the class 0 given the data, probability that the class 1 given the data, etc. just by computing the probability of the data for each class (how well the data fits a model of each class).

In many machine learning applications, crowdsourcing has become the primary means for label collection. In this paper, we study the optimal error rate for aggregating labels provided by a set of non-expert workers. Under the classic Dawid-Skene model, we establish matching upper and lower bounds with an exact exponent $mI(\pi)$ in which $m$ is the number of workers and $I(\pi)$ the average Chernoff information that characterizes the workers' collective ability. Such an exact characterization of the error exponent allows us to state a precise sample size requirement $m>\frac{1}{I(\pi)}\log\frac{1}{\epsilon}$ in order to achieve an $\epsilon$ misclassification error. In addition, our results imply the optimality of various EM algorithms for crowdsourcing initialized by consistent estimators.

Partition functions of probability distributions are important quantities for model evaluation and comparisons. We present a new method to compute partition functions of complex and multimodal distributions. Such distributions are often sampled using simulated tempering, which augments the target space with an auxiliary inverse temperature variable. Our method exploits the multinomial probability law of the inverse temperatures, and provides estimates of the partition function in terms of a simple quotient of Rao-Blackwellized marginal inverse temperature probability estimates, which are updated while sampling. We show that the method has interesting connections with several alternative popular methods, and offers some significant advantages. In particular, we empirically find that the new method provides more accurate estimates than Annealed Importance Sampling when calculating partition functions of large Restricted Boltzmann Machines (RBM); moreover, the method is sufficiently accurate to track training and validation log-likelihoods during learning of RBMs, at minimal computational cost.

Zoubin Ghahramani We show that a neural network with arbitrary depth and non-linearities, with dropout applied before every weight layer, is mathematically equivalent to an approximation to a well known Bayesian model. This interpretation might offer an explanation to some of dropout's key properties, such as its robustness to overfitting. Our interpretation allows us to reason about uncertainty in deep learning, and allows the introduction of the Bayesian machinery into existing deep learning frameworks in a principled way. This document is an appendix for the main paper "Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning" by Gal and Ghahramani, 2015 (http://arxiv.org/abs/1506.02142).

A Markov chain is a random process with the property that the next state depends only on the current state. For example: If you have the choice of red or blue twice the process would be Markovian if each time you chose the decision had nothing to do with your choice previously (see diagram below). How can Markov Chains help us? To start with we need to define some basic terminology. The changes of state within the system are called transitions, and the probabilities associated with various state-changes are called transition probabilities.

Probabilistic models can be defined by an energy function, where the probability of each state is proportional to the exponential of the state's negative energy. This paper considers a generalization of energy-based models in which the probability of a state is proportional to an arbitrary positive, strictly decreasing, and twice differentiable function of the state's energy. The precise shape of the nonlinear map from energies to unnormalized probabilities has to be learned from data together with the parameters of the energy function. As a case study we show that the above generalization of a fully visible Boltzmann machine yields an accurate model of neural activity of retinal ganglion cells. We attribute this success to the model's ability to easily capture distributions whose probabilities span a large dynamic range, a possible consequence of latent variables that globally couple the system. Similar features have recently been observed in many datasets, suggesting that our new method has wide applicability.

Efficient unsupervised training and inference in deep generative models remains a challenging problem. One basic approach, called Helmholtz machine, involves training a top-down directed generative model together with a bottom-up auxiliary model used for approximate inference. Recent results indicate that better generative models can be obtained with better approximate inference procedures. Instead of improving the inference procedure, we here propose a new model which guarantees that the top-down and bottom-up distributions can efficiently invert each other. We achieve this by interpreting both the top-down and the bottom-up directed models as approximate inference distributions and by defining the model distribution to be the geometric mean of these two. We present a lower-bound for the likelihood of this model and we show that optimizing this bound regularizes the model so that the Bhattacharyya distance between the bottom-up and top-down approximate distributions is minimized. This approach results in state of the art generative models which prefer significantly deeper architectures while it allows for orders of magnitude more efficient approximate inference.