Lecture 1. Free Independence and Free Harmonic Analysis. 2 1.1. Probability spaces. 2 1.2. Non-commutative probability spaces. 3 1.3. Classical independence. 4 1.4. Free products of non-commutative probability spaces. 5 1.5. Free Fock space. 6 1.6. Free Central Limit Theorem. 8 1.7. Free Harmonic Analysis. 10 1.8. Further topics. 16 Lecture 2. Random Matrices and Free Probability. 17 2.1. Rando...

In the first half of the course, we covered topics from probability theory. The difference between statistics and probability theory is the following: In probability theory, we know everything about the underlying process that generates random variables. We then try to infer characteristics of these random variables. In statistics, however, we do not know the underlying process at all. To the c...

To treat probability rigorously, we define a sample space S whose elements are the possible outcomes of some process or experiment. For example, the sample space might be the outcomes of the roll of a die, or flips of a coin. To each element x of the sample space, we assign a probability, which will be a non-negative number between 0 and 1, which we will denote by p(x). We require that p(x) = ...

In a recent paper, Deutsch [1] claims to derive the “probabilistic predictions of quantum theory” from the “non-probabilistic axioms of quantum theory” and the “non-probabilistic part of classical decision theory.” We show that his derivation fails because it includes hidden probabilistic assumptions.

We first define Algorithmic Probability, an extremely powerful method of inductive inference. We discuss its completeness, incomputability, diversity and subjectivity and show that its incomputability in no way inhibits its use for practical prediction. Applications to Bernoulli sequence prediction and grammar discovery are described. We conclude with a note on its employment in a very strong A...

Perhaps it is not surprising that mathematics has always been popular among anti-evolutionists. Math is unique in its ability to bamboozle a lay audience, making it well-suited to their purposes. William Dembski, of Baylor University, represents the cutting edge in anti-Darwinian mathematics. His bailiwick is probability and information theory, which he fashions into a formidable, but ultimatel...

In this tutorial, I will discuss the concepts behind generalizing ordering to measuring and apply these ideas to the derivation of probability theory. The fundamental concept is that anything that can be ordered can be measured. Since we are in the business of making statements about the world around us, we focus on ordering logical statements according to implication. This results in a Boolean...

This is sometimes denoted simply “X−1(B) ⊂ F.” Since the probability measure P is only defined on sets F ∈ F, a random variable must satisfy this condition if we are to be able to find the probability Pr[X ∈ B] for each Borel set B, or even if we want to find the distribution function (DF) FX(b) ≡ Pr[X ≤ b] for each rational number b. Note that set-inverses are rather well-behaved functions fro...

In recent years books on probability theory have mushroomed. The two books under review have this in common : they are concerned with probability theory proper, rather than with its foundations (like the recent elegant but austere book by Neveu), and they give a balanced view of the theory rather than dwelling at length on some specialized topics. What is probability theory? Chung's spirited pr...