Markov Chains - Department of Mathematical Sciences
... i1 , X0 = i0 is the same as the conditional probability Xn+1 = j given only the previous state Xn = i. This is what we mean when we say that “given the current state any other information about the past is irrelevant for predicting ...
... i1 , X0 = i0 is the same as the conditional probability Xn+1 = j given only the previous state Xn = i. This is what we mean when we say that “given the current state any other information about the past is irrelevant for predicting ...
Lecture 5: Hashing with real numbers and their big-data applications
... Strictly speaking, one cannot hash to a real number since computers lack infinite precision. Instead, one hashes to rational numbers in [0, 1]. For instance, hash IP addresses to the set [p] as before, and then think of number “i mod p”as the rational number i/p. This works OK so long as our method ...
... Strictly speaking, one cannot hash to a real number since computers lack infinite precision. Instead, one hashes to rational numbers in [0, 1]. For instance, hash IP addresses to the set [p] as before, and then think of number “i mod p”as the rational number i/p. This works OK so long as our method ...
Lower Bounds for the Complexity of Reliable
... the size needed for circuits using only correct gates. By the noisy complexity of a function we mean the minimum number of gates needed for the reliable computation of the function. Note that in this model the circuit cannot be more reliable than its last gate. For a given function, the ratio of it ...
... the size needed for circuits using only correct gates. By the noisy complexity of a function we mean the minimum number of gates needed for the reliable computation of the function. Note that in this model the circuit cannot be more reliable than its last gate. For a given function, the ratio of it ...
CHAPTER III - MARKOV CHAINS 1. General Theory of Markov
... associated with the Markov chain is ergodic if the chain is indecomposable. Next we wish to show that under certain conditions on the transition matrix T the sequence is strong mixing. In order to do this we first prove a classical theorem in the theory of positive matrices. Theorem 2.1 (Perron-Frob ...
... associated with the Markov chain is ergodic if the chain is indecomposable. Next we wish to show that under certain conditions on the transition matrix T the sequence is strong mixing. In order to do this we first prove a classical theorem in the theory of positive matrices. Theorem 2.1 (Perron-Frob ...
The Multiplication Game The game
... like, with as many digits as you like. Your number is printed on the slip of paper along with the product of the two numbers. The dealer shows you the slip so that you can verify that the product is correct. You win if the first digit of the product is 4 through 9; you lose if it is 1, 2, or 3. The ...
... like, with as many digits as you like. Your number is printed on the slip of paper along with the product of the two numbers. The dealer shows you the slip so that you can verify that the product is correct. You win if the first digit of the product is 4 through 9; you lose if it is 1, 2, or 3. The ...
Full text in PDF form
... includes also Shannon’s entropy H. Considerations of choice of the value of α imply that exp(H) appears to be the most appropriate measure of Ess. Entropy and Ess can be viewed thanks to their log / exp relationship as two aspects of the same thing. In Probability and Statistics the Ess aspect could ...
... includes also Shannon’s entropy H. Considerations of choice of the value of α imply that exp(H) appears to be the most appropriate measure of Ess. Entropy and Ess can be viewed thanks to their log / exp relationship as two aspects of the same thing. In Probability and Statistics the Ess aspect could ...
PDF
... well. Model selection makes an arbitrary choice between these models, and therefore we cannot be confident that the model is a true representation of the underlying process. Given that there are many qualitatively different structures that are approximately equally good, we cannot learn a unique str ...
... well. Model selection makes an arbitrary choice between these models, and therefore we cannot be confident that the model is a true representation of the underlying process. Given that there are many qualitatively different structures that are approximately equally good, we cannot learn a unique str ...
Outline of the Monte Carlo Strategy Chapter 11
... This formula will become useful when transforming simple pseudo random number generators to more general ones. All the PDFs above have been written as functions of only one stochastic variable. Such PDFs are called univariate. A PDF may well consist of any number of variables, in which case we call ...
... This formula will become useful when transforming simple pseudo random number generators to more general ones. All the PDFs above have been written as functions of only one stochastic variable. Such PDFs are called univariate. A PDF may well consist of any number of variables, in which case we call ...
"Bayesian Data Analysis"(pdf)
... The abandonment of superstitious beliefs about...Fairies and Witches was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is not less a misleading misconception, an illusory attempt to exteriorize or ma ...
... The abandonment of superstitious beliefs about...Fairies and Witches was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is not less a misleading misconception, an illusory attempt to exteriorize or ma ...
Solutions to Problems for Math. H90 Issued 19 Oct. 2007
... Problem 4: This problem explores elementary probabilistic ideas. On a computer, a Random Number Generator U() can be construed as a “function” that takes no argument but produces, each time it is invoked, a random number independent of those it produces at other invocations. These random numbers are ...
... Problem 4: This problem explores elementary probabilistic ideas. On a computer, a Random Number Generator U() can be construed as a “function” that takes no argument but produces, each time it is invoked, a random number independent of those it produces at other invocations. These random numbers are ...
Proceedings of the Sixteenth Annual Conference on Uncertainty in Artificial... pages 201-210, Stanford, California, June 2000
... well. Model selection makes an arbitrary choice between these models, and therefore we cannot be confident that the model is a true representation of the underlying process. Given that there are many qualitatively different structures that are approximately equally good, we cannot learn a unique str ...
... well. Model selection makes an arbitrary choice between these models, and therefore we cannot be confident that the model is a true representation of the underlying process. Given that there are many qualitatively different structures that are approximately equally good, we cannot learn a unique str ...
Modeling Data Dissemination in Online Social Networks: A
... Recent years have witnessed a dramatic growth of user population of online social networks (OSNs). For example, according to the report in March 2013, Facebook has 1.11 billion people using the site each month, which represents a 23 percent growth from a year earlier [3]. OSNs are organized around u ...
... Recent years have witnessed a dramatic growth of user population of online social networks (OSNs). For example, according to the report in March 2013, Facebook has 1.11 billion people using the site each month, which represents a 23 percent growth from a year earlier [3]. OSNs are organized around u ...
Probability box
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A probability box (or p-box) is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes.An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds may have almost any shapes, including step functions, so long as they are monotonically increasing and do not cross each other. A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.