Probabilistic programming languages (PPLs) are a powerful modeling tool, able to represent any computable probability distribution. Unfortunately, probabilistic program inference is often intractable, and existing PPLs mostly rely on expensive, approximate sampling-based methods. To alleviate this problem, one could try to learn from past inferences, so that future inferences run faster. This strategy is known as amortized inference; it has recently been applied to Bayesian networks and deep generative models. This paper proposes a system for amortized inference in PPLs. In our system, amortization comes in the form of a parameterized guide program. Guide programs have similar structure to the original program, but can have richer data flow, including neural network components. These networks can be optimized so that the guide approximately samples from the posterior distribution defined by the original program. We present a flexible interface for defining guide programs and a stochastic gradient-based scheme for optimizing guide parameters, as well as some preliminary results on automatically deriving guide programs. We explore in detail the common machine learning pattern in which a 'local' model is specified by 'global' random values and used to generate independent observed data points; this gives rise to amortized local inference supporting global model learning.

Learning has become an online activity - people routinely use a wide variety of online learning platforms, ranging from wikis and question answering (Q&A) sites to online communities and blogs, to learn about a large range of topics. In this context, people find solutions to their problems by looking for closely related pieces of information, executing a sequence of queries or, more generally, performing a series of online actions. For example, a high school student may study several closely related wiki pages to prepare an essay about a historical event; a software developer may read several answers within a Q&A site to solve a specific programming problem; and, a researcher may check a specialized blog written by one of her peers to learn about a new concept or technique. All the above are examples of learning patterns, in which people perform a series of online actions - reading a wiki page, an answer, or a blog - to achieve a predefined goal - writing an essay, solving a programming problem, or learning about a new concept or technique. In this context, one may expect that people with similar goals undertake similar sequences of online actions and thus adopt similar learning patterns. Therefore, one could leverage the vast availability of online traces of users' learning activity to disambiguate among interleaved learning patterns adopted by individuals over time, as well as to automatically identify and track those people's interests and goals over time. In this work, we introduce a novel probabilistic model, the Hierarchical Dirichlet Hawkes Process (HDHP), for clustering continuous-time grouped streaming data, which we use to uncover the dynamics of learning activity on the web. The HDHP leverages the properties of the Hierarchical Dirichlet Process (HDP) [18], a popular Bayesian nonparametric model for clustering problems involving multiple groups of data, combined with the Hawkes process [13], a temporal point process particularly well fitted to model social activity [11, 19, 20]. In particular, the former is used to account for an infinite number of learning patterns, which are shared across users (groups) of an online learning platform.

Dynamic text networks have been widely studied in recent years, primarily because the Internet stores textual data in a way that allows links between different documents. Articles on the Wikipedia (Hoffman et al., 2010), citation networks in journal articles (Moody, 2004), and linked blog posts (Latouche et al., 2011) are examples of dynamic text networks, or networks of documents that are generated over time. But each application has idiosyncratic features, such as the structure of the links and the nature of the time varying documents, so analysis typically requires bespoke models that directly address those aspects.

We study model evaluation and model selection from the perspective of generalization ability (GA): the ability of a model to predict outcomes in new samples from the same population. We believe that GA is one way formally to address concerns about the external validity of a model. The GA of a model estimated on a sample can be measured by its empirical out-of-sample errors, called the generalization errors (GE). We derive upper bounds for the GE, which depend on sample sizes, model complexity and the distribution of the loss function. The upper bounds can be used to evaluate the GA of a model, ex ante. We propose using generalization error minimization (GEM) as a framework for model selection. Using GEM, we are able to unify a big class of penalized regression estimators, including lasso, ridge and bridge, under the same set of assumptions. We establish finite-sample and asymptotic properties (including $\mathcal{L}_2$-consistency) of the GEM estimator for both the $n \geqslant p$ and the $n < p$ cases. We also derive the $\mathcal{L}_2$-distance between the penalized and corresponding unpenalized regression estimates. In practice, GEM can be implemented by validation or cross-validation. We show that the GE bounds can be used for selecting the optimal number of folds in $K$-fold cross-validation. We propose a variant of $R^2$, the $GR^2$, as a measure of GA, which considers both both in-sample and out-of-sample goodness of fit. Simulations are used to demonstrate our key results.

Bayesian max-margin models have shown superiority in various practical applications, such as text categorization, collaborative prediction, social network link prediction and crowdsourcing, and they conjoin the flexibility of Bayesian modeling and predictive strengths of max-margin learning. However, Monte Carlo sampling for these models still remains challenging, especially for applications that involve large-scale datasets. In this paper, we present the stochastic subgradient Hamiltonian Monte Carlo (HMC) methods, which are easy to implement and computationally efficient. We show the approximate detailed balance property of subgradient HMC which reveals a natural and validated generalization of the ordinary HMC. Furthermore, we investigate the variants that use stochastic subsampling and thermostats for better scalability and mixing. Using stochastic subgradient Markov Chain Monte Carlo (MCMC), we efficiently solve the posterior inference task of various Bayesian max-margin models and extensive experimental results demonstrate the effectiveness of our approach.

Motivation: Recent advances in technology for brain imaging and high-throughput genotyping have motivated studies examining the influence of genetic variation on brain structure. Wang et al. (Bioinformatics, 2012) have developed an approach for the analysis of imaging genomic studies using penalized multi-task regression with regularization based on a novel group $l_{2,1}$-norm penalty which encourages structured sparsity at both the gene level and SNP level. While incorporating a number of useful features, the proposed method only furnishes a point estimate of the regression coefficients; techniques for conducting statistical inference are not provided. A new Bayesian method is proposed here to overcome this limitation. Results: We develop a Bayesian hierarchical modeling formulation where the posterior mode corresponds to the estimator proposed by Wang et al. (Bioinformatics, 2012), and an approach that allows for full posterior inference including the construction of interval estimates for the regression parameters. We show that the proposed hierarchical model can be expressed as a three-level Gaussian scale mixture and this representation facilitates the use of a Gibbs sampling algorithm for posterior simulation. Simulation studies demonstrate that the interval estimates obtained using our approach achieve adequate coverage probabilities that outperform those obtained from the nonparametric bootstrap. Our proposed methodology is applied to the analysis of neuroimaging and genetic data collected as part of the Alzheimer's Disease Neuroimaging Initiative (ADNI), and this analysis of the ADNI cohort demonstrates clearly the value added of incorporating interval estimation beyond only point estimation when relating SNPs to brain imaging endophenotypes.

Linear regression is the most basic and the most widely used technique in machine learning; yet for all its simplicity, studying it can unlock some of the most important concepts in statistics. If you have a basic undestanding of linear regression expressed as \hat{Y} \theta_0 \theta_1X, but don't have a background in statistics and find statements like "ridge regression is equivalent to the maximum a posteriori (MAP) estimate with a zero-mean Gaussian prior" bewildering, then this post is for you. With a superficial goal of understanding that somewhat obtuse statement, its main objective is to explore the topic, starting from the standard formulation of linear regression, moving on to the probabilistic approach (maximum likelihood formulation) and from there to Bayesian linear regression. I'll use the \theta character throughout to refer to the coefficients (weights) of a regression model, either explicitly broken out as \theta_0 and \theta_1 for intercept and slope respectively, or just \theta referring to the vector of coefficients. I'll usually use the expression \theta Tx_i for the prediction a model gives at x_i, the assumption being that a 1 has been added to the vector of values at x_i . 1 In the single predictor case, we know that the least squares fit is the line that minimizes the sum of the squared distances between observed data and predicted values, i.e. it minimizes the Residual Sum of Squares (RSS): These residuals are pretty important in how we reason about our model.

This course is about the fundamental concepts of machine learning, focusing on neural networks, SVM and decision trees. These topics are getting very hot nowadays because these learning algorithms can be used in several fields from software engineering to investment banking. Learning algorithms can recognize patterns which can help detect cancer for example or we may construct algorithms that can have a very very good guess about stock prices movement in the market. In each section we will talk about the theoretical background for all of these algorithms then we are going to implement these problems together. The first chapter is about regression: very easy yet very powerful and widely used machine learning technique.

Feature learning and deep learning have drawn great attention in recent years as a way of transforming input data into more effective representations using learning algorithms. Such interest has grown in the area of music information retrieval (MIR) as well, particularly in music audio classification tasks such as auto-tagging. In this paper, we present a two-stage learning model to effectively predict multiple labels from music audio. The first stage learns to project local spectral patterns of an audio track onto a high-dimensional sparse space in an unsupervised manner and summarizes the audio track as a bag-of-features. The second stage successively performs the unsupervised learning on the bag-of-features in a layer-by-layer manner to initialize a deep neural network and finally fine-tunes it with the tag labels. Through the experiment, we rigorously examine training choices and tuning parameters, and show that the model achieves high performance on Magnatagatune, a popularly used dataset in music auto-tagging.

Obtaining labels can be costly and time-consuming. Active learning allows a learning algorithm to intelligently query samples to be labeled for efficient learning. Fisher information ratio (FIR) has been used as an objective for selecting queries in active learning. However, little is known about the theory behind the use of FIR for active learning. There is a gap between the underlying theory and the motivation of its usage in practice. In this paper, we attempt to fill this gap and provide a rigorous framework for analyzing existing FIR-based active learning methods. In particular, we show that FIR can be asymptotically viewed as an upper bound of the expected variance of the log-likelihood ratio. Additionally, our analysis suggests a unifying framework that not only enables us to make theoretical comparisons among the existing querying methods based on FIR, but also allows us to give insight into the development of new active learning approaches based on this objective.