|
Plate Notation
In Bayesian inference, plate notation is a method of representing variables that repeat in a graphical model. Instead of drawing each repeated variable individually, a plate or rectangle is used to group variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate. The assumptions are that the subgraph is duplicated that many times, the variables in the subgraph are indexed by the repetition number, and any links that cross a plate boundary are replicated once for each subgraph repetition. Example In this example, we consider Latent Dirichlet allocation, a Bayesian network that models how documents in a corpus are topically related. There are two variables not in any plate; ''α'' is the parameter of the uniform Dirichlet prior on the per-document topic distributions, and ''β'' is the parameter of the uniform Dirichlet prior on the per-topic word distribution. The outermost plate represen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Bayes
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes: #To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables. #To derive a lower bound for the marginal likelihood (sometimes called the ''evidence'') of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphical Models
''Graphical Models'' is an academic journal in computer graphics and geometry processing publisher by Elsevier. , its editor-in-chief is Jorg Peters of the University of Florida. History This journal has gone through multiple names. Founded in 1972 as ''Computer Graphics and Image Processing'' by Azriel Rosenfeld, it became the first journal to focus on computer image analysis. Its first change of name came in 1983, when it became ''Computer Vision, Graphics, and Image Processing''. In 1991 it split into two journals, ''CVGIP: Graphical Models and Image Processing'', and ''CVGIP: Image Understanding'', which later became ''Computer Vision and Image Understanding''. Meanwhile, in 1995, the journal ''Graphical Models and Image Processing'' removed the "CVGIP" prefix from its former name, and finally took its current title, ''Graphical Models'', in 2002. Ranking Although initially ranked by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Inference Using Gibbs Sampling
Bayesian inference using Gibbs sampling (BUGS) is a statistical software for performing Bayesian inference using Markov chain Monte Carlo (MCMC) methods. It was developed by David Spiegelhalter at the Medical Research Council Biostatistics Unit in Cambridge in 1989 and released as free software in 1991. The BUGS project has evolved through four main versions: ClassicBUGS, WinBUGS, OpenBUGS anMultiBUGS MultiBUGS is built on the existing algorithms and tools in OpenBUGS and WinBUGS, which are no longer developed, and implements parallelization to speed up computation. Several R packages are availableR2MultiBUGSacts as an interface to MultiBUGS, whilNimbleis an extension of the BUGS language. Alternative implementations of the BUGS language include JAGS and Stan. See also * Spike and slab variable selection * Bayesian structural time series Bayesian structural time series (BSTS) model is a statistical technique used for feature selection, time series forecasting, nowcasting, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LaTeX
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms). It is a complex emulsion that coagulates on exposure to air, consisting of proteins, alkaloids, starches, sugars, oils, tannins, resins, and gums. It is usually exuded after tissue injury. In most plants, latex is white, but some have yellow, orange, or scarlet latex. Since the 17th century, latex has been used as a term for the fluid substance in plants, deriving from the Latin word for "liquid". It serves mainly as defense against herbivorous insects. Latex is not to be confused with plant sap; it is a distinct substance, separately produced, and with different functions. The word latex is also used to refer to natural latex rubber, particularly non-vulcanized rubber. Such is the case in products like latex gloves, latex condoms ... [...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] |
Random Matrices
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of ''an unspecified person'' from within a group would be a random vector. Normally each element of a random vector is a real number. Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. More formally, a multivariate random variable is a column vector \mathbf = (X_1,\dots,X_n)^\mathsf (or its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Bayes
Empirical Bayes methods are procedures for statistical inference in which the prior probability distribution is estimated from the data. This approach stands in contrast to standard Bayesian methods, for which the prior distribution is fixed before any data are observed. Despite this difference in perspective, empirical Bayes may be viewed as an approximation to a fully Bayesian treatment of a hierarchical model wherein the parameters at the highest level of the hierarchy are set to their most likely values, instead of being integrated out. Empirical Bayes, also known as maximum marginal likelihood, represents a convenient approach for setting hyperparameters, but has been mostly supplanted by fully Bayesian hierarchical analyses since the 2000s with the increasing availability of well-performing computation techniques. Introduction Empirical Bayes methods can be seen as an approximation to a fully Bayesian treatment of a hierarchical Bayes model. In, for example, a two-sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphical Model
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a Graph (discrete mathematics), graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning. Types of graphical models Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or Factor graph, factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperparameter
In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis. For example, if one is using a beta distribution to model the distribution of the parameter ''p'' of a Bernoulli distribution, then: * ''p'' is a parameter of the underlying system (Bernoulli distribution), and * ''α'' and ''β'' are parameters of the prior distribution (beta distribution), hence ''hyper''parameters. One may take a single value for a given hyperparameter, or one can iterate and take a probability distribution on the hyperparameter itself, called a hyperprior. Purpose One often uses a prior which comes from a parametric family of probability distributions – this is done partly for explicitness (so one can write down a distribution, and choose the form by varying the hyperparameter, rather than trying to produce an arbitrary function), and partly so that one can ''vary'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
