An introduction to Markov chains

... Z × Z is infinite makes it a more complicated but at the same time more interesting mathematical object to study. We may still (with some work) answer simple questions of the form • What is the probability that the random walk is in state (2, 2) at time n = 10? • What is the probability that the ran ...

... Z × Z is infinite makes it a more complicated but at the same time more interesting mathematical object to study. We may still (with some work) answer simple questions of the form • What is the probability that the random walk is in state (2, 2) at time n = 10? • What is the probability that the ran ...

An Introduction to Statistical Signal Processing

... teaching experience. The emphasis of the book shifted increasingly towards examples and a viewpoint that better reflected the title of the courses we taught using the book for many years at Stanford University and at the University of Maryland: An Introduction to Statistical Signal Processing. Much ...

... teaching experience. The emphasis of the book shifted increasingly towards examples and a viewpoint that better reflected the title of the courses we taught using the book for many years at Stanford University and at the University of Maryland: An Introduction to Statistical Signal Processing. Much ...

Nucleation and growth in two dimensions

... different droplets. This is most important (and obvious) when k ≪ log n, when the process behaves almost exactly like bootstrap percolation (see Section 2.2), and nucleations are likely to be swallowed by a large ‘critical droplet’ before they have time to grow on their own. Above k = Θ(log n) this ...

... different droplets. This is most important (and obvious) when k ≪ log n, when the process behaves almost exactly like bootstrap percolation (see Section 2.2), and nucleations are likely to be swallowed by a large ‘critical droplet’ before they have time to grow on their own. Above k = Θ(log n) this ...

Plausibility Measures: A User`s Guide

... dual, although it does induce a dual ordering d on subsets of W , where A d B iff Pl(A) Pl(B ). Given the simplicity and generality of plausibility measures, we were not surprised to discover that Weber [1991] recently defined a notion of uncertainty measures, which is a slight generalization of ...

... dual, although it does induce a dual ordering d on subsets of W , where A d B iff Pl(A) Pl(B ). Given the simplicity and generality of plausibility measures, we were not surprised to discover that Weber [1991] recently defined a notion of uncertainty measures, which is a slight generalization of ...

How to Delegate Computations: The Power of No

... The study of MIPs that are secure against no-signaling provers was motivated by the study of MIPs with provers that share entangled quantum states. Recall that no-signaling provers are allowed to use arbitrary strategies, as long as their strategies cannot be used for communication between any two d ...

... The study of MIPs that are secure against no-signaling provers was motivated by the study of MIPs with provers that share entangled quantum states. Recall that no-signaling provers are allowed to use arbitrary strategies, as long as their strategies cannot be used for communication between any two d ...

CS 208: Automata Theory and Logic

... program, i.e. there are some undecidable formal languages. ...

... program, i.e. there are some undecidable formal languages. ...

Self-Referential Probability

... So T(w) is a (usually maximally consistent) set of sentences. Analogous to p. ...

... So T(w) is a (usually maximally consistent) set of sentences. Analogous to p. ...

Probability Essentials. Springer, Berlin, 2004.

... work of Kolmogorov, the French mathematician Paul Lévy (1886–1971) set the tone for modern Probability with his seminal work on Stochastic Processes as well as characteristic functions and limit theorems. We think of Probability Theory as a mathematical model of chance, or random events. The idea i ...

... work of Kolmogorov, the French mathematician Paul Lévy (1886–1971) set the tone for modern Probability with his seminal work on Stochastic Processes as well as characteristic functions and limit theorems. We think of Probability Theory as a mathematical model of chance, or random events. The idea i ...

Stable Beliefs and Conditional Probability Spaces

... about them. In chapter 4 we will define our notion of r-stable beliefs. Our definition will be similar to Leitgeb’s, but we will allow conditioning on sets with measure 0 as well. We will also prove some important properties of r-stable sets. We will show that our r-stable sets are well-founded w.r. ...

... about them. In chapter 4 we will define our notion of r-stable beliefs. Our definition will be similar to Leitgeb’s, but we will allow conditioning on sets with measure 0 as well. We will also prove some important properties of r-stable sets. We will show that our r-stable sets are well-founded w.r. ...

An Invitation to Sample Paths of Brownian Motion

... As a corollary of this definition, one can already remark that for all t, s ∈ I: Cov(Bt , Bs ) = s ∧ t. Indeed, assume that t ≥ s. Then Cov(Bt , Bs ) = Cov(Bt − Bs , Bs ) + Cov(Bs , Bs ) by bilinearity of the covariance. The first term vanishes by the independence of increments, and the second term ...

... As a corollary of this definition, one can already remark that for all t, s ∈ I: Cov(Bt , Bs ) = s ∧ t. Indeed, assume that t ≥ s. Then Cov(Bt , Bs ) = Cov(Bt − Bs , Bs ) + Cov(Bs , Bs ) by bilinearity of the covariance. The first term vanishes by the independence of increments, and the second term ...

A LINEAR ALGORITHM FOR THE RANDOM GENERATION OF REGULAR LANGUAGES

... regular languages and then describe the divide-and-conquer RGA. In Section 3, we analyze and compare the complexity of the two RGAs. In Section 4, we discuss implementation issues and compare experimental performances of both RGAs. We conclude in Section 5 with some remarks and possible extensions. ...

... regular languages and then describe the divide-and-conquer RGA. In Section 3, we analyze and compare the complexity of the two RGAs. In Section 4, we discuss implementation issues and compare experimental performances of both RGAs. We conclude in Section 5 with some remarks and possible extensions. ...

When Did Bayesian Inference Become “Bayesian”? Stephen E. Fienberg

... Problem in the Doctrine of Chances,” (14) contains what is arguably the first detailed description of the theorem from elementary probability theory now associated with his name. Bayes’ paper, which was submitted for publication by Richard Price, is remarkably opaque in its notation and details, and ...

... Problem in the Doctrine of Chances,” (14) contains what is arguably the first detailed description of the theorem from elementary probability theory now associated with his name. Bayes’ paper, which was submitted for publication by Richard Price, is remarkably opaque in its notation and details, and ...

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.