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Imprecise Dirichlet Process
In probability theory and statistics, the Dirichlet process (DP) is one of the most popular Bayesian nonparametric models. It was introduced by Thomas Ferguson as a prior over probability distributions. A Dirichlet process \mathrm\left(s,G_0\right) is completely defined by its parameters: G_0 (the ''base distribution'' or ''base measure'') is an arbitrary distribution and s (the ''concentration parameter'') is a positive real number (it is often denoted as \alpha). According to the Bayesian paradigm these parameters should be chosen based on the available prior information on the domain. The question is: how should we choose the prior parameters \left(s,G_0\right) of the DP, in particular the infinite dimensional one G_0, in case of lack of prior information? To address this issue, the only prior that has been proposed so far is the limiting DP obtained for s\rightarrow 0, which has been introduced under the name of Bootstrapping, Bayesian bootstrap by Rubin;Rubin D (1981). The Ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Process
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution. As an example, a bag of 100 real-world dice is a ''random probability mass function (random pmf)'' - to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Distribution Function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by at each of the data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function. Definition Let be independent, identically distributed real random variables with the common cumulative distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imprecise Probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with expert elicitation, because: * People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval. * As an interval is compatible with a range of opinions, the analysis ought to be more convincing to a range of different people. Introduction Uncertainty is traditionally modelled by a probability distribution, as developed by Kolmogorov, Laplace, de Finetti, Ramsey, Cox, Lindley, and many others. However, this has not been unanimously accepted by scientists, statisticians, and probabilists: it has been argued that some modification or broadening of probabili ... [...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 \boldsymbol\alpha 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 ''K'' ≥ 2 with parameters ''α''1, ..., ''α''''K'' > 0 has a probability density function with respect to Lebesgue m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Variable
In statistics, a categorical variable (also called qualitative variable) is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property. In computer science and some branches of mathematics, categorical variables are referred to as enumerations or enumerated types. Commonly (though not in this article), each of the possible values of a categorical variable is referred to as a level. The probability distribution associated with a random categorical variable is called a categorical distribution. Categorical data is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cross tabulations, or from observations o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas ''beta distribution of the second kind'' is an alternative name for the beta prime distribution. The generalization to mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regularized Incomplete Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution For The Lower (red) And Upper (blue) Probability Of The Hypothesis
Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiodental fricative while in borrowed words is instead commonly transcribed as μπ. Letters that arose from beta include the Roman letter and the Cyrillic letters and . Name Like the names of most other Greek letters, the name of beta was adopted from the acrophonic name of the corresponding letter in Phoenician, which was the common Semitic word ''*bait'' ('house'). In Greek, the name was ''bêta'', pronounced in Ancient Greek. It is spelled βήτα in modern monotonic orthography and pronounced . History The letter beta was derived from the Phoenician letter beth . Uses Algebraic numerals In the system of Greek numerals, beta has a value of 2. Such use is denoted by a number mark: Β′. Computing Finance Beta is used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
