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Sunrise Problem
The sunrise problem can be expressed as follows: "What is the probability that the sun will rise tomorrow?" The sunrise problem illustrates the difficulty of using probability theory when evaluating the plausibility of statements or beliefs. According to the Bayesian interpretation of probability, probability theory can be used to evaluate the plausibility of the statement, "The sun will rise tomorrow." The sunrise problem was first introduced publicly in 1763 by Richard Price in his famous coverage of Thomas Bayes' foundational work in Bayesianism. Laplace's approach Pierre-Simon Laplace, who treated it by means of his rule of succession. Let ''p'' be the long-run frequency of sunrises, i.e., the sun rises on 100 × ''p''% of days. ''Prior'' to knowing of any sunrises, one is completely ignorant of the value of ''p''. Laplace represented this prior ignorance by means of a uniform probability distribution on ''p''. For instance, the probability that ''p'' is betw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buck Creek IN - Sunrise
Buck may refer to: Common meanings * A colloquialism for a United States, Canadian, or Australian dollar * An adult male animal in some species - see List of animal names - e.g.: ** Goat buck ** Deer buck ** Sheep buck * Derby shoes, nicknamed "bucks" in modern colloquial English, for the common use of buckskin in their making Arts and entertainment * BUCK, a ''My Little Pony'' fan convention in Manchester, UK * Buck, someone who excels in the krump dance style * ''Buck'' (film), a 2011 documentary * ''Buck'' (magazine), a defunct UK publication (2008–2011) * '' Buck: A Memoir'', a 2013 book by MK Asante Companies * Buck (design company), a design-driven creative commercial production company * Buck (human resources consulting company), a human resources consulting company * Buck Knives, an American knife manufacturer Fictional characters * Buck, a character in the 2004 American animated Western musical comedy movie '' Home on the Range'' * Buck, a character in the fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Probability
In probability theory, conditional probability is a measure of the probability of an Event (probability theory), event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This particular method relies on event A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as or occasionally . This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac. For example, the probability that any given person has a cough on any given day ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Problems
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Smoothing
In statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth count data, eliminating issues caused by certain values having 0 occurrences. Given a set of observation counts \mathbf = \langle x_1, x_2, \ldots, x_d \rangle from a d-dimensional multinomial distribution with N trials, a "smoothed" version of the counts gives the estimator : \hat\theta_i = \frac \qquad (i = 1, \ldots, d), where the smoothed count \hat x_i = N \hat\theta_i, and the "pseudocount" ''α'' > 0 is a smoothing parameter, with ''α'' = 0 corresponding to no smoothing (this parameter is explained in below). Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability ( relative frequency) x_i/N and the uniform probability 1/d. Common choices for ''α'' are 0 (no smoothing), (the Jeffreys prior), or 1 (Laplace's rule of succession), but the parameter may also be set empi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unsolved Problems In Statistics
There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The notable unsolved problems in statistics are generally of a different flavor; according to John Tukey, "difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." A list of "one or two open problems" (in fact 22 of them) was given by David Cox. Inference and testing * How to detect and correct for systematic errors, especially in sciences where random errors are large (a situation Tukey termed uncomfortable science). * The Graybill–Deal estimator is often used to estimate the common mean of two normal populations with unknown and possibly unequal variances. Though this estimator is generally unbiased, its admissibility remains to be shown. * Meta-analysis: Though independent p-values can be combined using Fisher's method, techniques are still being developed to handle the case of dependent p-values. * Behrens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newcomb's Paradox
In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future with near-certainty. Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore Laboratory. However, it was first analyzed in a philosophy paper by Robert Nozick in 1969 and appeared in the March 1973 issue of ''Scientific American'', in Martin Gardner's " Mathematical Games". Reprinted with an addendum and annotated bibliography in his book ''The Colossal Book of Mathematics'' (). Today it is a much debated problem in the philosophical branch of decision theory. The problem There are two agents: a reliable predictor and a player. Two boxes are designated A and B. The player is given a choice between taking only box B or taking both boxes A and B. The player knows the following: * Box A is transparent and always contains a visible $1,000. * Box B is opa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doomsday Argument
The doomsday argument (DA), or Carter catastrophe, is a probabilistic argument that claims to predict the future population of the human species based on an estimation of the number of humans born to date. The doomsday argument was originally proposed by the astrophysicist Brandon Carter in 1983, leading to the initial name of the Carter catastrophe. The argument was subsequently championed by the philosopher John A. Leslie and has since been independently conceived by J. Richard Gott and Holger Bech Nielsen. Similar principles of eschatology were proposed earlier by Heinz von Foerster, among others. A more general form was given earlier in the Lindy effect, which proposes that for certain phenomena, the future life expectancy is proportional to (though not necessarily equal to) the current age and is based on a decreasing mortality rate over time. Summary The premise of the argument is as follows: suppose that the total number of human beings who will ever exist is fixed. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problem Of Induction
The problem of induction is a philosophical problem that questions the rationality of predictions about unobserved things based on previous observations. These inferences from the observed to the unobserved are known as "inductive inferences". David Hume, who first formulated the problem in 1739, argued that there is no non-circular way to justify inductive inferences, while he acknowledged that everyone does and must make such inferences. The traditional Inductivism, inductivist view is that all claimed Empirical evidence, empirical laws, either in everyday life or through the scientific method, can be justified through some form of reasoning. The problem is that many philosophers tried to find such a justification but their proposals were not accepted by others. Identifying the inductivist view as the scientific view, C. D. Broad once said that induction is "the glory of science and the scandal of philosophy". In contrast, Karl Popper's critical rationalism claimed that indu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Succession
In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. The formula is still used, particularly to estimate underlying probabilities when there are few observations or events that have not been observed to occur at all in (finite) sample data. Statement of the rule of succession If we repeat an experiment that we know can result in a success or failure, ''n'' times independently, and get ''s'' successes, and ''n − s'' failures, then what is the probability that the next repetition will succeed? More abstractly: If ''X''1, ..., ''X''''n''+1 are conditionally independent random variables that each can assume the value 0 or 1, then, if we know nothing more about them, :P(X_=1 \mid X_1+\cdots+X_n=s)=. Interpretation Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had observ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reference Class Problem
In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case. For example, to estimate the probability of an aircraft crashing, we could refer to the frequency of crashes among various different sets of aircraft: all aircraft, this make of aircraft, aircraft flown by this company in the last ten years, etc. In this example, the aircraft for which we wish to calculate the probability of a crash is a member of many different classes, in which the frequency of crashes differs. It is not obvious which class we should refer to for this aircraft. In general, any case is a member of very many classes among which the frequency of the attribute of interest differs. The reference class problem discusses which class is the most appropriate to use. More formally, many arguments in statistics take the form of a statistical syllogism: #X proportion of F are G #I is an F #Therefore, the chance that I is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edwin Thompson Jaynes
Edwin Thompson Jaynes (July 5, 1922 – April 30, 1998) was the Wayman Crow Distinguished Professor of Physics at Washington University in St. Louis. He wrote extensively on statistical mechanics and on foundations of probability and statistical inference, initiating in 1957 the maximum entropy interpretation of thermodynamics as being a particular application of more general Bayesian/ information theory techniques (although he argued this was already implicit in the works of Josiah Willard Gibbs). Jaynes strongly promoted the interpretation of probability theory as an extension of logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o .... In 1963, together with his doctoral student Fred Cummings, he modeled the evolution of a two-level atom in an electromagnetic field, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayes' Theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole. Based on Bayes' law, both the prevalence of a disease in a given population and the error rate of an infectious disease test must be taken into account to evaluate the meaning of a positive test result and avoid the ''base-rate fallacy''. One of Bayes' theorem's many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of Realization (probability), observations given a model configuration (i.e., th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
