# Chapter 1. Probabilistic Reasoning

Among all the predictions that were made about the 21^{st} century, maybe the most unexpected one was that we would collect such a formidable amount of data about everything, everyday, and everywhere in the world. Recent years have seen an incredible explosion of data collection about our world, our lives, and technology; this is the main driver of what we can certainly call a revolution. We live in the Age of Information. But collecting data is nothing if we don't exploit it and try to extract knowledge out of it.

At the beginning of the 20^{th} century, with the birth of statistics, the world was all about collecting data and making statistics. In that time, the only reliable tools were pencils and paper and of course, the eyes and ears of the observers. Scientific observation was still in its infancy, despite the prodigious development of the 19^{th} century.

More than a hundred years later, we have computers, we have electronic sensors, we have massive data storage and we are able to store huge amounts of data continuously about, not only our physical world, but also our lives, mainly through the use of social networks, the Internet, and mobile phones. Moreover, the density of our storage technology has increased so much that we can, nowadays, store months if not years of data into a very small volume that can fit in the palm of our hand.

But storing data is not acquiring knowledge. Storing data is just keeping it somewhere for future use. At the same time as our storage capacity dramatically evolved, the capacity of modern computers increased too, at a pace that is sometimes hard to believe. When I was a doctoral student, I remember how proud I was when in the laboratory I received that brand-new, shiny, all-powerful PC for carrying my research work. Today, my old smart phone, which fits in my pocket, is more than 20 times faster.

Therefore in this book, you will learn one of the most advanced techniques to transform data into knowledge: **machine learning**. This technology is used in every aspect of modern life now, from search engines, to stock market predictions, from speech recognition to autonomous vehicles. Moreover it is used in many fields where one would not suspect it at all, from quality assurance in product chains to optimizing the placement of antennas for mobile phone networks.

Machine learning is the marriage between computer science and probabilities and statistics. A central theme in machine learning is the problem of inference or how to produce knowledge or predictions using an algorithm fed with data and examples. And this brings us to the two fundamental aspects of machine learning: the design of algorithms that can extract patterns and high-level knowledge from vast amounts of data and also the design of algorithms that can use this knowledge—or, in scientific terms: learning and inference.

Pierre-Simon Laplace (1749-1827) a French mathematician and one of the greatest scientists of all time, was presumably among the first to understand an important aspect of data collection: data is unreliable, uncertain and, as we say today, noisy. He was also the first to develop the use of probabilities to deal with such aspects of uncertainty and to represent one's degree of belief about an event or information.

In his *Essai philosophique sur les probabilités* (1814), Laplace formulated an original mathematical system for reasoning about new and old data, in which one's belief about something could be updated and improved as soon as new data where available. Today we call that Bayesian reasoning. Indeed Thomas Bayes was the first, toward the end of the 18^{th} century, to discover this principle. Without any knowledge about Bayes' work, Pierre-Simon Laplace rediscovered the same principle and formulated the modern form of the Bayes theorem. It is interesting to note that Laplace eventually learned about Bayes' posthumous publications and acknowledged Bayes to be the first to describe the principle of this inductive reasoning system. Today, we speak about Laplacian reasoning instead of Bayesian reasoning and we call it the Bayes-Price-Laplace theorem.

More than a century later, this mathematical technique was reborn thanks to new discoveries in computing probabilities and gave birth to one of the most important and used techniques in machine learning: the probabilistic graphical model.

From now on, it is important to note that the term **graphical** refers to the theory of graphs—that is, a mathematical object with nodes and edges (and not graphics or drawings). You know that, when you want to explain to someone the relationships between different objects or entities, you take a sheet of paper and draw boxes that you connect with lines or arrows. It is an easy and neat way to show relationships, whatever they are, between different elements.

**Probabilistic Graphical Models** (**PGM** for short) are exactly that: you want to describe relationships between variables. However, you don't have any certainty about your variables, but rather beliefs or uncertain knowledge. And we know now that probabilities are the way to represent and deal with such uncertainties, in a mathematical and rigorous way.

A probabilistic graphical model is a tool to represent beliefs and uncertain knowledge about facts and events using probabilities. It is also one of the most advanced machine learning techniques nowadays and has many industrial success stories.

Probabilistic graphical models can deal with our imperfect knowledge about the world because our knowledge is always limited. We can't observe everything, we can't represent all the universe in a computer. We are intrinsically limited as human beings, as are our computers. With probabilistic graphical models, we can build simple learning algorithms or complex expert systems. With new data, we can improve those models and refine them as much as we can and also we can infer new information or make predictions about unseen situations and events.

In this first chapter you will learn about the fundamentals needed to understand probabilistic graphical models; that is, probabilities and the simple rules of calculus on which they are based. We will have an overview of what we can do with probabilistic graphical models and the related R packages. These techniques are so successful that we will have to restrict ourselves to just the most important R packages.

We will see how to develop simple models, piece by piece, like a brick game and how to connect models together to develop even more advanced expert systems. We will cover the following concepts and applications and each section will contain numerical examples that you can directly use with R:

- Machine learning
- Representing uncertainty with probabilities
- Notions of probabilistic expert systems
- Representing knowledge with graphs
- Probabilistic graphical models
- Examples and applications