arXiv subjects

James Stein

Publications and source records attributed to James Stein.

3 recordsLinked to original sources

arXiv2026

Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials

Blackwell and Cover both proposed the same method for guessing the larger of a pair of two numbers when one can only observe one and the other of the two numbers remains unobserved. As originally stated, the procedure ended with a computation showing that the probability of correctly guessing the larger number was greater than one-half. In this paper, we investigate what we can say when about the number in the pair that was not observed we employ the Blackwell/Cover procedure. Let α1 and α2 be two different numbers in (0,1]. Let G be the uniform distribution on [0, α1] and H be the uniform distribution on [0, α2]. We think of G and H as being represented by the heads probabilities of coins. We use a variant of the Blackwell-Cover strategy ro take pairs of coins chosen from distributions created from G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. We show that there are situations in which recursive application of this process results in a set consisting almost completely of coins from one of the two distributions G and H, and another set consisting almost completely of coins from the other distribution.

math.PR↗

arXiv2026

Blackwells Demon: Postdiction and Prediction in Random Walks

Maxwells Demon is a mythical being, first described by the physicist James Clerk Maxwell (although named Maxwells Demon by Lord Kelvin). Maxwell used it in a thought experiment to potentially violate the Second Law of Thermodynamics by exploiting inhomogeneities existing in a statistically homogeneous system. Blackwells Demon, making (as far as is known) its first appearance in this paper, illustrates a counterintuitive situation occurring in a random walk variation of the Two Envelope problem[1], that it is possible under restrictive conditions to predict with success probability > 1/2 the direction of a random walk generated by the flip of a fair coin. Like Maxwells Demon, Blackwells Demon operates by exploiting inhomogeneities that exist in a statistically homogeneous system. Maxwells Demon achieves its results by knowing when a molecule is moving rapidly and when it is not. Blackwells Demon achieves its results by knowing when a prediction strategy is successful and when it is not. At the time Maxwell proposed his Demon, it confronted a technological Everest, the ability to open and close a gate permitting the passage of a single molecule, and the ability to gauge the speed of an approaching molecule. Blackwells Demon merely has to turn a light on and off, conduct visual observations and keep simple statistical records. It should be noted that the analysis in this paper does not demonstrate the ability to predict the flip of a fair coin ab initio with success probability > 1/2, as it is necessary to embed the fair coin in an environment of some complexity in order to achieve this result.

math.HO↗

arXiv2017

A Coin-Tossing Conundrum

It is shown that an equiprobability hypothesis leads to a scenario in which it is possible to predict the outcome of a single toss of a fair coin with a success probability greater than 50%. We discuss whether this hypothesis might be independent of the usual hypotheses governing probability, as well as whether this hypothesis might be assumed as a result of the Principle of Indifference. Also discussed are ways to implement or circumvent the hypothesis.

stat.OT↗