
Lecture 9: Indistinguishability and Pseudorandomness (Sep 27, Anthony Chang)
... Hni is uniformly random, whereas bit i + 1 of Hni+1 is drawn from Xn . Define another hybrid H̃ni+1 = {l ← Xn , r ← Um(n) : l0→i ||1 − li+1 ||ri+2→m }. H̃ni+1 is exactly Hni+1 with bit i + 1 flipped; if D can distinguish Hni from Hni+1 , then it can certainly distinguish Hni+1 from H̃ni+1 . We can s ...
... Hni is uniformly random, whereas bit i + 1 of Hni+1 is drawn from Xn . Define another hybrid H̃ni+1 = {l ← Xn , r ← Um(n) : l0→i ||1 − li+1 ||ri+2→m }. H̃ni+1 is exactly Hni+1 with bit i + 1 flipped; if D can distinguish Hni from Hni+1 , then it can certainly distinguish Hni+1 from H̃ni+1 . We can s ...
Solutions
... Solution: Suppose instead f (σn ) = k. Let L be a list of numbers. Note that each permutation may be decomposed into a product of disjoint cycles. In order to determine the smallest n such that a given permutation has order k, we want to minimize the sum of all the numbers in L while making the gcd ...
... Solution: Suppose instead f (σn ) = k. Let L be a list of numbers. Note that each permutation may be decomposed into a product of disjoint cycles. In order to determine the smallest n such that a given permutation has order k, we want to minimize the sum of all the numbers in L while making the gcd ...
12_ClassNotes
... Examining Repeated Measures Data on Individuals Generalized linear mixed models (GLMM) also provide a means of incorporating longitudinal designs with categorical outcomes into situations where there are clustered data structures. One of the attractive properties of the GLMM is that it allows for li ...
... Examining Repeated Measures Data on Individuals Generalized linear mixed models (GLMM) also provide a means of incorporating longitudinal designs with categorical outcomes into situations where there are clustered data structures. One of the attractive properties of the GLMM is that it allows for li ...
pdf
... efficiently obtained (with some noise), even though f is the only function we have access to. Even having query access to f ∗ itself would not seem to help much at first glance, since the location of the relevant variables of f ∗ is unknown to us, and cannot be found without introducing a dependence ...
... efficiently obtained (with some noise), even though f is the only function we have access to. Even having query access to f ∗ itself would not seem to help much at first glance, since the location of the relevant variables of f ∗ is unknown to us, and cannot be found without introducing a dependence ...
Slides - Rutgers Statistics
... It is curious that this axiomatization is restrictive on the range of all probability functions: the real numbers in [0,1], and not a richer set; yet it is almost completely permissive about their domains: can be any set you like, however large, and F can be any field on , however large. ...
... It is curious that this axiomatization is restrictive on the range of all probability functions: the real numbers in [0,1], and not a richer set; yet it is almost completely permissive about their domains: can be any set you like, however large, and F can be any field on , however large. ...
Introduction to Artificial Intelligence – Course 67842
... Probabilities relate propositions to agent’s own beliefs, not directly about the world: P(A25 | no reported accidents) = 0.06 The beliefs depend on the percepts the agent has received. These are not claims of a “probabilistic tendency” in the current situation (but might be learned from past exper ...
... Probabilities relate propositions to agent’s own beliefs, not directly about the world: P(A25 | no reported accidents) = 0.06 The beliefs depend on the percepts the agent has received. These are not claims of a “probabilistic tendency” in the current situation (but might be learned from past exper ...
1. Discrete Distributions
... Recall that in the standard model of structural reliability, a systems consists of n components, each of which, independently of the others, is either working for failed. Let X i denote the state of component i, where 1 means working and 0 means failed. Thus, the state vector is X = ( X 1 , X 2 , .. ...
... Recall that in the standard model of structural reliability, a systems consists of n components, each of which, independently of the others, is either working for failed. Let X i denote the state of component i, where 1 means working and 0 means failed. Thus, the state vector is X = ( X 1 , X 2 , .. ...
Montana Curriculum Organizer: High School Mathematics Statistics
... High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the sy ...
... High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the sy ...
Probability interpretations

The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical tendency of something to occur or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory.There are two broad categories of probability interpretations which can be called ""physical"" and ""evidential"" probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as the dice yielding a six) tends to occur at a persistent rate, or ""relative frequency"", in a long run of trials. Physical probabilities either explain, or are invoked to explain, these stable frequencies. Thus talking about physical probability makes sense only when dealing with well defined random experiments. The two main kinds of theory of physical probability are frequentist accounts (such as those of Venn, Reichenbach and von Mises) and propensity accounts (such as those of Popper, Miller, Giere and Fetzer).Evidential probability, also called Bayesian probability (or subjectivist probability), can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility, or the degree to which the statement is supported by the available evidence. On most accounts, evidential probabilities are considered to be degrees of belief, defined in terms of dispositions to gamble at certain odds. The four main evidential interpretations are the classical (e.g. Laplace's) interpretation, the subjective interpretation (de Finetti and Savage), the epistemic or inductive interpretation (Ramsey, Cox) and the logical interpretation (Keynes and Carnap).Some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. The physical interpretation, for example, is taken by followers of ""frequentist"" statistical methods, such as R. A. Fisher, Jerzy Neyman and Egon Pearson. Statisticians of the opposing Bayesian school typically accept the existence and importance of physical probabilities, but also consider the calculation of evidential probabilities to be both valid and necessary in statistics. This article, however, focuses on the interpretations of probability rather than theories of statistical inference.The terminology of this topic is rather confusing, in part because probabilities are studied within a variety of academic fields. The word ""frequentist"" is especially tricky. To philosophers it refers to a particular theory of physical probability, one that has more or less been abandoned. To scientists, on the other hand, ""frequentist probability"" is just another name for physical (or objective) probability. Those who promote Bayesian inference view ""frequentist statistics"" as an approach to statistical inference that recognises only physical probabilities. Also the word ""objective"", as applied to probability, sometimes means exactly what ""physical"" means here, but is also used of evidential probabilities that are fixed by rational constraints, such as logical and epistemic probabilities.It is unanimously agreed that statistics depends somehow on probability. But, as to what probability is and how it is connected with statistics, there has seldom been such complete disagreement and breakdown of communication since the Tower of Babel. Doubtless, much of the disagreement is merely terminological and would disappear under sufficiently sharp analysis.