In this paper, we address the fundamental problem of line spectral estimation in a Bayesian framework. We target model order and parameter estimation via variational inference in a probabilistic model in which the frequencies are continuous-valued, i.e., not restricted to a grid; and the coefficients are governed by a Bernoulli-Gaussian prior model turning model order selection into binary sequence detection. Unlike earlier works which retain only point estimates of the frequencies, we undertake a more complete Bayesian treatment by estimating the posterior probability density functions (pdfs) of the frequencies and computing expectations over them. Thus, we additionally capture and operate with the uncertainty of the frequency estimates. Aiming to maximize the model evidence, variational optimization provides analytic approximations of the posterior pdfs and also gives estimates of the additional parameters. We propose an accurate representation of the pdfs of the frequencies by mixtures of von Mises pdfs, which yields closed-form expectations. We define the algorithm VALSE in which the estimates of the pdfs and parameters are iteratively updated. VALSE is a gridless, convergent method, does not require parameter tuning, can easily include prior knowledge about the frequencies and provides approximate posterior pdfs based on which the uncertainty in line spectral estimation can be quantified. Simulation results show that accounting for the uncertainty of frequency estimates, rather than computing just point estimates, significantly improves the performance. The performance of VALSE is superior to that of state-of-the-art methods and closely approaches the Cram\'er-Rao bound computed for the true model order.

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.

We study probability measures induced by set functions with constraints. Such measures arise in a variety of real-world settings, where prior knowledge, resource limitations, or other pragmatic considerations impose constraints. We consider the task of rapidly sampling from such constrained measures, and develop fast Markov chain samplers for them. Our first main result is for MCMC sampling from Strongly Rayleigh (SR) measures, for which we present sharp polynomial bounds on the mixing time. As a corollary, this result yields a fast mixing sampler for Determinantal Point Processes (DPPs), yielding (to our knowledge) the first provably fast MCMC sampler for DPPs since their inception over four decades ago. Beyond SR measures, we develop MCMC samplers for probabilistic models with hard constraints and identify sufficient conditions under which their chains mix rapidly. We illustrate our claims by empirically verifying the dependence of mixing times on the key factors governing our theoretical bounds.

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.

Like the course I just released on Hidden Markov Models, Recurrent Neural Networks are all about learning sequences - but whereas Markov Models are limited by the Markov assumption, Recurrent Neural Networks are not - and as a result, they are more expressive, and more powerful than anything we've seen on tasks that we haven't made progress on in decades. So what's going to be in this course and how will it build on the previous neural network courses and Hidden Markov Models? In the first section of the course we are going to add the concept of time to our neural networks. I'll introduce you to the Simple Recurrent Unit, also known as the Elman unit. We are going to revisit the XOR problem, but we're going to extend it so that it becomes the parity problem - you'll see that regular feedforward neural networks will have trouble solving this problem but recurrent networks will work because the key is to treat the input as a sequence.

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.

A complex activity consists of a set of temporally-composed events of atomic actions, which are the lowest-level events that can be directly detected from sensors. In other words, a complex activity is usually composed of multiple atomic actions occurring consecutively and concurrently over a duration of time. Modeling and recognizing complex activities remains an open research question as it faces several challenges: First, understanding complex activities calls for not only the inference of atomic actions, but also the interpretation of their rich temporal dependencies. Second, individuals often possess diverse styles of performing the same complex activity. As a result, a complex activity recognition model should be capable of capturing and propagating the underlying uncertainties over atomic actions and their temporal relationships. Third, a complex activity recognition model should also tolerate errors introduced from atomic action level, due to sensor noise or low-level prediction errors. A. Related Work Currently, a lot of research focuses on semantic-based complex activity modeling. Many semantic-based models such as context-free grammar (CFG) [26] and Markov logic network (MLN) [11], [18]) are used to represent complex activities, which can handle rich temporal relations.