Statistics, models, and this book's approach

Statistics is about collecting, organizing, analyzing, and interpreting data, and hence statistical knowledge is essential for data analysis. Two main statistical methods are used in data analysis:

• Exploratory Data Analysis (EDA): This is about numerical summaries, such as the mean, mode, standard deviation, and interquartile ranges (this part of EDA is also known as descriptive statistics). EDA is also about visually inspecting the data, using tools you may be already familiar with, such as histograms and scatter plots.
• Inferential statistics: This is about making statements beyond the current data. We may want to understand some particular phenomenon, or maybe we want to make predictions for future (as yet unobserved) data points, or we need to choose among several competing explanations for the same observations. Inferential statistics is the set of methods and tools that will help us to answer these types of questions.

The focus of this book is upon how to perform Bayesian inferential statistics, and then how to use EDA to summarize, interpret, check, and communicate the results of Bayesian inference.

Most introductory statistical courses, at least for non-statisticians, are taught as a collection of recipes that more or less go like this: go to the statistical pantry, pick one tin can and open it, add data to taste, and stir until you obtain a consistent p-value, preferably under 0.05. The main goal in this type of course is to teach you how to pick the proper can. I never liked this approach, mainly because the most common result is a bunch of confused people unable to grasp, even at the conceptual level, the unity of the different learned methods. We will take a different approach: we will also learn some recipes, but this will be homemade rather than canned food; we will learn how to mix fresh ingredients that will suit different gastronomic occasions and, more importantly, that will let you to apply concepts far beyond the examples in this book.

Taking this approach is possible for two reasons:

• Ontological: Statistics is a form of modeling unified under the mathematical framework of probability theory. Using a probabilistic approach provides a unified view of what may seem like very disparate methods; statistical methods and machine learning (ML) methods look much more similar under the probabilistic lens.
• Technical: Modern software, such as PyMC3, allows practitioners, just like you and me, to define and solve models in a relative easy way. Many of these models were unsolvable just a few years ago or required a high level of mathematical and technical sophistication.

Working with data

Data is an essential ingredient in statistics and data science. Data comes from several sources, such as experiments, computer simulations, surveys, and field observations. If we are the ones in charge of generating or gathering the data, it is always a good idea to first think carefully about the questions we want to answer and which methods we will use, and only then proceed to get the data. In fact, there is a whole branch of statistics dealing with data collection, known as experimental design. In the era of the data deluge, we can sometimes forget that gathering data is not always cheap. For example, while it is true that the Large Hadron Collider (LHC) produces hundreds of terabytes a day, its construction took years of manual and intellectual labor.

As a general rule, we can think of the process generating the data as stochastic, because there is ontological, technical, and/or epistemic uncertainty, that is, the system is intrinsically stochastic, there are technical issues adding noise or restricting us from measuring with arbitrary precision, and/or there are conceptual limitations veiling details from us. For all these reasons, we always need to interpret data in the context of models, including mental and formal ones. Data does not speak but through models.

In this book, we will assume that we already have collected the data. Our data will also be clean and tidy, something rarely true in the real world. We will make these assumptions in order to focus on the subject of this book. I just want to emphasize, especially for newcomers to data analysis, that even when not covered in this book, these are important skills that you should learn and practice in order to successfully work with data.

A very useful skill when analyzing data is knowing how to write code in a programming language, such as Python. Manipulating data is usually necessary given that we live in a messy world with even messier data, and coding helps to get things done. Even if you are lucky and your data is very clean and tidy, coding will still be very useful since modern Bayesian statistics is done mostly through programming languages such as Python or R.

If you want to learn how to use Python for cleaning and manipulating data, I recommend reading the excellent book, Python Data Science Handbook, by Jake VanderPlas.

Bayesian modeling

Models are simplified descriptions of a given system or process that, for some reason, we are interested in. Those descriptions are deliberately designed to capture only the most relevant aspects of the system and not to explain every minor detail. This is one reason a more complex model is not always a better one.

There are many different kinds of models; in this book, we will restrict ourselves to Bayesian models. We can summarize the Bayesian modeling process using three steps:

1. Given some data and some assumptions on how this data could have been generated, we design a model by combining building blocks known as probability distributions. Most of the time these models are crude approximations, but most of the time is all we need.
2. We use Bayes' theorem to add data to our models and derive the logical consequences of combining the data and our assumptions. We say we are conditioning the model on our data.
3. We criticize the model by checking whether the model makes sense according to different criteria, including the data, our expertise on the subject, and sometimes by comparing several models.

In general, we will find ourselves performing these three steps in an iterative non-linear fashion. We will retrace our steps at any given point: maybe we made a silly coding mistake, or we found a way to change the model and improve it, or we realized that we need to add more data or collect a different kind of data.

Bayesian models are also known as probabilistic models because they are built using probabilities. Why probabilities? Because probabilities are the correct mathematical tool to model uncertainty, so let's take a walk through the garden of forking paths.