This chapter covers principal component analysis and low-rank models, which are popular techniques to process high-dimensional datasets with many features. We begin by defining the mean of random vectors and random matrices. Then, we introduce the covariance matrix which encodes the variance of any linear combination of the entries in a random vector, and explain how to estimate it from data. We model the geographic location of Canadian cities as a running example. Next, we present principal component analysis (PCA), a method to extract the directions of maximum variance in a dataset. We explain how to use PCA to find optimal low-dimensional representations of high-dimensional data and apply it to a dataset of human faces. Then, we introduce low-rank models for matrix-valued data and describe how to fit them using the singular-value decomposition. We show that this approach is able to automatically identify meaningful patterns in real-world weather data. Finally, we explain how to estimate missing entries in a matrix under a low-rank assumption and apply this methodology to predict movie ratings via collaborative filtering.

This chapter introduces continuous random variables which enable us to model uncertain continuous quantities. We again begin with a formal definition, but quickly move on to describe how to manipulate continuous random variables in practice. We define the cumulative distribution function and quantiles (including the median) and explain how to estimate them from data. We then introduce the concept of probability density and describe its main properties. We present two approaches to obtain nonparametric models of probability densities from data: The histogram and kernel density estimation. Next, we define two celebrated continuous parametric distributions – the exponential and the Gaussian – and show how to fit them to data using maximum-likelihood estimation. We use these distributions to model the interarrival time of calls at a call center, and height in a population, respectively. Finally, we discuss how to simulate continuous random variables via inverse transform sampling.

This chapter provides an overview of the book, describing the contents of each chapter.

This chapter introduces probability. We begin with an informal definition which enables us to build intuition about the properties of probability. Then, we present a more rigorous definition, based on the mathematical framework of probability spaces. Next, we describe conditional probability, a concept that makes it possible to update probabilities when additional information is revealed. In our first encounter with statistics, we explain how to estimate probabilities and conditional probabilities from data, as illustrated by an analysis of votes in the United States Congress. Building upon the concept of conditional probability, we define independence and conditional independence, which are critical concepts in probabilistic modeling. The chapter ends with a surprising twist: In practice, probabilities are often impossible to compute analytically! Fortunately, the Monte Carlo method provides a pragmatic solution to this challenge, allowing us to approximate probabilities very accurately using computer simulations. We apply w 3 × 3 basketball tournament from the 2020 Tokyo Olympics.

In this chapter we present two spatial dependent models: one based on defining a latent variable for each area, and the other by defining one latent variable for each pair of latent areas. We call the latter the latent edges model. We compare both models with a real data set. Extensions to spatio-temporal constructions are also considered.

In this chapter we define what a conjugate family is in a Bayesian analysis context and develop detailed examples of some cases; in particular, we review the beta and binomial case, the Pareto and inverse Pareto case, the gamma and gamma case and the gamma and Poisson case. We conclude by providing a list of conjugate models.

In this chapter we show how to define temporal dependent sequences using a moving average type of construction. We compare the performance of this construction with a Markov-process type. We finally extend the models to include seasonal and periodic dependencies.

In this chapter we start with some attempts to construct dependence sequences with order larger than one and present a general result to achieve an invariant distribution via a three-level hierarchical model. We finally present some results involving exponential families.

In this chapter we describe a general procedure to construct Markov sequences with invariant distributions. The procedure can be used with conjugate and non-conjugate models and with parametric and nonparametric distributions. We derive several examples in detail and finish with some applications in survival analysis.

In this chapter we introduce the concept of exchangeability and show how to construct exchangeable sequences; we present our first result of how to construct exchangeable sequences and maintain a desirable marginal distribution and provide detailed examples. We finish with an application of exchangeable constructions in a meta analysis. Bugs and R code are provided.