In this article, we investigate large sample properties of model selection procedures in a general Bayesian framework when a closed form expression of the marginal likelihood function is not available or a local asymptotic quadratic approximation of the log-likelihood function does not exist. Under appropriate identifiability assumptions on the true model, we provide sufficient conditions for a Bayesian model selection procedure to be consistent and obey the Occam's razor phenomenon, i.e., the probability of selecting the "smallest" model that contains the truth tends to one as the sample size goes to infinity. In order to show that a Bayesian model selection procedure selects the smallest model containing the truth, we impose a prior anti-concentration condition, requiring the prior mass assigned by large models to a neighborhood of the truth to be sufficiently small. In a more general setting where the strong model identifiability assumption may not hold, we introduce the notion of local Bayesian complexity and develop oracle inequalities for Bayesian model selection procedures. Our Bayesian oracle inequality characterizes a trade-off between the approximation error and a Bayesian characterization of the local complexity of the model, illustrating the adaptive nature of averaging-based Bayesian procedures towards achieving an optimal rate of posterior convergence. Specific applications of the model selection theory are discussed in the context of high-dimensional nonparametric regression and density regression where the regression function or the conditional density is assumed to depend on a fixed subset of predictors. As a result of independent interest, we propose a general technique for obtaining upper bounds of certain small ball probability of stationary Gaussian processes.

In many real applications of statistical learning, a decision made from misclassification can be too costly to afford; in this case, a reject option, which defers the decision until further investigation is conducted, is often preferred. In recent years, there has been much development for binary classification with a reject option. Yet, little progress has been made for the multicategory case. In this article, we propose margin-based multicategory classification methods with a reject option. In addition, and more importantly, we introduce a new and unique refine option for the multicategory problem, where the class of an observation is predicted to be from a set of class labels, whose cardinality is not necessarily one. The main advantage of both options lies in their capacity of identifying error-prone observations. Moreover, the refine option can provide more constructive information for classification by effectively ruling out implausible classes. Efficient implementations have been developed for the proposed methods. On the theoretical side, we offer a novel statistical learning theory and show a fast convergence rate of the excess $\ell$-risk of our methods with emphasis on diverging dimensionality and number of classes. The results can be further improved under a low noise assumption. A set of comprehensive simulation and real data studies has shown the usefulness of the new learning tools compared to regular multicategory classifiers. Detailed proofs of theorems and extended numerical results are included in the supplemental materials available online.

David Freedman is the author of an excellent book: "Statistical Models: Theory and Practice" which discusses the issue of causation. It's a very unique stat book in that it really gets into the issue of model assumptions. It claims to be introductory but I believe that a semester or two of math stat as a pre-req would be helpful. In the time series context, you can run a VAR and then do tests for Granger Causality to see if one variable is really "causing" the other where "causing" is defined by Granger. R has a nice package called vars which makes building VAR models and doing testing extremely straightforward.

Large labeled training sets are the critical building blocks of supervised learning methods and are key enablers of deep learning techniques. For some applications, creating labeled training sets is the most time-consuming and expensive part of applying machine learning. We therefore propose a paradigm for the programmatic creation of training sets called data programming in which users express weak supervision strategies or domain heuristics as labeling functions, which are programs that label subsets of the data, but that are noisy and may conflict. We show that by explicitly representing this training set labeling process as a generative model, we can "denoise" the generated training set, and establish theoretically that we can recover the parameters of these generative models in a handful of settings. We then show how to modify a discriminative loss function to make it noise-aware, and demonstrate our method over a range of discriminative models including logistic regression and LSTMs. Experimentally, on the 2014 TAC-KBP Slot Filling challenge, we show that data programming would have led to a new winning score, and also show that applying data programming to an LSTM model leads to a TAC-KBP score almost 6 F1 points over a state-of-the-art LSTM baseline (and into second place in the competition). Additionally, in initial user studies we observed that data programming may be an easier way for non-experts to create machine learning models when training data is limited or unavailable.

Most tasks in Machine Learning can be reduced to classification tasks. For example, we have a medical dataset and we want to classify who has diabetes (positive class) and who doesn't (negative class). We have a dataset from the financial world and want to know which customers will default on their credit (positive class) and which customers will not (negative class). To do this, we can train a Classifier with a'training dataset' and after such a Classifier is trained (we have determined its model parameters) and can accurately classify the training set, we can use it to classify new data (test set). If the training is done properly, the Classifier should predict the class probabilities of the new data with a similar accuracy.

Machine learning has a high profile currently and is riding a wave of exposure in the media that includes articles about subjects from self-driving cars and self-landing rockets, to computers beating the world's best players at Go, the most computationally complex board game in the world. Is there an opportunity for your organisation, and the marketers within it, to make use of this "new" technology? Machine learning techniques were developed as long ago as the 1950s, but with the advent of big data and large analytical engines, the prevalence and the ease of applying the techniques has increased. Additionally, organisations now understand the value that analytics can bring, so are willing to place it front and center in their plans and invest more time and resources in exploring new and better techniques. Segmentation and predictive models, for instance, have proven themselves time and again in the marketing world, but to a certain extent, they require a higher degree of knowledge to understand. In some cases, a machine learning technique unburdens the user of the statistical work, but provides just as good an answer as a traditional technique.

Here is possible to find the application of the Naive Bayes approach to a specific problem: the classification of SMS into spam ("an undesired messages, e.g. The supporting code can be found here. The data used for such playground activity is the SMS Spam Collection v. 1, a public set of SMS messages that have been collected for mobile phone spam research where each message has been properly labeled as spam or ham. 'In machine learning and statistics, classification is the problem of identifying to which of a set of categories (sub-populations) a new observation belongs, on the basis of a training set of data containing observations (or instances) whose category membership is known. An example would be assigning a given email into "spam" or "non-spam" classes or assigning a diagnosis to a given patient as described by observed characteristics of the patient (gender, blood pressure, presence or absence of certain symptoms, etc.).

Processes such as disease propagation and information diffusion often spread over some latent network structure which must be learned from observation. Given a set of unlabeled training examples representing occurrences of an event type of interest (e.g., a disease outbreak), our goal is to learn a graph structure that can be used to accurately detect future events of that type. Motivated by new theoretical results on the consistency of constrained and unconstrained subset scans, we propose a novel framework for learning graph structure from unlabeled data by comparing the most anomalous subsets detected with and without the graph constraints. Our framework uses the mean normalized log-likelihood ratio score to measure the quality of a graph structure, and efficiently searches for the highest-scoring graph structure. Using simulated disease outbreaks injected into real-world Emergency Department data from Allegheny County, we show that our method learns a structure similar to the true underlying graph, but enables faster and more accurate detection.

Complex activity recognition is challenging due to the inherent uncertainty and diversity of performing a complex activity. Normally, each instance of a complex activity has its own configuration of atomic actions and their temporal dependencies. We propose in this paper an atomic action-based Bayesian model that constructs Allen's interval relation networks to characterize complex activities with structural varieties in a probabilistic generative way: By introducing latent variables from the Chinese restaurant process, our approach is able to capture all possible styles of a particular complex activity as a unique set of distributions over atomic actions and relations. We also show that local temporal dependencies can be retained and are globally consistent in the resulting interval network. Moreover, network structure can be learned from empirical data. A new dataset of complex hand activities has been constructed and made publicly available, which is much larger in size than any existing datasets. Empirical evaluations on benchmark datasets as well as our in-house dataset demonstrate the competitiveness of our approach.

Testing whether a probability distribution is compatible with a given Bayesian network is a fundamental task in the field of causal inference, where Bayesian networks model causal relations. Here we consider the class of causal structures where all correlations between observed quantities are solely due to the influence from latent variables. We show that each model of this type imposes a certain signature on the observable covariance matrix in terms of a particular decomposition into positive semidefinite components. This signature, and thus the underlying hypothetical latent structure, can be tested in a computationally efficient manner via semidefinite programming. This stands in stark contrast with the algebraic geometric tools required if the full observable probability distribution is taken into account. The semidefinite test is compared with tests based on entropic inequalities.