What follows is a collection of nice posts on the Bayesian interpretation of probability and its relevance to quantum theory. It turns out that a lot of arguments about the interpretation of quantum theory are at least partially arguments about the meaning of the probability! For example, suppose you have an electron in a state where the probability to measure its spin being "up" along the z axis is 50%. Then you measure its spin and find it is indeed up. The probability now jumps to 100%. What has happened? Did we "collapse the wavefunction" of the electron by means of some mysterious physical process? Or did we just change our prior based on new information? Bayesianism suggests the latter. This seems to imply that the "wavefunction" of the electron is just a summary of our assumptions about it, not someone we can ever measure. Some people find this infuriating; I find it explains a lot of things that otherwise seem mysterious.

There are a lot of tricky issues here. Quantum theory is more general than classical probability theory. It presents a lot of new puzzles of its own. But, at the very least, we need a clear understanding of what "probability" means before we can tackle these quantum quandaries.

I believe the frequentist interpretation just isn't good enough for understanding the role of probability in quantum theory. This is especially clear in quantum cosmology, where we apply quantum theory to the entire universe. We can't prepare a large number of identical copies of the whole universe to run experiments on!

This page dedicates to a general-purpose machine learning technique called Maximum Entropy Modeling (MaxEnt for short). On this page you will find:

Maximum Entropy Modeling tutorials
Maxent related software
Annotated papers on Maxent
Other Maxent resources on the web

Various approximations for distributions are studied, especially those involving the Binomial, Poisson, gamma, and Gaussian (normal) distributions. m-procedures are used to make comparisons. A simple approximation to a continuous random variable is obtained by subdividing an interval which includes the range (the set of possible values) into small enough subintervals that the density is approximately constant over each subinterval. A point in each subinterval is selected and is assigned the probability mass in its subinterval. The combination of the selected points and the corresponding probabilities describes the distribution of an approximating simple random variable. Calculations based on this distribution approximate corresponding calculations on the continuous distribution.

Edwin T. Jaynes was one of the first people to realize that probability theory, as originated by Laplace, is a generalization of Aristotelian logic that reduces to deductive logic in the special case that our hypotheses are either true or false. This web site has been established to help promote this interpretation of probability theory by distributing articles, books and related material. As Ed Jaynes originated this interpretation of probability theory we have a large selection of his articles, as well as articles by a number of other people who use probability theory in this way

In statistics, Bessel's correction, named after Friedrich Bessel, is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the number of observations in a sample: it corrects the bias in the estimation of the population variance, and some (but not all) of the bias in the estimation of the population standard deviation.

This introductory probability book, published by the American Mathematical Society, is available from AMS bookshop. It has, since publication, also been available for download here in pdf format. We are pleased that this has made our book more widely available.

Linearity of expectation basically says that the expected value of a sum of random variables
is equal to the sum of the individual expectations

Here you will find an attempt to offer an intuitive explanation of Bayesian reasoning - an excruciatingly gentle introduction that invokes all the human ways of grasping numbers, from natural frequencies to spatial visualization.  The intent is to convey, not abstract rules for manipulating numbers, but what the numbers mean, and why the rules are what they are (and cannot possibly be anything else).  When you are finished reading this page, you will see Bayesian problems in your dreams.

The uncertainties package handles calculations that involve numbers with uncertainties (like 3.14±0.01). It also transparently yields the derivatives of any expression (these derivatives are used for calculating uncertainties).

In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions.

This Note provides practical operational formulae to be used when one wishes to transform a probability density function of a random variable X to a random variable f(X) without affecting the underlying probability distribution. Such a process, often referred to as change of scale or transformation of coordinates, has nothing to do with the way the distribution function is displayed in a graph. This, along with the fact that the terms scale, axis and even coordinates are sometimes used interchangeably, is a frequent source of confusion.

Despite the apparent triviality of the whole matter, lengthy discussions often arise from the fact that probability density functions for f(X) are sometimes plotted in graphs with horizontal axis reporting a different function g(x). This, strictly speaking, is not illegal and, occasionally, it may be even justified by graph-appearance reasons.

What’s wrong with the probability notation used in Bayesian stats papers? The triple whammy of

1. overloading p() for every probability function,
2. using bound variables named after random variables, and
3. using the bound variable names to distinguish probability functions.

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. For reasons described below, real-world measurements are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly).

A set of nontransitive dice is a set of dice for which the relation "is more likely to roll a higher number" is not transitive. See also intransitivity. This situation is similar to that in the game Rock, Paper, Scissors, in which each element has an advantage over one choice and a disadvantage to the other.