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Variable Elimination
Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields.Zhang, N.L., Poole, D.:A Simple Approach to Bayesian Network Computations.In: 7th Canadian Conference on Artificial Intelligence, pp. 171--178. Springer, New York (1994) It can be used for inference of maximum a posteriori (MAP) state or estimation of conditional or marginal distributions over a subset of variables. The algorithm has exponential time complexity, but could be efficient in practice for low- treewidth graphs, if the proper elimination order is used. Factors Enabling a key reduction in algorithmic complexity, a factor f, also known as a potential, of variables V is a relation between each instantiation of v of variables f to a non-negative number, commonly denoted as f(x). A factor does not necessarily have a set interpretation. One may perform operations on factors of different representations such as a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Inference
Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian inference uses a prior distribution to estimate posterior probabilities. 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" derive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic 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. Graphical models 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Network
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (''e.g.'' speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Random Field
In the domain of physics and probability, a Markov random field (MRF), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call .... In other words, a random field is said to be a Andrey Markov, Markov random field if it satisfies Markov properties. The concept originates from the Spin glass#Sherrington–Kirkpatrick model, Sherrington–Kirkpatrick model. A Markov network or MRF is similar to a Bayesian network in its representation of dependencies; the differences being that Bayesian networks are directed acyclic graph, directed and acyclic, whereas Markov networks are undirected and may be cyclic. Thus, a Markov network can represent certain dependencies th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum A Posteriori
An estimation procedure that is often claimed to be part of Bayesian statistics is the maximum a posteriori (MAP) estimate of an unknown quantity, that equals the mode of the posterior density with respect to some reference measure, typically the Lebesgue measure. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior density over the quantity one wants to estimate. MAP estimation is therefore a regularization of maximum likelihood estimation, so is not a well-defined statistic of the Bayesian posterior distribution. Description Assume that we want to estimate an unobserved population parameter \theta on the basis of observations x. Let f be the sampling distribution of x, so that f(x\mid\theta) is the probability of x when the underlying population parameter is \theta. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Distribution
Conditional (if then) may refer to: * Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B * Conditional proof, in logic: a proof that asserts a conditional, and proves that the antecedent leads to the consequent *Material conditional, in propositional calculus, or logical calculus in mathematics * Relevance conditional, in relevance logic * Conditional (computer programming), a statement or expression in computer programming languages *A conditional expression in computer programming languages such as ?: *Conditions in a contract A contract is an agreement that specifies certain legally enforceable rights and obligations pertaining to two or more parties. A contract typically involves consent to transfer of goods, services, money, or promise to transfer any of thos ... Grammar and linguistics * Conditional mood (or conditional tense), a verb form in many languages * Conditional se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoreti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Treewidth
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests A forest is an ecosystem characterized by a dense community of trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological functio .... An example of graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly are called '' -trees'', and the graphs with treewidth at most are called '' partial -trees''. Many other well-studied graph families also have bounded treewidth. Treewidth may be formally defined in several equivalent ways: in terms of the size of the largest vertex set in a tree decomposition of the graph, in terms of the size of the largest clique in a chordal completi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability Distribution
A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw-Hill Connect. Webp.274/ref> They are constructed to allow for different degrees and types of movement. Some joints, such as the knee, elbow, and shoulder, are self-lubricating, almost frictionless, and are able to withstand compression and maintain heavy loads while still executing smooth and precise movements. Other joints such as suture (joint), sutures between the bones of the skull permit very little movement (only during birth) in order to protect the brain and the sense organs. The connection between a tooth and the jawbone is also called a joint, and is described as a fibrous joint known as a gomphosis. Joints are classified both structurally and functionally. Joints play a vital role in the human body, contributing to movement, sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Degree Algorithm
In numerical analysis, the minimum degree algorithm is an algorithm used to permute the rows and columns of a symmetric sparse matrix before applying the Cholesky decomposition, to reduce the number of non-zeros in the Cholesky factor. This results in reduced storage requirements and means that the Cholesky factor can be applied with fewer arithmetic operations. (Sometimes it may also pertain to an incomplete Cholesky factor used as a preconditioner—for example, in the preconditioned conjugate gradient algorithm.) Minimum degree algorithms are often used in the finite element method where the reordering of nodes can be carried out depending only on the topology of the mesh, rather than on the coefficients in the partial differential equation, resulting in efficiency savings when the same mesh is used for a variety of coefficient values. Given a linear system : \mathbf\mathbf = \mathbf where A is an n \times n real symmetric sparse square matrix. The Cholesky factor L will typical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
