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Graphical Model
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a 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 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. Undirected Graphical Model The un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" ( Herman Adèr quoting Kenneth Bollen). All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. Introduction Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D-separation
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). 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 that can represent and solve decision problems under uncertainty are called influence diagrams. Graphical mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Junction Tree Algorithm
The junction tree algorithm (also known as 'Clique Tree') is a method used in machine learning to extract marginalization in general graphs. In essence, it entails performing belief propagation on a modified graph called a junction tree. The graph is called a tree because it branches into different sections of data; nodes of variables are the branches. The basic premise is to eliminate cycles by clustering them into single nodes. Multiple extensive classes of queries can be compiled at the same time into larger structures of data. There are different algorithms to meet specific needs and for what needs to be calculated. Inference algorithms gather new developments in the data and calculate it based on the new information provided. Junction tree algorithm Hugin algorithm * If the graph is directed then moralize it to make it un-directed. *Introduce the evidence. * Triangulate the graph to make it chordal. *Construct a junction tree from the triangulated graph (we will call t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Tree
In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by . Later it was rediscovered by and has since been studied by many other authors. Definition Intuitively, a tree decomposition represents the vertices of a given graph as subtrees of a tree, in such a way that vertices in are adjacent only when the corresponding subtrees intersect. Thus, forms a subgraph of the intersection graph of the subtrees. The full intersection graph is a chordal graph. Each subtree associates a graph vertex with a set of tree nodes. To define this formally, we represent each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belief Propagation
A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take it to be true; for instance, to believe that snow is white is comparable to accepting the truth of the proposition "snow is white". However, holding a belief does not require active introspection. For example, few carefully consider whether or not the sun will rise tomorrow, simply assuming that it will. Moreover, beliefs need not be ''occurrent'' (e.g. a person actively thinking "snow is white"), but can instead be ''dispositional'' (e.g. a person who if asked about the color of snow would assert "snow is white"). There are various different ways that contemporary philosophers have tried to describe beliefs, including as representations of ways that the world could be (Jerry Fodor), as dispositions to act as if certain things are true (Rod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Graph
A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to represent factorization of a probability distribution function, enabling efficient computations, such as the computation of marginal distributions through the sum-product algorithm. One of the important success stories of factor graphs and the sum-product algorithm is the decoding of capacity-approaching error-correcting codes, such as LDPC and turbo codes. Factor graphs generalize constraint graphs. A factor whose value is either 0 or 1 is called a constraint. A constraint graph is a factor graph where all factors are constraints. The max-product algorithm for factor graphs can be viewed as a generalization of the arc-consistency algorithm for constraint processing. Definition A factor graph is a bipartite graph representing the factorization of a function. Given a factorization of a function g(X_1,X_2,\dots,X_n), :g(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dependency Network (graphical Model)
Dependency networks (DNs) are graphical models, similar to Markov networks, wherein each vertex (node) corresponds to a random variable and each edge captures dependencies among variables. Unlike Bayesian networks, DNs may contain cycles. Each node is associated to a conditional probability table, which determines the realization of the random variable given its parents. Markov blanket In a Bayesian network, the Markov blanket of a node is the set of parents and children of that node, together with the children's parents. The values of the parents and children of a node evidently give information about that node. However, its children's parents also have to be included in the Markov blanket, because they can be used to explain away the node in question. In a Markov random field, the Markov blanket for a node is simply its adjacent (or neighboring) nodes. In a dependency network, the Markov blanket for a node is simply the set of its parents. Dependency network versus Baye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naive Bayes Classifier
In statistics, naive Bayes classifiers are a family of simple " probabilistic classifiers" based on applying Bayes' theorem with strong (naive) independence assumptions between the features (see Bayes classifier). They are among the simplest Bayesian network models, but coupled with kernel density estimation, they can achieve high accuracy levels. Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression, which takes linear time, rather than by expensive iterative approximation as used for many other types of classifiers. In the statistics literature, naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes. All these names reference the use of Bayes' theorem in the classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method. Introduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Model
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discrete mathematics *Graph of a function * Graph of a relation *Graph paper *Chart, a means of representing data (also called a graph) Computing *Graph (abstract data type), an abstract data type representing relations or connections *graph (Unix), Unix command-line utility *Conceptual graph, a model for knowledge representation and reasoning Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also *Complex network *Graf *Graff (other) *Graph database *Grapheme, in linguistics *Graphemics *Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") *List of information graphics software *Statistical graphics Statistical graphics, also known as statistical graphical techniques, are graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
