What does "causality" mean, and how can you represent it mathematically? How can you encode causal assumptions, and what bearing do they have on data analysis? These types of questions are at the core of the practice of data science, but deep knowledge about them is surprisingly uncommon. If you analyze data without regard to causality, you open your results up for the possibility of enormous biases. This includes everything from recommendation system results, to post-hoc reports on observational data, to experiments run without proper holdout groups. I've been blogging a lot recently about causality, and wanted to go through some of the material at a more technical level. Recent posts have been aimed at a more general audience. This one will be aimed at practitioners, and will assume a basic working knowledge of math and data analysis. To get the most from this post you should have a reasonable understanding of linear regression and probability (although we'll review a lot of probability). Prior knowledge of graphical models will make some concepts more familiar, but is not required. Judea Pearl, in his book Causality, constantly remarks that until very recently, causality was a concept in search of a language.

Probabilities can be consistently estimated using random forests. It is, however, unclear how random forests should be updated to make predictions for other centers or at different time points. The first method has been proposed by Elkan and may be used for updating any machine learning approach yielding consistent probabilities, so-called probability machines. The second approach is a new strategy specifically developed for random forests. Using the terminal nodes, which represent conditional probabilities, the random forest is first translated to logistic regression models.

In this paper, we study the performance of extremum estimators from the perspective of generalization ability (GA): the ability of a model to predict outcomes in new samples from the same population. By adapting the classical concentration inequalities, we derive upper bounds on the empirical out-of-sample prediction errors as a function of the in-sample errors, in-sample data size, heaviness in the tails of the error distribution, and model complexity. We show that the error bounds may be used for tuning key estimation hyper-parameters, such as the number of folds K in cross-validation. We also show how K affects the bias-variance tradeoff for cross-validation. Simulations are used to demonstrate key results. We would also like to acknowledge participants at the 12th International Symposium on Econometric Theory and Applications and the 26th New Zealand Econometric Study Group as well as seminar participants at Utah, UNSW, and University of Melbourne for useful questions and comments. Fisher would like to acknowledge the financial support of the Australian Research Council, grant DP0663477. 1 1 Introduction Traditionally in econometrics, an estimation method is implemented on sample data in order to infer patterns in a population. Put another way, inference centers on generalizing to the population the pattern learned from the sample and evaluating how well the sample pattern fits the population. An alternative perspective is to consider how well a sample pattern fits another sample. In this paper, we study the ability of a model estimated from a given sample to fit new samples from the same population, referred to as the generalization ability (GA) of the model. As a way of evaluating the external validity of sample estimates, the concept of GA has been implemented in recent empirical research. For example, in the policy evaluation literature [Belloni et al., 2013, Gechter, 2015, Dolton, 2006, Blundell et al., 2004], the central question is whether any treatment effect estimated from a pilot program can be generalized to out-of-sample individuals.

ABSTRACT We study the relationship between online Gaussian process (GP) regression and kernel least mean squares (KLMS) algorithms. While the latter have no capacity of storing the entire posterior distribution during online learning, we discover that their operation corresponds to the assumption of a fixed posterior covariance that follows a simple parametric model. Interestingly, several well-known KLMS algorithms correspond to specific cases of this model. The probabilistic perspective allows us to understand how each of them handles uncertainty, which could explain some of their performance differences. Index Terms-- online learning, regression, Gaussian processes, kernel least-mean squares 1. INTRODUCTION Gaussian Process (GP) regression is a state-of-the-art Bayesian technique for nonlinear regression [1].

Sorry if this is too trivial, but let me start at the "very beginning:" Linear regression. The goal of (ordinary least-squares) linear regression is to find the optimal weights that -- when linearly combined with the inputs -- result in a model that minimizes the vertical offsets between the target and explanatory variables, but let's not get distracted by model fitting, which is a different topic;). So, in linear regression, we compute a linear combination of weights and inputs (let's call this function the "net input function"). Next, let's consider logistic regression. Here, we put the net input z through a non-linear "activation function" -- the logistic sigmoid function where.

Robust Bayesian models are appealing alternatives to standard models, providing protection from data that contains outliers or other departures from the model assumptions. Historically, robust models were mostly developed on a case-by-case basis; examples include robust linear regression, robust mixture models, and bursty topic models. In this paper we develop a general approach to robust Bayesian modeling. We show how to turn an existing Bayesian model into a robust model, and then develop a generic strategy for computing with it. We use our method to study robust variants of several models, including linear regression, Poisson regression, logistic regression, and probabilistic topic models. We discuss the connections between our methods and existing approaches, especially empirical Bayes and James-Stein estimation.

We describe an application of machine learning to the problem of predicting preterm birth. We conduct a secondary analysis on a clinical trial dataset collected by the National In- stitute of Child Health and Human Development (NICHD) while focusing our attention on predicting different classes of preterm birth. We compare three approaches for deriving predictive models: a support vector machine (SVM) approach with linear and non-linear kernels, logistic regression with different model selection along with a model based on decision rules prescribed by physician experts for prediction of preterm birth. Our approach highlights the pre-processing methods applied to handle the inherent dynamics, noise and gaps in the data and describe techniques used to handle skewed class distributions. Empirical experiments demonstrate significant improvement in predicting preterm birth compared to past work.

We describe GTApprox - a new tool for medium-scale surrogate modeling in industrial design. Compared to existing software, GTApprox brings several innovations: a few novel approximation algorithms, several advanced methods of automated model selection, novel options in the form of hints. We demonstrate the efficiency of GTApprox on a large collection of test problems. In addition, we describe several applications of GTApprox to real engineering problems.

We develop an automated variational method for inference in models with Gaussian process (GP) priors and general likelihoods. The method supports multiple outputs and multiple latent functions and does not require detailed knowledge of the conditional likelihood, only needing its evaluation as a black-box function. Using a mixture of Gaussians as the variational distribution, we show that the evidence lower bound and its gradients can be estimated efficiently using empirical expectations over univariate Gaussian distributions. Furthermore, the method is scalable to large datasets which is achieved by using an augmented prior via the inducing-variable approach underpinning most sparse GP approximations, along with parallel computation and stochastic optimization. We evaluate our method with experiments on small datasets, medium-scale datasets and a large dataset, showing its competitiveness under different likelihood models and sparsity levels. Moreover, we analyze learning in our model under batch and stochastic settings, and study the effect of optimizing the inducing inputs. Finally, in the large-scale experiment, we investigate the problem of predicting airline delays and show that our method is on par with the state-of-the-art hard-coded approach for scalable GP regression.

The SLOPE estimates regression coefficients by minimizing a regularized residual sum of squares using a sorted-$\ell_1$-norm penalty. The SLOPE combines testing and estimation in regression problems. It exhibits suitable variable selection and prediction properties, as well as minimax optimality. This paper introduces the Bayesian SLOPE procedure for linear regression. The classical SLOPE estimate is the posterior mode in the normal regression problem with an appropriate prior on the coefficients. The Bayesian SLOPE considers the full Bayesian model and has the advantage of offering credible sets and standard error estimates for the parameters. Moreover, the hierarchical Bayesian framework allows for full Bayesian and empirical Bayes treatment of the penalty coefficients; whereas it is not clear how to choose these coefficients when using the SLOPE on a general design matrix. A direct characterization of the posterior is provided which suggests a Gibbs sampler that does not involve latent variables. An efficient hybrid Gibbs sampler for the Bayesian SLOPE is introduced. Point estimation using the posterior mean is highlighted, which automatically facilitates the Bayesian prediction of future observations. These are demonstrated on real and synthetic data.