Centered on multivariate probabilities, this chapter unravels the intricacies of joint probabilities, covariance matrices, and multivariate normal distributions. The discussion on conditional probability and Bayes’ theorem provides a robust foundation for modeling complex relationships between several variables.

Diving into stochastic processes, this chapter explores stationarity, scaling laws, and fractal dimensions. It delves into diverse processes, from random walks to more general self-affine processes, unraveling their implications in modeling complex phenomena. In line with the rest of the book, it discusses the modeling of stochastic processes as an instance of multivariate sets of random variables.

This chapter navigates the construction of networks from data. Various network-building tools from thresholding to information filtering are introduced and discussed. The reader is guided through the use of network representations for the construction of effective multivariate probabilistic models.

This chapter probes the intricate relationship between cause and effect. It navigates through Wiener–Granger causality and introduces transfer entropy, providing the tools to dissect causal relationships in data-driven models.

Within this chapter, the focus is on delving into univariate probability distributions. It dissects the anatomy of the normal distribution, explores characteristic functions, and analyzes stable distributions. This chapter establishes a comprehensive understanding of various probability distributions, setting the stage for more advanced discussions.

This chapter delves into methods for comparing probability estimates and assessing the goodness of models. Tools and methodologies such as null models, p-values, and Bayesian model selections are introduced and discussed. From regression to classification, this chapter illuminates model evaluation via likelihood and model selection techniques.

Unveiling the intricate world of networks and graphs, this chapter unearths the significance of adjacency matrices, centrality, and propagation. From paths to higher-order networks, readers gain insights into employing network structures to model complexities inherent in diverse systems.

Focusing on networks as potent representations of complex, real systems, this chapter unearths network construction, information filtering, and higher-order networks. Readers are empowered to capture the essence of intricate relationships within complex systems by using network representations.

The concluding chapter reflects on the scientific method, automated model construction, and the evolving landscape of modeling. It contemplates the balance between complexity and interpretability, providing insights into the future of data-driven modeling.

This chapter is dedicated to nonparametric estimation methods, dissecting the sample mean, moments, and probability mass functions. It delves into the convergence laws for the sample means, enabling the construction of probability distributions from empirical data.