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29
Updating Probabilities
, 2002
"... As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a "naive space", which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR ("coarsening a ..."
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Cited by 69 (4 self)
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As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a "naive space", which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR ("coarsening at random") in the statistical literature characterizes when "naive" conditioning in a naive space works. We show that the CAR condition holds rather infrequently, and we provide a procedural characterization of it, by giving a randomized algorithm that generates all and only distributions for which CAR holds. This substantially extends previous characterizations of CAR. We also consider more generalized notions of update such as Jeffrey conditioning and minimizing relative entropy (MRE). We give a generalization of the CAR condition that characterizes when Jeffrey conditioning leads to appropriate answers, and show that there exist some very simple settings in which MRE essentially never gives the right results. This generalizes and interconnects previous results obtained in the literature on CAR and MRE.
The rendezvous problem on discrete locations
 J. Appl. Probab
, 1990
"... Two friends have become separated in a building or shopping mall and and wish to meet as quickly as possible. There are n possible locations where they might meet. However, the locations are identical and there has been no prior agreement where to meet or how to search. Hence they must use identical ..."
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Cited by 54 (4 self)
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Two friends have become separated in a building or shopping mall and and wish to meet as quickly as possible. There are n possible locations where they might meet. However, the locations are identical and there has been no prior agreement where to meet or how to search. Hence they must use identical strategies and must treat all locations in a symmetrical fashion. Suppose their search proceeds in discrete time. Since they wish to avoid the possibility of never meeting, they will wish to use some randomizing strategy. If each person searches one of the n locations at random at each step, then rendezvous will require n steps on average. It is possible to do better than this: although the optimal strategy is difficult to characterize for general n, there is a strategy with an expected time until rendezvous of less than 0.829 n for large enough n. For n = 2 and 3 the optimal strategy can be established and on average 2 and 8/3 steps are required respectively. There are many tantalizing variations on this problem, which we discuss with some conjectures.
Asymmetric Rendezvous on the Plane
, 1997
"... We consider rendezvous problems in which two players move on the plane and wish to cooperate in order to minimise their first meeting time. We begin by considering the case when they know that they are a distance d apart, but they do not know the direction in which they should travel. We also cons ..."
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Cited by 22 (0 self)
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We consider rendezvous problems in which two players move on the plane and wish to cooperate in order to minimise their first meeting time. We begin by considering the case when they know that they are a distance d apart, but they do not know the direction in which they should travel. We also consider a situation in which player 1 knows the initial position of player 2, while player 2 is only given information on the initial distance of player 1. Finally we give some results for the case where one of the players is placed at an initial position chosen equiprobably from a finite set of points.
Dynamical bias in the coin toss
, 2004
"... We analyze the natural process of flipping a coin which is caught in the hand. We prove that vigorouslyflipped coins are biased to come up the same way they started. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measureme ..."
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Cited by 16 (0 self)
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We analyze the natural process of flipping a coin which is caught in the hand. We prove that vigorouslyflipped coins are biased to come up the same way they started. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on highspeed photography are reported. For natural flips, the chance of coming up as started is about.51.
Searching for the Next Best Mate
 In
, 1997
"... . How do we humans go about choosing a mate? Do we shop for them, checking prices and values and selecting the best? Do we apply for them, wooing several and taking the best that accepts us in return? Or do we screen them, testing one after another in succession until the right one comes along? Econ ..."
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Cited by 15 (4 self)
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. How do we humans go about choosing a mate? Do we shop for them, checking prices and values and selecting the best? Do we apply for them, wooing several and taking the best that accepts us in return? Or do we screen them, testing one after another in succession until the right one comes along? Economists and other behavioral scientists have analyzed these matechoice approaches to find their optimal algorithmic solutions; but what people really do is often quite different from these optima. In this paper, we analyze the third approach of mate choice as applicant screening and show through simulation analyses that a traditional optimal solution to this problemthe 37% rulecan be beaten along several dimensions by a class of simple "satisficing" algorithms we call the Take the Next Best mate choice rules. Thus, human mate search behavior should not necessarily be compared to the lofty optimal ideal, but instead may be more usefully studied through the development and analysis of poss...
The Lure of Choice
"... The contributors have asserted their moral rights. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior permission in writing of the publisher, nor be circulated in any form of binding or cover ..."
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Cited by 9 (0 self)
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The contributors have asserted their moral rights. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior permission in writing of the publisher, nor be circulated in any form of binding or cover other than that in which it is published. Typeset, printed and bound by:
Stochastic Theory of Early Viral Infection: Continuous versus Burst Production of Virions
, 2010
"... Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of production have been suggested. Standard viral dynamic models formulated as sets of ordinary differential equations can not distinguish between these two modes of ..."
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Cited by 9 (0 self)
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Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of production have been suggested. Standard viral dynamic models formulated as sets of ordinary differential equations can not distinguish between these two modes of viral production, as the predicted dynamics is identical as long as infected cells produce the same total number of virions over their lifespan. Here we show that in stochastic models of viral infection the two modes of viral production yield different early term dynamics. Further, we analytically determine the probability that infections initiated with any number of virions and infected cells reach extinction, the state when both the population of virions and infected cells vanish, and show this too has different solutions for continuous and burst production. We also compute the distributions of times to establish infection as well as the distribution of times to extinction starting from both a single virion as well as from a single infected cell for both modes of virion production.
The Monty Hall Problem is not a Probability Puzzle ∗ (It’s a challenge in mathematical modelling)
, 2010
"... Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door ..."
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Cited by 8 (1 self)
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Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2? ” Is it to your advantage to switch your choice? The answer is “yes ” but the literature offers many reasons why this is the correct answer. The present paper argues that the most common reasoning found in introductory statistics texts, depending on making a number of “obvious ” or “natural ” assumptions and then computing a conditional probability, is a classical example of solution driven science. The best reason to switch is to be found in von Neumann’s minimax theorem from game theory, rather than in Bayes’ theorem. 1
A Combinatorial Identity And The World Series
, 1993
"... In this note the author gives a simple probabilistic proof of a combinatorial identity by calculating the winning probability in the World Series. The winning probabilities and the expected length of the championship series are given by the applications of the identity and its generalization. ..."
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Cited by 5 (0 self)
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In this note the author gives a simple probabilistic proof of a combinatorial identity by calculating the winning probability in the World Series. The winning probabilities and the expected length of the championship series are given by the applications of the identity and its generalization.
Techniques for maintaining connectivity in wireless adhoc networks under energy constraints
 ACM Transaction on Embedded Computing Systems
, 2007
"... Distributed wireless systems (DWSs) are emerging as the enabler for nextgeneration wireless applications. There is a consensus that DWSbased applications, such as pervasive computing, sensor networks, wireless information networks, and speech and data communication networks, will form the backbone ..."
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Cited by 4 (4 self)
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Distributed wireless systems (DWSs) are emerging as the enabler for nextgeneration wireless applications. There is a consensus that DWSbased applications, such as pervasive computing, sensor networks, wireless information networks, and speech and data communication networks, will form the backbone of the next technological revolution. Simultaneously, with great economic, industrial, consumer, and scientific potential, DWSs pose numerous technical challenges. Among them, two are widely considered as crucial: autonomous localized operation and minimization of energy consumption. We address the fundamental problem of how to maximize the lifetime of the network using only local information, while preserving network connectivity. We start by introducing the carefree sleep (CS) Theorem that provides provably optimal conditions for a node to go into sleep mode while ensuring that global connectivity is not affected. The CS theorem is the basis for an efficient localized algorithm that decides which nodes will go to into sleep mode and for how long. We have also developed mechanisms for collecting neighborhood information and for the coordination of distributed energy minimization protocols. The effectiveness of the approach is demonstrated using a comprehensive study of the performance of the algorithm over a wide range of network parameters. Another important highlight is the first mathematical and Monte Carlo analysis that establishes the importance of considering nodes within a small number of hops in order to preserve energy.