Yield curve forecasting is an important problem in finance. In this work we explore the use of Gaussian Processes in conjunction with a dynamic modeling strategy, much like the Kalman Filter, to model the yield curve. Gaussian Processes have been successfully applied to model functional data in a variety of applications. A Gaussian Process is used to model the yield curve. The hyper-parameters of the Gaussian Process model are updated as the algorithm receives yield curve data. Yield curve data is typically available as a time series with a frequency of one day. We compare existing methods to forecast the yield curve with the proposed method. The results of this study showed that while a competing method (a multivariate time series method) performed well in forecasting the yields at the short term structure region of the yield curve, Gaussian Processes perform well in the medium and long term structure regions of the yield curve. Accuracy in the long term structure region of the yield curve has important practical implications. The Gaussian Process framework yields uncertainty and probability estimates directly in contrast to other competing methods. Analysts are frequently interested in this information. In this study the proposed method has been applied to yield curve forecasting, however it can be applied to model high frequency time series data or data streams in other domains.

Topic models are one of the most popular methods for learning representations of text, but a major challenge is that any change to the topic model requires mathematically deriving a new inference algorithm. A promising approach to address this problem is autoencoding variational Bayes (AEVB), but it has proven diffi- cult to apply to topic models in practice. We present what is to our knowledge the first effective AEVB based inference method for latent Dirichlet allocation (LDA), which we call Autoencoded Variational Inference For Topic Model (AVITM). This model tackles the problems caused for AEVB by the Dirichlet prior and by component collapsing. We find that AVITM matches traditional methods in accuracy with much better inference time. Indeed, because of the inference network, we find that it is unnecessary to pay the computational cost of running variational optimization on test data. Because AVITM is black box, it is readily applied to new topic models. As a dramatic illustration of this, we present a new topic model called ProdLDA, that replaces the mixture model in LDA with a product of experts. By changing only one line of code from LDA, we find that ProdLDA yields much more interpretable topics, even if LDA is trained via collapsed Gibbs sampling.

In this paper, we study the information-theoretic limits of learning the structure of Bayesian networks (BNs), on discrete as well as continuous random variables, from a finite number of samples. We show that the minimum number of samples required by any procedure to recover the correct structure grows as $\Omega(m)$ and $\Omega(k \log m + (k^2/m))$ for non-sparse and sparse BNs respectively, where $m$ is the number of variables and $k$ is the maximum number of parents per node. We provide a simple recipe, based on an extension of the Fano's inequality, to obtain information-theoretic limits of structure recovery for any exponential family BN. We instantiate our result for specific conditional distributions in the exponential family to characterize the fundamental limits of learning various commonly used BNs, such as conditional probability table based networks, gaussian BNs, noisy-OR networks, and logistic regression networks. En route to obtaining our main results, we obtain tight bounds on the number of sparse and non-sparse essential-DAGs. Finally, as a byproduct, we recover the information-theoretic limits of sparse variable selection for logistic regression.

Increasingly complex generative models are being used across disciplines as they allow for realistic characterization of data, but a common difficulty with them is the prohibitively large computational cost to evaluate the likelihood function and thus to perform likelihood-based statistical inference. A likelihood-free inference framework has emerged where the parameters are identified by finding values that yield simulated data resembling the observed data. While widely applicable, a major difficulty in this framework is how to measure the discrepancy between the simulated and observed data. Transforming the original problem into a problem of classifying the data into simulated versus observed, we find that classification accuracy can be used to assess the discrepancy. The complete arsenal of classification methods becomes thereby available for inference of intractable generative models. We validate our approach using theory and simulations for both point estimation and Bayesian inference, and demonstrate its use on real data by inferring an individual-based epidemiological model for bacterial infections in child care centers.

We harness the power of Bayesian emulation techniques, designed to aid the analysis of complex computer models, to examine the structure of complex Bayesian analyses themselves. These techniques facilitate robust Bayesian analyses and/or sensitivity analyses of complex problems, and hence allow global exploration of the impacts of choices made in both the likelihood and prior specification. We show how previously intractable problems in robustness studies can be overcome using emulation techniques, and how these methods allow other scientists to quickly extract approximations to posterior results corresponding to their own particular subjective specification. The utility and flexibility of our method is demonstrated on a reanalysis of a real application where Bayesian methods were employed to capture beliefs about river flow. We discuss the obvious extensions and directions of future research that such an approach opens up.

Learning the directed acyclic graph (DAG) structure of a Bayesian network from observational data is a notoriously difficult problem for which many hardness results are known. In this paper we propose a provably polynomial-time algorithm for learning sparse Gaussian Bayesian networks with equal noise variance --- a class of Bayesian networks for which the DAG structure can be uniquely identified from observational data --- under high-dimensional settings. We show that $O(k^4 \log p)$ number of samples suffices for our method to recover the true DAG structure with high probability, where $p$ is the number of variables and $k$ is the maximum Markov blanket size. We obtain our theoretical guarantees under a condition called Restricted Strong Adjacency Faithfulness, which is strictly weaker than strong faithfulness --- a condition that other methods based on conditional independence testing need for their success. The sample complexity of our method matches the information-theoretic limits in terms of the dependence on $p$. We show that our method out-performs existing state-of-the-art methods for learning Gaussian Bayesian networks in terms of recovering the true DAG structure while being comparable in speed to heuristic methods.

So, you have been thinking about picking up machine learning, but given the confusing state of the web you don't know where to begin? Or maybe you have finished the first 7 steps and are looking for some follow-up material, beyond the introductory? This post is the second installment of the 7 Steps to Mastering Machine Learning in Python series (since there are 2 parts, I guess it now qualifies as a series). If you have started with the original post, you should already be satisfactorily up to speed, skill-wise. If not, you may want to review that post first, which may take some time, depending on your current level of understanding; however, I assure you that doing so will be worth your effort.

We draw a formal connection between using synthetic training data to optimize neural network parameters and approximate, Bayesian, model-based reasoning. In particular, training a neural network using synthetic data can be viewed as learning a proposal distribution generator for approximate inference in the synthetic-data generative model. We demonstrate this connection in a recognition task where we develop a novel Captcha-breaking architecture and train it using synthetic data, demonstrating both state-of-the-art performance and a way of computing task-specific posterior uncertainty. Using a neural network trained this way, we also demonstrate successful breaking of real-world Captchas currently used by Facebook and Wikipedia. Reasoning from these empirical results and drawing connections with Bayesian modeling, we discuss the robustness of synthetic data results and suggest important considerations for ensuring good neural network generalization when training with synthetic data.

In this paper we introduce a new classification algorithm called Optimization of Distributions Differences (ODD). The algorithm aims to find a transformation from the feature space to a new space where the instances in the same class are as close as possible to one another while the gravity centers of these classes are as far as possible from one another. This aim is formulated as a multiobjective optimization problem that is solved by a hybrid of an evolutionary strategy and the Quasi-Newton method. The choice of the transformation function is flexible and could be any continuous space function. We experiment with a linear and a non-linear transformation in this paper. We show that the algorithm can outperform 6 other state-of-the-art classification methods, namely naive Bayes, support vector machines, linear discriminant analysis, multi-layer perceptrons, decision trees, and k-nearest neighbors, in 12 standard classification datasets. Our results show that the method is less sensitive to the imbalanced number of instances comparing to these methods. We also show that ODD maintains its performance better than other classification methods in these datasets, hence, offers a better generalization ability.

Restricted Boltzmann machines~(RBMs) and conditional RBMs~(CRBMs) are popular models for a wide range of applications. In previous work, learning on such models has been dominated by contrastive divergence~(CD) and its variants. Belief propagation~(BP) algorithms are believed to be slow for structured prediction on conditional RBMs~(e.g., Mnih et al. [2011]), and not as good as CD when applied in learning~(e.g., Larochelle et al. [2012]). In this work, we present a matrix-based implementation of belief propagation algorithms on CRBMs, which is easily scalable to tens of thousands of visible and hidden units. We demonstrate that, in both maximum likelihood and max-margin learning, training conditional RBMs with BP as the inference routine can provide significantly better results than current state-of-the-art CD methods on structured prediction problems. We also include practical guidelines on training CRBMs with BP, and some insights on the interaction of learning and inference algorithms for CRBMs.