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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] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normalizing Constant
In probability theory, a normalizing constant or normalizing factor is used to reduce any probability function to a probability density function with total probability of one. For example, a Gaussian function can be normalized into a probability density function, which gives the standard normal distribution. In Bayes' theorem, a normalizing constant is used to ensure that the sum of all possible hypotheses equals 1. Other uses of normalizing constants include making the value of a Legendre polynomial at 1 and in the orthogonality of orthonormal functions. A similar concept has been used in areas other than probability, such as for polynomials. Definition In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function. Examples If we start from the simple Gaussian function p(x) = e^, \quad x\in(-\infty,\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cromwell's Rule
Cromwell's rule, named by statistician Dennis Lindley, states that the use of prior probabilities of 1 ("the event will definitely occur") or 0 ("the event will definitely not occur") should be avoided, except when applied to statements that are logically true or false, such as equaling 4. The reference is to Oliver Cromwell, who wrote to the General Assembly of the Church of Scotland on 3 August 1650, shortly before the Battle of Dunbar, including a phrase that has become well known and frequently quoted: As Lindley puts it, assigning a probability should "leave a little probability for the moon being made of green cheese; it can be as small as 1 in a million, but have it there since otherwise an army of astronauts returning with samples of the said cheese will leave you unmoved". Similarly, in assessing the likelihood that tossing a coin will result in either a head or a tail facing upwards, there is a possibility, albeit remote, that the coin will land on its edge a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace
Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summarized and extended the work of his predecessors in his five-volume Traité de mécanique céleste, ''Mécanique céleste'' (''Celestial Mechanics'') (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. Laplace also popularized and further confirmed Isaac Newton, Sir Isaac Newton's work. In statistics, the Bayesian probability, Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplace operator, Laplacian differential operator, widely used in mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Prior
In Bayesian probability theory, if, given a likelihood function p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function p(x \mid \theta). A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary. Further, conjugate priors may clarify how a likelihood function updates a prior distribution. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.Howard Raiffa and Robert Schlaifer. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961. A similar c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, (Chapter 49: Dirichlet and Inverted Dirichlet Distributions) hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet distribution is the '' Dirichlet process''. Definitions Probability density function The Dirichlet distribution of order with parameters has a probability density function with respect to Lebesgue measure on the Euclidean space given by f \left(x_1,\ldots, x_; \alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multinomial Distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided die rolled ''n'' times. For ''n'' statistical independence, independent trials each of which leads to a success for exactly one of ''k'' categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories. When ''k'' is 2 and ''n'' is 1, the multinomial distribution is the Bernoulli distribution. When ''k'' is 2 and ''n'' is bigger than 1, it is the binomial distribution. When ''k'' is bigger than 2 and ''n'' is 1, it is the categorical distribution. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so ''n'' determines the suffix, and ''k'' the prefix). The Bernoulli distribution models the outcome of a si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , then is the logarithm of to base , written , so . As a single-variable function, the logarithm to base is the inverse of exponentiation with base . The logarithm base is called the ''decimal'' or ''common'' logarithm and is commonly used in science and engineering. The ''natural'' logarithm has the number as its base; its use is widespread in mathematics and physics because of its very simple derivative. The ''binary'' logarithm uses base and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergeometric Distribution
In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without'' replacement, from a finite Statistical population, population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of k successes in n draws ''with'' replacement. Definitions Probability mass function The following conditions characterize the hypergeometric distribution: * The result of each draw (the elements of the population being sampled) can be classified into one of Binary variable, two mutually exclusive categories (e.g. Pass/Fail or Employed/Unemployed). * The probability of a success changes on each draw, as each draw decreases the population ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior Probability
A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable. In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern. There are many ways to const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Probability
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name. Statement The law of total probability isZwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. page 31. a theorem that states, in its discrete case, if \left\ is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event A :P(A)=\sum_n P(A\cap B_n) or, alternatively, :P(A)=\sum_n P(A\mid B_n)P(B_n), where, for any n, if P(B_n) = 0 , then these terms are simply omitted from the summation since P(A\mid B_n) is finite. The summation can be interpreted as a weighted average, and consequently the marginal probability, P(A), is sometimes called "average probability"; "overall probability" is sometimes used i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
