Evidence and Probability Data

This chapter discusses evidence and probability data with particular attention on Bayesian estimation. The Protestant ethic slowed probability developments in the United States, but the idea of quantification continued apace in England and on the Continent. In particular, Thomas Bayes invented a simple but profound mathematical means to connect outcomes with causes with conditional probabilities and Bayesian estimation. The chapter explains conditional probabilities and Bayesian logic, giving several examples, including incidence of accurate cancer diagnosis with inexact diagnostics. The chapter also introduces Bayes’s magnum opus An Essay Toward Solving a Problem in the Doctrine of Chances and gives his example of rolling billiard balls on a billiard table to show Bayes’s theorem.

REFLECTIONS ON THE HYPERBOLIC PLANE

The most general solution to the Einstein equations in 4 = 3 + 1 dimensions in the asymptotic limit close to the cosmological singularity under the BKL (Belinskii–Khalatnikov–Lifshitz) hypothesis can be visualized by the behavior of a billiard ball in a triangular domain on the Upper Poincaré Half Plane (UPHP). The billiard system (named "big billiard") can be schematized by dividing the successions of trajectories according to Poincaré return map on the sides of the billiard table, according to the paradigms implemented by the BKL investigation and by the CB–LKSKS (Chernoff–Barrow–Lifshitz–Khalatnikov–Sinai–Khanin–Shchur) one. Different maps are obtained, according to different symmetry-quotienting mechanisms used to analyze the dynamics. In the inhomogeneous case, new structures have been uncovered, such that, in this framework, the billiard table (named "small billiard") consists of 1/6 of the previous one. The connections between the symmetry-quotienting mechanisms are further investigated on the UPHP. The relation between the complete billiard and the small billiard are also further explained according to the role of Weyl reflections. The quantum properties of the system are sketched as well, and the physical interpretation of the wave function is further developed. In particular, a physical interpretation for the symmetry-quotienting maps is proposed.