Neutral Vector

	Neutral Vector In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type of statistical independence amongst its elements. In particular, when elements of the random vector must add up to certain sum, then an element in the vector is neutral with respect to the others if the distribution of the vector created by expressing the remaining elements as proportions of their total is independent of the element that was omitted. Definition A single element X_i of a random vector X_1,X_2,\ldots,X_k is neutral if the ''relative'' proportions of all the other elements are independent of X_i. Formally, consider the vector of random variables :X=(X_1,\ldots,X_k) where :\sum_^k X_i=1. The values X_i are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say X_1, and consider the distribution of the remaining intervals within the remaining length. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
picture info	Statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	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] [Amazon]
picture info	Random Variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
picture info	Statistical Independence Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Generalized Dirichlet Distribution In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral. The density function of p_1,\ldots,p_ is : \left \prod_^B(a_i,b_i)\right p_k^ \prod_^\left p_i^\left(\sum_^kp_j\right)^\right where we define p_k= 1- \sum_^p_i. Here B(x,y) denotes the Beta function. This reduces to the standard Dirichlet distribution if b_=a_i+b_i for 2\leqslant i\leqslant k-1 (b_0 is arbitrary). For example, if ''k=4'', then the density function of p_1,p_2,p_3 is : \left prod_^B(a_i,b_i)\right p_1^p_2^p_3^p_4^\left(p_2+p_3+p_4\right)^\left(p_3+p_4\right)^ where p_1+p_2+p_3<1 and $p_4=1-p_1-p_2-p_3$ . Connor and Mosimann define the PDF as they did for the following reason. Define random variables $z_1,\ldots,z_$ with $z_1=p_1, z_2=p_2/\left(1-p_1\right), z_3=p_3/\left ...$ [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Theory Of Probability Distributions A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction (" falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]