10.4: Probabilistic Reasoning: Rules of Probability
... • And strength in turn was characterized in the last lecture in terms of probability: A strong argument is one in which it is probable (but not necessary) that if the premises are true, then the conclusion is true. ...
... • And strength in turn was characterized in the last lecture in terms of probability: A strong argument is one in which it is probable (but not necessary) that if the premises are true, then the conclusion is true. ...
chapter 13_uncertainty
... unequivocal belief that the sentence is false, Assigning a probability of 1 corresponds to an unequivocal belief that the sentence is true. Probabilities between 0 and 1 correspond to intermediate degrees of belief in the truth of the sentence. The sentence itself is in fact either true or fal ...
... unequivocal belief that the sentence is false, Assigning a probability of 1 corresponds to an unequivocal belief that the sentence is true. Probabilities between 0 and 1 correspond to intermediate degrees of belief in the truth of the sentence. The sentence itself is in fact either true or fal ...
one - Celia Green
... trial - having to guess a green sweet, or having to guess a not-green sweet – are disjunctive, or mutually exclusive. So we have to add the probabilities of the two possible results. This again produces what may seem a rather paradoxical result: in neither situation – guessing a green sweet, and gue ...
... trial - having to guess a green sweet, or having to guess a not-green sweet – are disjunctive, or mutually exclusive. So we have to add the probabilities of the two possible results. This again produces what may seem a rather paradoxical result: in neither situation – guessing a green sweet, and gue ...
ON THE NUMBER OF VERTICES OF RANDOM CONVEX POLYHEDRA 1 Introduction
... Thus if we know E(ν M ) for every M = 1, 2, ..., then we also know N and this would solve a hard problem unsolved until now: what is the maximum number of vertices of the convex polyhedron (3.23) if m, n are fixed and a1 , . . . , an , b vary arbitrarily in the m-dimensional space. We only prove the ...
... Thus if we know E(ν M ) for every M = 1, 2, ..., then we also know N and this would solve a hard problem unsolved until now: what is the maximum number of vertices of the convex polyhedron (3.23) if m, n are fixed and a1 , . . . , an , b vary arbitrarily in the m-dimensional space. We only prove the ...
Doob: Half a century on - Imperial College London
... his background in pure mathematics (originally analytic number theory), Cramér was able to take up the lead of the Grundbegriffe and present, at greater length, a synthesis of much of the subject as it then stood. The title of the present piece is in part a tribute to Cramér (1976). Also coming to ...
... his background in pure mathematics (originally analytic number theory), Cramér was able to take up the lead of the Grundbegriffe and present, at greater length, a synthesis of much of the subject as it then stood. The title of the present piece is in part a tribute to Cramér (1976). Also coming to ...
Handling Uncertainties - using Probability Theory to
... The language system of probability theory, as most other systems, contains a set of axioms (un-provable statements which are known to be true in all cases) that are used to constrain the probabilities assigned to events. Three axioms of probability are as follows: 1. All values of probabilities are ...
... The language system of probability theory, as most other systems, contains a set of axioms (un-provable statements which are known to be true in all cases) that are used to constrain the probabilities assigned to events. Three axioms of probability are as follows: 1. All values of probabilities are ...
this paper - William M. Briggs
... alternative—is used to guide a user to select a particular model. Model correctness is not examined here. I take models as given, and instead look at the second question of probability assignment. That problem is huge, so here only a small piece of it is taken in the context of logical probability: ...
... alternative—is used to guide a user to select a particular model. Model correctness is not examined here. I take models as given, and instead look at the second question of probability assignment. That problem is huge, so here only a small piece of it is taken in the context of logical probability: ...
Entropy Measures vs. Kolmogorov Complexity
... The Kolmogorov complexity K(x) measures the amount of information contained in an individual object (usually a string) x, by the size of the smallest program that generates it. It naturally characterizes a probability distribution over Σ∗ (the set of all finite binary strings), assigning a probabili ...
... The Kolmogorov complexity K(x) measures the amount of information contained in an individual object (usually a string) x, by the size of the smallest program that generates it. It naturally characterizes a probability distribution over Σ∗ (the set of all finite binary strings), assigning a probabili ...
The Dynamics Of Projecting Confidence in Decision Making
... will win. She purchases an insurance policy out of fear of loss. In each case, she has a preference for probability distributions over gambling and or insurance. It is known that the cardinal utility function for gambling is convex, and that for insurance is concave. For example, Friedman and Savage ...
... will win. She purchases an insurance policy out of fear of loss. In each case, she has a preference for probability distributions over gambling and or insurance. It is known that the cardinal utility function for gambling is convex, and that for insurance is concave. For example, Friedman and Savage ...
Chinese-Whispers-Bas.. - Bayes
... judgements. In science we must always be as objective as possible. Probability judgements are like all the other judgements that a scientist necessarily makes, and should be argued for in the same careful, honest, informed way. ...
... judgements. In science we must always be as objective as possible. Probability judgements are like all the other judgements that a scientist necessarily makes, and should be argued for in the same careful, honest, informed way. ...
Here
... them. An attractive position is that all of these are reducible to one of them (think about how this might work). One position that has been defended is that temporal asymmetries are only apparent; to beings like us, with a temporally asymmetric perspective on the world, there seems to be a distinct ...
... them. An attractive position is that all of these are reducible to one of them (think about how this might work). One position that has been defended is that temporal asymmetries are only apparent; to beings like us, with a temporally asymmetric perspective on the world, there seems to be a distinct ...
Recherches sur la probabilité des jugements, principalement en
... the problem, with other points which it will be difficult to indicate in this preamble, but which will be examined scrupulously in the continuation of the work. The different solutions what one finds, either in the Traité des Probabilities2 , or in the first Supplément to this great work3 , have a ...
... the problem, with other points which it will be difficult to indicate in this preamble, but which will be examined scrupulously in the continuation of the work. The different solutions what one finds, either in the Traité des Probabilities2 , or in the first Supplément to this great work3 , have a ...
A Puzzle About Degree of Belief
... ‘correspond to’ the facts about L.A.’s location, they do not ‘fit’ the way things actually are. And it is a desideratum of a belief—perhaps the most important one—that it be true. For short, beliefs are governed by a norm of veracity. ...
... ‘correspond to’ the facts about L.A.’s location, they do not ‘fit’ the way things actually are. And it is a desideratum of a belief—perhaps the most important one—that it be true. For short, beliefs are governed by a norm of veracity. ...
A and B
... know what outcomes could happen, but we don’t know which particular outcome did or will happen. EX: chance the light will be red everyday on your way to work Light probably follows a timed pattern Chance of you hitting red light is random ...
... know what outcomes could happen, but we don’t know which particular outcome did or will happen. EX: chance the light will be red everyday on your way to work Light probably follows a timed pattern Chance of you hitting red light is random ...
Frequentism as a positivism: a three-tiered interpretation of probability
... There are two rather different senses in which physical chance appears in accounts of probability. One is the existence of physical theories, for example the Copenhagen and objective collapse interpretations of quantum mechanics, in which reality itself is nondeterministic and thus the existence of ...
... There are two rather different senses in which physical chance appears in accounts of probability. One is the existence of physical theories, for example the Copenhagen and objective collapse interpretations of quantum mechanics, in which reality itself is nondeterministic and thus the existence of ...
Entropy and Uncertainty
... and engineering but also in economics, linguistics, music, architecture, urban planning, social and cultural theory and even in creationism. Many of the scientific applications will be described in the lectures over the next three weeks. In this brief introductory lecture we describe some of the the ...
... and engineering but also in economics, linguistics, music, architecture, urban planning, social and cultural theory and even in creationism. Many of the scientific applications will be described in the lectures over the next three weeks. In this brief introductory lecture we describe some of the the ...
Chapter 6
... When the same chance process is repeated several times, we are often interested whether a particular outcome does or does not happen on each repetition. Here are some examples: - To test whether someone has extrasensory perception (ESP), choose one of four cards at random – a star, wave, cross, or c ...
... When the same chance process is repeated several times, we are often interested whether a particular outcome does or does not happen on each repetition. Here are some examples: - To test whether someone has extrasensory perception (ESP), choose one of four cards at random – a star, wave, cross, or c ...
the mathematical facts of games of chance between
... The slot games have gained and maintained top popularity despite their nontransparency with respect to parametric configuration, as this information is not exposed. Slots remains the only game in which players are not aware of the essential parameters of the game, such as number of stops of the reel ...
... The slot games have gained and maintained top popularity despite their nontransparency with respect to parametric configuration, as this information is not exposed. Slots remains the only game in which players are not aware of the essential parameters of the game, such as number of stops of the reel ...
What does it mean for something to be random? An event is called
... • a 999,999 in 1,000,000 chance of losing $1 and a 1 in a 1,000,000 chance of winning $1,000,000. In all three of these bets the expected value is 0, but the outcomes are very different. A lot of information about these bets is lost by considering only the expected values of the bets. In each case, ...
... • a 999,999 in 1,000,000 chance of losing $1 and a 1 in a 1,000,000 chance of winning $1,000,000. In all three of these bets the expected value is 0, but the outcomes are very different. A lot of information about these bets is lost by considering only the expected values of the bets. In each case, ...
Name Date ______ AP Biology Chi
... • The Rule of Addition: The chance of an event occurring when that event can occur two or more different ways is equal to the sum of the probabilities of each individual event Question: If 2 coins are tossed, what is the chance that the toss will yield 2 unmatched coins (1 head & 1 tail)? Answer: 1/ ...
... • The Rule of Addition: The chance of an event occurring when that event can occur two or more different ways is equal to the sum of the probabilities of each individual event Question: If 2 coins are tossed, what is the chance that the toss will yield 2 unmatched coins (1 head & 1 tail)? Answer: 1/ ...
A Characterization of Entropy in Terms of Information Loss
... f : {a, b} → {c}. Suppose p is the probability measure on {a, b} such that each point has measure 1/2, while q is the unique probability measure on the set {c}. Then H(p) = ln 2, while H(q) = 0. The information loss associated with the map f is defined to be H(p) − H(q), which in this case equals ln ...
... f : {a, b} → {c}. Suppose p is the probability measure on {a, b} such that each point has measure 1/2, while q is the unique probability measure on the set {c}. Then H(p) = ln 2, while H(q) = 0. The information loss associated with the map f is defined to be H(p) − H(q), which in this case equals ln ...
Exact upper tail probabilities of random series
... variables, but those estimates are not exact. The first exact upper tail probability was derived in [19] with i.i.d. nonnegative {ξj } having regular variation at infinity, where the coefficients {aj } could be random. This result was later generalized in [8], [9] and [14]. Recently there are severa ...
... variables, but those estimates are not exact. The first exact upper tail probability was derived in [19] with i.i.d. nonnegative {ξj } having regular variation at infinity, where the coefficients {aj } could be random. This result was later generalized in [8], [9] and [14]. Recently there are severa ...
3. Probability Measure
... is the relative frequency of A in the first n runs (it is also a random variable in the compound experiment). If we have chosen the correct probability measure for the experiment, then in some sense we expect that the relative frequency of each event should converge to the probability of the event: ...
... is the relative frequency of A in the first n runs (it is also a random variable in the compound experiment). If we have chosen the correct probability measure for the experiment, then in some sense we expect that the relative frequency of each event should converge to the probability of the event: ...
Math 411 Solutions to Exam 1 October 2, 2001 1. (10) A large basket
... One way to see that this is correct is to think of an n-tuple whose first k slots are filled with s’s and whose last (n − k) slots are filled with f’s. This represents the event that the first k tosses were heads and the last n − k tosses were tails. Since each toss is independent of the outcome of ...
... One way to see that this is correct is to think of an n-tuple whose first k slots are filled with s’s and whose last (n − k) slots are filled with f’s. This represents the event that the first k tosses were heads and the last n − k tosses were tails. Since each toss is independent of the outcome of ...
Lec2
... • What is he most probable classification of the new instance given the training data? • The most probable classification of the new instance is obtained by combining the prediction of all hypothesis, weighted by their posterior probabilities ...
... • What is he most probable classification of the new instance given the training data? • The most probable classification of the new instance is obtained by combining the prediction of all hypothesis, weighted by their posterior probabilities ...
History of randomness
In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. At the same time, most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.The Chinese were perhaps the earliest people to formalize odds and chance 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the sixteenth century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of modern calculus had a positive impact on the formal study of randomness. In the 19th century the concept of entropy was introduced in physics.The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, and mathematical foundations for probability were introduced, leading to its axiomatization in 1933. At the same time, the advent of quantum mechanics changed the scientific perspective on determinacy. In the mid to late 20th-century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness.Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the twentieth century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases, such randomized algorithms are able to outperform the best deterministic methods.