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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." Laplace's approach The sunrise problem was first introduced in the 18th century by 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''. Thus the probability that ''p'' is between 20% and 50% is just 30%. This must not be interpreted to mean that in 30% of all cases, ''p'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buck Creek IN - Sunrise
Buck may refer to: Common meanings * A colloquialism for a dollar or similar currency * An adult male in some animal species - see List of animal names * Derby shoes, nicknamed "bucks" for the common use of buckskin in their making People *Buck (nickname) *Buck Pierce (born 1981), Canadian football quarterback * Buck (surname), a list of people *Buck 65, stage name of Canadian hip hop artist Richard Terfry *Buck Angel, stage name of American trans man, adult film producer and performer Jake Miller (born 1972) *Buck Dharma, stage name of American guitarist Donald Roeser (born 1947) *Buck Ellison (born 1987), American artist *Buck Henry, stage name of American actor, writer, and director Henry Zuckerman (1930–2020) *Buck Jones, stage name of American film actor Charles Gebhart (1891–1942) *Buck Owens, stage name of American singer and guitarist Alvis Owens Jr. (1929–2006) *Young Buck, stage name of American rapper David Darnell Brown (born 1981) * David Paul Grove (born 1958), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayes' Theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in the theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence. Bayesian inference is fundamental to Bayesia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Problems
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."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, . The higher the probability of an event, the more likely it is that the event will occur. 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 conc ... [...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 categorical data. Given a set of observation counts \textstyle from a \textstyle -dimensional multinomial distribution with \textstyle trials, a "smoothed" version of the counts gives the estimator: :\hat\theta_i= \frac \qquad (i=1,\ldots,d), where the smoothed count \textstyle and the "pseudocount" ''α'' > 0 is a smoothing parameter. ''α'' = 0 corresponds 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) \textstyle , and the uniform probability \textstyle . Invoking Laplace's rule of succession, some authors have argued that ''α'' should be 1 (in which case the term add-one smoothing is also used), though in practice a smaller value is typically chosen. From a Bayesian point of ... [...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. * Behren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problem Of Induction
First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inference from the observed to the unobserved is known as "inductive inferences", and Hume, while acknowledging that everyone does and must make such inferences, argued that there is no non-circular way to justify them, thereby undermining one of the Enlightenment pillars of rationality. While David Hume is credited with raising the issue in Western analytic philosophy in the 18th century, the Pyrrhonist school of Hellenistic philosophy and the Cārvāka school of ancient Indian philosophy had expressed skepticism about inductive justification long prior to that. The traditional inductivist view is that all claimed empirical laws, either in everyday life or through the scientific method, can be justified through some form of reasoning. The pr ... [...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. 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 is a reliable predictor, another player, and two boxes 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 opaque, and its content has alrea ... [...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 discovered 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. If the total number of humans who were born or will ever be born is denoted by ''N'', then the Copernican principle sugg ... [...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] |
Conditional Probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. 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: P(A \mid B) = \frac. For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone unwell (sick) is coughing might b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bible
The Bible (from Koine Greek , , 'the books') is a collection of religious texts or scriptures that are held to be sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of texts of a variety of forms originally written in Hebrew, Aramaic, and Koine Greek. These texts include instructions, stories, poetry, and prophecies, among other genres. The collection of materials that are accepted as part of the Bible by a particular religious tradition or community is called a biblical canon. Believers in the Bible generally consider it to be a product of divine inspiration, but the way they understand what that means and interpret the text can vary. The religious texts were compiled by different religious communities into various official collections. The earliest contained the first five books of the Bible. It is called the Torah in Hebrew and the Pentateuch (meaning ''five books'') in Greek; the second oldest part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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