Arbitrarily large randomness distillation
... The randomness of this process depends crucially on the model that one uses to describe it. 1) The quantum state and measurement cannot be derived from the outcome probability distribution. 2) Even if they could, one cannot exclude a supra-quantum theory with more predictive power. ...
... The randomness of this process depends crucially on the model that one uses to describe it. 1) The quantum state and measurement cannot be derived from the outcome probability distribution. 2) Even if they could, one cannot exclude a supra-quantum theory with more predictive power. ...
The Flawed Probabilistic Foundation of Law and Economics
... the law. By the same token, the effectiveness of a promised reward depends on a person’s probability of actually being rewarded for taking an action that the law favors. This dependency is crucial. Because of informational asymmetries and the high costs of law enforcement, the legal system often fai ...
... the law. By the same token, the effectiveness of a promised reward depends on a person’s probability of actually being rewarded for taking an action that the law favors. This dependency is crucial. Because of informational asymmetries and the high costs of law enforcement, the legal system often fai ...
CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
... CAT(0) cube complexes are a higher dimensional generalization of trees, which arise naturally in the splitting theory of groups with codimension-1 subgroups [Sag95, Sag97]. A group is a-T-menable or has the Haagerup property [CCJ+ 01] if it admits a proper isometric action on a Hilbert space. This p ...
... CAT(0) cube complexes are a higher dimensional generalization of trees, which arise naturally in the splitting theory of groups with codimension-1 subgroups [Sag95, Sag97]. A group is a-T-menable or has the Haagerup property [CCJ+ 01] if it admits a proper isometric action on a Hilbert space. This p ...
Chap 4 from Ross
... is ever 1 or less at any time within the following four months? Solution: To find the desired probability, we consider a Markov chain with the state equal to the amount the pensioner has at the end of a month. Because we are interested in whether this amount ever falls as low as 1, we will let 1 mea ...
... is ever 1 or less at any time within the following four months? Solution: To find the desired probability, we consider a Markov chain with the state equal to the amount the pensioner has at the end of a month. Because we are interested in whether this amount ever falls as low as 1, we will let 1 mea ...
PLAUSIBILITY AND PROBABILITY IN SCENARIO
... In short, the two words have been confused with each other for centuries, and even although they became much more distinct in the last 300 years, today plausibility often still deploys probability to define itself. Plausibility, and its connections with the pliable notion of what can be applauded, q ...
... In short, the two words have been confused with each other for centuries, and even although they became much more distinct in the last 300 years, today plausibility often still deploys probability to define itself. Plausibility, and its connections with the pliable notion of what can be applauded, q ...
John Tabak-Probability and Statistics_ The Science of Uncertainty
... no physical explanation for the rain, or that the laws of cause and effect do not apply to weather. It does mean that, given the current state of meteorological theory and the current set of measurements available from satellites and weather stations, the meteorologist still has some uncertainty abo ...
... no physical explanation for the rain, or that the laws of cause and effect do not apply to weather. It does mean that, given the current state of meteorological theory and the current set of measurements available from satellites and weather stations, the meteorologist still has some uncertainty abo ...
On the round complexity of black-box constructions of
... The original version of this paper [21] claimed stronger versions of the results that were subsequently shown to be incorrect Ostrovsky et al. [18]. In particular, the original version claimed that 4 rounds (resp. 5 rounds) are necessary for PAR-CBCH (resp. PAR-SH), but this implicitly assumed that ...
... The original version of this paper [21] claimed stronger versions of the results that were subsequently shown to be incorrect Ostrovsky et al. [18]. In particular, the original version claimed that 4 rounds (resp. 5 rounds) are necessary for PAR-CBCH (resp. PAR-SH), but this implicitly assumed that ...
Distinguishing Hidden Markov Chains
... multi-monitor. If the two given HMCs are equivalent then even a multi-monitor can only guess. Now assume the two HMCs are not equivalent. It is known (see e.g. [28]) that then there exists a linear-length prefix of the observation sequence that is more likely in one HMC than in the other HMC. A mult ...
... multi-monitor. If the two given HMCs are equivalent then even a multi-monitor can only guess. Now assume the two HMCs are not equivalent. It is known (see e.g. [28]) that then there exists a linear-length prefix of the observation sequence that is more likely in one HMC than in the other HMC. A mult ...
1. Markov chains
... Notice how the probability is spreading out away from its initial concentration on the state 2. We could keep calculating n for more values of n, but it is intuitively clear what will happen: the probability will continue to spread out, and n will approach the uniform distribution: n ! ( 16 ; 61 ...
... Notice how the probability is spreading out away from its initial concentration on the state 2. We could keep calculating n for more values of n, but it is intuitively clear what will happen: the probability will continue to spread out, and n will approach the uniform distribution: n ! ( 16 ; 61 ...
A Joint Characterization of Belief Revision Rules
... function from D to P. Since we focus on belief revision, states are subjective probability functions. Speci…cally, we consider a …xed, non-empty set of worlds which for expositional simplicity is …nite or countably in…nite.7 Subsets of are called events. Let P be the set of probability measures ove ...
... function from D to P. Since we focus on belief revision, states are subjective probability functions. Speci…cally, we consider a …xed, non-empty set of worlds which for expositional simplicity is …nite or countably in…nite.7 Subsets of are called events. Let P be the set of probability measures ove ...
Twenty-One Arguments Against Propensity Analyses of Probability
... the single case; and (iii) how probability is counterfactually robust. But satisfying these three desiderata is not enough to uniquely fix the content of propensity approaches. As with any substantive philosophical project, Popper’s proposal soon splintered into many different subprojects. Popper’s ...
... the single case; and (iii) how probability is counterfactually robust. But satisfying these three desiderata is not enough to uniquely fix the content of propensity approaches. As with any substantive philosophical project, Popper’s proposal soon splintered into many different subprojects. Popper’s ...
Sample_Chapter - McGraw Hill Higher Education
... produce constant outputs. Stochastic simulations have random inputs and produce random outputs. Inputs might include activity times, arrival intervals, and routing sequences. Outputs include metrics such as average flow time, flow rate, and resource utilization. Any output impacted by a random input ...
... produce constant outputs. Stochastic simulations have random inputs and produce random outputs. Inputs might include activity times, arrival intervals, and routing sequences. Outputs include metrics such as average flow time, flow rate, and resource utilization. Any output impacted by a random input ...
A version of this paper appeared in Statistical Science (vol
... unlike games of chance, fair prices could not be deduced from assumptions about equal chances. Chances might not be equal. Probabilities in practical problems would have to be found from observation. Bernoulli proved, within his theory, that this would be possible. He proved that if a large number o ...
... unlike games of chance, fair prices could not be deduced from assumptions about equal chances. Chances might not be equal. Probabilities in practical problems would have to be found from observation. Bernoulli proved, within his theory, that this would be possible. He proved that if a large number o ...
Probability and Statistics
... individuals originating from identical ova, become very different from each other. In addition, the environment influences the conditions of selection; it thus changes the type both directly and obliquely. As long as such changes are reversible, their study is guided by the principles described abov ...
... individuals originating from identical ova, become very different from each other. In addition, the environment influences the conditions of selection; it thus changes the type both directly and obliquely. As long as such changes are reversible, their study is guided by the principles described abov ...
Dilation for Sets of Probabilities
... Ck+l,.. . , Cn) where C + = {C,, . . . , C,). Consider the events E i j = (C+- Ci) u (C-- Cj) for Ci E C+ and Cj E C-. Case 1: If there exists i, j such that Sp(A, Eij) < 1, then by Lemmas 3.2 and 3.4, Sp(A, (Ci u Cj)) > 1 since E,", = (Ci u Cj). We arrive at the following contradiction. Because the ...
... Ck+l,.. . , Cn) where C + = {C,, . . . , C,). Consider the events E i j = (C+- Ci) u (C-- Cj) for Ci E C+ and Cj E C-. Case 1: If there exists i, j such that Sp(A, Eij) < 1, then by Lemmas 3.2 and 3.4, Sp(A, (Ci u Cj)) > 1 since E,", = (Ci u Cj). We arrive at the following contradiction. Because the ...
RANDOM WALKS AND AN O∗(n5) VOLUME ALGORITHM FOR
... in O∗ (n4 ) steps. In order not to obscure our arguments by numerical considerations, we assume throughout that we can do exact real arithmetic, and that the separation oracle gives an exact answer. Neither of these assumptions is essential; we could do all our computations with a precision (say) ε∗ ...
... in O∗ (n4 ) steps. In order not to obscure our arguments by numerical considerations, we assume throughout that we can do exact real arithmetic, and that the separation oracle gives an exact answer. Neither of these assumptions is essential; we could do all our computations with a precision (say) ε∗ ...
Probability Theory
... Let us show that one of the events (1.3) and (1.4) will always take place, which would imply that the sum of their probabilities is at least 1, and hence prove (1.2). Indeed, assume that the event (1.3) does not take place, that is, some column contains only ...
... Let us show that one of the events (1.3) and (1.4) will always take place, which would imply that the sum of their probabilities is at least 1, and hence prove (1.2). Indeed, assume that the event (1.3) does not take place, that is, some column contains only ...
Infinite monkey theorem
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.