Introduction to Bayesian Estimation and Copula Models of Dependence

3 Background for Markov Chain Monte Carlo

3.1 Randomization

3.1.1 Rolling Dice

It is a very old idea, even an ancient one: The idea that sometimes we lack information needed to make informed deterministic decisions, and therefore we resort to higher authorities. It happened in ancient Egypt and China, in pre-Columbian South and Central America [12], and was well-documented in Greece (e.g., the Oracle at Delphi) [6]. People used to consult the gods before making important decisions: to go to war or not to go to war, when to start harvesting, how to plan families, friendships, and strategic alliances. This communication with the higher powers was usually established through the institutes of priests equipped with special knowledge and special communication devices for communication with gods.

One of the older and more venerable devices for such communication is a die or a set of dice. In its modern version, a die is a six-sided cube with numbers or symbols on each of its sides. Older versions could be five sided (pichica of Incas [12]) or even double sided (a coin). Putting aside the esoteric nature of these objects, we can also consider them to be simple and handy pre-computer era tools of randomization.

Imagine a story from the old times of tribal wars. The Chief of a tribe, before raiding the neighbors’ lands, seeks the advice of the High Priest. Priests might not talk directly to the gods, but they are capable of collecting and analyzing information. Let us say that from a ...