Of course as an introductory book, we can only leave it at that: an introductory book. For the mathematically trained, they may cure the curiosity this text generates with other texts designed with mathematical analysis in mind. For the enthusiast with less mathematical background, or one who is not interested in the mathematics but simply the practice of Bayesian methods, this text should be sufficient and entertaining. The choice of PyMC as the probabilistic programming language is two-fold. As of this writing, there is currently no central resource for examples and explanations in the PyMC universe.

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.

We consider maximum likelihood estimation for Gaussian Mixture Models (Gmms). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which is its ability to fulfill positive definiteness constraints in closed form is of key importance. We propose an alternative to EM by appealing to the rich Riemannian geometry of positive definite matrices, using which we cast Gmm parameter estimation as a Riemannian optimization problem. Surprisingly, such an out-of-the-box Riemannian formulation completely fails and proves much inferior to EM. This motivates us to take a closer look at the problem geometry, and derive a better formulation that is much more amenable to Riemannian optimization. We then develop (Riemannian) batch and stochastic gradient algorithms that outperform EM, often substantially. We provide a non-asymptotic convergence analysis for our stochastic method, which is also the first (to our knowledge) such global analysis for Riemannian stochastic gradient. Numerous empirical results are included to demonstrate the effectiveness of our methods.

Recently, impressive denoising results have been achieved by Bayesian approaches which assume Gaussian models for the image patches. This improvement in performance can be attributed to the use of per-patch models. Unfortunately such an approach is particularly unstable for most inverse problems beyond denoising. In this work, we propose the use of a hyperprior to model image patches, in order to stabilize the estimation procedure. There are two main advantages to the proposed restoration scheme: Firstly it is adapted to diagonal degradation matrices, and in particular to missing data problems (e.g. inpainting of missing pixels or zooming). Secondly it can deal with signal dependent noise models, particularly suited to digital cameras. As such, the scheme is especially adapted to computational photography. In order to illustrate this point, we provide an application to high dynamic range imaging from a single image taken with a modified sensor, which shows the effectiveness of the proposed scheme.

Nonlinear kernel regression models are often used in statistics and machine learning because they are more accurate than linear models. Variable selection for kernel regression models is a challenge partly because, unlike the linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog of each explanatory variable for Bayesian kernel regression models when the kernel is shift-invariant --- for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert spaces (RKHS) to define a linear vector space that: (i) captures nonlinear structure, and (ii) can be projected onto the original explanatory variables. The projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based on the random Fourier expansion we propose a computationally efficient class of Bayesian approximate kernel regression (BAKR) models for both nonlinear regression and binary classification for which one can compute an analog of effect sizes. We illustrate the utility of BAKR by examining two important problems in statistical genetics: genomic selection (i.e. phenotypic prediction) and association mapping (i.e. inference of significant variants or loci). State-of-the-art methods for genomic selection and association mapping are based on kernel regression and linear models, respectively. BAKR is the first method that is competitive in both settings.

We tackle the problem of collaborative filtering (CF) with side information, through the lens of Gaussian Process (GP) regression. Driven by the idea of using the kernel to explicitly model user-item similarities, we formulate the GP in a way that allows the incorporation of low-rank matrix factorisation, arriving at our model, the Tucker Gaussian Process (TGP). Consequently, TGP generalises classical Bayesian matrix factorisation models, and goes beyond them to give a natural and elegant method for incorporating side information, giving enhanced predictive performance for CF problems. Moreover we show that it is a novel model for regression, especially well-suited to grid-structured data and problems where the dependence on covariates is close to being separable.

Learning the structure of dependencies among multiple random variables is a problem of considerable theoretical and practical interest. In practice, score optimisation with multiple restarts provides a practical and surprisingly successful solution, yet the conditions under which this may be a well founded strategy are poorly understood. In this paper, we prove that the problem of identifying the structure of a Bayesian Network via regularised score optimisation can be recast, in expectation, as a submodular optimisation problem, thus guaranteeing optimality with high probability. This result both explains the practical success of optimisation heuristics, and suggests a way to improve on such algorithms by artificially simulating multiple data sets via a bootstrap procedure. We show on several synthetic data sets that the resulting algorithm yields better recovery performance than the state of the art, and illustrate in a real cancer genomic study how such an approach can lead to valuable practical insights.

Advanced driver assistance systems (ADAS) can be significantly improved with effective driver action prediction (DAP). Predicting driver actions early and accurately can help mitigate the effects of potentially unsafe driving behaviors and avoid possible accidents. In this paper, we formulate driver action prediction as a timeseries anomaly prediction problem. While the anomaly (driver actions of interest) detection might be trivial in this context, finding patterns that consistently precede an anomaly requires searching for or extracting features across multi-modal sensory inputs. We present such a driver action prediction system, including a real-time data acquisition, processing and learning framework for predicting future or impending driver action. The proposed system incorporates camera-based knowledge of the driving environment and the driver themselves, in addition to traditional vehicle dynamics. It then uses a deep bidirectional recurrent neural network (DBRNN) to learn the correlation between sensory inputs and impending driver behavior achieving accurate and high horizon action prediction. The proposed system performs better than other existing systems on driver action prediction tasks and can accurately predict key driver actions including acceleration, braking, lane change and turning at durations of 5sec before the action is executed by the driver.

Here we discuss general applications of statistical models, whether they arise from data science, operations research, engineering, machine learning or statistics. We do not discuss specific algorithms such as decision trees, logistic regression, Bayesian modeling, Markov models, data reduction or feature selection. Instead, I discuss frameworks - each one using its own types of techniques and algorithms - to solve real life problems. Most of the entries below are found in Wikipedia, and I have used a few definitions or extracts from the relevant Wikipedia articles, in addition to personal contributions. Spatial dependency is the co-variation of properties within geographic space: characteristics at proximal locations appear to be correlated, either positively or negatively.

Written by Chris Fonnesbeck, Assistant Professor of Biostatistics, Vanderbilt University Medical Center. You can view, fork, and play with this project on the Domino data science platform. A common applied statistics task involves building regression models to characterize non-linear relationships between variables. It is possible to fit such models by assuming a particular non-linear functional form, such as a sinusoidal, exponential, or polynomial function, to describe one variable's response to the variation in another. Unless this relationship is obvious from the outset, however, it involves possibly extensive model selection procedures to ensure the most appropriate model is retained. Alternatively, a non-parametric approach can be adopted by defining a set of knots across the variable space and use a spline or kernel regression to describe arbitrary non-linear relationships.