ABSTRACT
Researchers would agree that statistical methods are of fundamental importance to their scientific investigations, which explains the substantial developments in statistical methodology and computing in recent years. On the other hand, there has been very little progress on the foundations of statistical inference. These foundational considerations are of the utmost importance, as they distinguish Statistics as a branch of Science and not just a collection of methods for scientists to use. Moreover, there are now alternatives-machine learning, data science, etc-poised to challenge Statistics, so setting a solid theoretical foundation is an urgent matter. This book attempts to lay out a new set of foundations that satisfy the goals of statistical inference, as we understand them, along with an approach, called inferential models (IMs), that fits within this new framework. Before we can get into these new details, we must first set some preliminary notation and concepts, which is the goal of this chapter. Section 1.2 fixes some basic ideas from probability theory and statistics that the reader is assumed to be familiar with. Section 1.3 attempts to lay out what, in our opinion, is the fundamental goal of scientific inference, and Section 1.4 gives a preview of one of the main ideas of the book, namely, that prediction plays a fundamental role in probabilistic inference. It is taken as the basic technique for propagating uncertainties specified in the sampling model to the space of quantities of interest, summarizing the knowledge gained about these quantities through observations. Finally, Section 1.5 provides an outline of the rest of this book.
