ApproximateBayesiancomputation ABC is a family of computational techniques in Bayesian statistics . These simulation techniques operate on summary data such as population mean, or variance to make broad inferences with less computation than might be required if all available data were analyzed in detail. They are especially useful in situations where evaluation of the likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. ABC methods originated in population and evolutionary genetics ref name Pritchard1999 cite journal last Pritchard first J. K. authorlink ... 2009 title ApproximateBayesiancomputation scheme for parameter inference and model selection in dynamical ..., K. Kirk, P. Toni, T. Stumpf, M.P.H. year 2010 title ABC SysBio approximateBayesiancomputation ... Beaumont first M. A. coauthors Zhang, W. and David Balding Balding, D. J. title ApproximateBayesiancomputation in population genetics journal Genetics volume 162 pages 2025 2035 url http www.genetics.org ... pmc 307566 ref ref name Plagnol cite journal last Plagnol first V. coauthors Tavar , S. title ApproximateBayesiancomputation and MCMC journal Monte Carlo and Quasi Monte Carlo Methods 2002 year 2004 ..., T. and Estoup, A. title Inferring population history with DIY ABC a user friendly approach to approximateBayesiancomputation journal Bioinformatics year 2008 url http bioinformatics.oxfordjournals.org ..., S. Excoffier, L. year 2010 title ABCtoolbox a versatile toolkit for approximateBayesian computations ... . http msbayes.sourceforge.net msBayes Comparative phylogeographic inference DEFAULTSORT ApproximateBayesianComputation Category Bayesian statistics Category Statistical approximations ... Overview In standard Bayesian inference the posterior distribution is given by math P theta D propto ... procedures can be combined with the standard computational approaches used in Bayesian inference such as Markov ... selection , as the whole apparatus of Bayesian model selection can be adapted to the ABC framework ... more details
Bayesian refers to methods in probability and statistics named after the Reverend Thomas Bayes ca. 1702&ndash 1761 , in particular methods related to statistical inference the Bayesian probability or degree of belief interpretation of probability, as opposed to Frequency probability frequency or proportion or Propensity probability propensity interpretations see probability interpretation Bayes theorem on conditional probability Bayesian inference These methods include Bayes estimator Bayes factor Bayesian average Bayesian spam filtering Bayesian game Bayesian inference Bayesian information criterion Bayesian multivariate linear regression Bayesian linear regression , a special case Bayesian network Empirical Bayes method Naive Bayes classifier Bayesian additive regression kernels Bayesian econometrics Bayesian experimental design Bayesian inference in phylogeny Bayesian search theory Bayesian VAR &mdash Bayesian vector autoregression Bayesian also refers to the application of this probability theory to the functioning of the brain Bayesian brain disambig Category Bayesian statistics su Bayesian ... more details
Refimprove date May 2011 Computation is defined as any type of calculation . ref http www.merriam webster.com dictionary computation ref Also defined as use of computer technology in Information processing . ref http dictionary.reference.com browse computation ref ref http www.answers.com topic computation ref Computation is a process following a well defined Model abstract model understood and expressed in an algorithm , Protocol computing protocol , network topology , etc. Computation is also a major subject matter of computer science it investigates what can or cannot be done in a computational manner. Wiktionary computation Classes of computationComputation can be classified by at least three orthogonal criteria digital vs analog electronics analog , sequential vs parallel computation parallel vs Concurrency computer science concurrent , batch processing batch vs interactive computation interactive . In practice, digital computation is often used to simulate natural processes for example, Evolutionary computation , including those that are more naturally described by analog models of computation for example, Artificial neural network . Computations as a physical phenomenon A computation can be seen as a purely physical phenomenon occurring inside a closed physical system called ... of view is the one adopted by the branch of theoretical physics called the physics of computation ... of the universe itself is a computation Pancomputationalism . Mathematical models of computation In the theory of computation , a diversity of mathematical models of computers have been developed. Typical mathematical Model of computation models of computers are the following State models including ... and process calculi History The word computation has an archaic meaning from its Latin language Latin ... computer science. Comparison to calculation See Calculation Comparison to computation See also Portal Computer Science Computing Physical information Real computation Reversible computation Hypercomputation ... more details
About the Banach algebra concept Approximation to the identity disambiguation Approximation to the identity In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring possibly without an identity that acts as a substitute for an identity element. More precisely, a right approximate identity in a Banach algebra , A , is a net mathematics net or a sequence math ,e lambda lambda in Lambda , math such that for every element, a , of A , the net or sequence math ,ae lambda lambda in Lambda , math has limit a . Similarly, a left approximate identity is a net math ,e lambda lambda in Lambda , math such that for every element, a , of A , the net or sequence math ,e lambda a lambda in Lambda , math has limit a . An approximate identity is a right approximate identity which is also a left approximate identity. For C algebra s, a right or left approximate identity is the same as an approximate identity. Every C algebra has an approximate identity of positive element positive elements of norm &le 1 indeed, the net of all positive elements of norm &le 1 in A with its natural order always suffices. This is called the canonical approximate identity of a C algebra. Approximate identities of C algebras are not unique. For example, for compact operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity. An approximate identity in a convolution algebra plays the same role as a sequence of function approximations to the Dirac delta function which is the identity element for convolution . For example the Fej r kernel s of Fourier series theory give rise to an approximate identity. Ring theory In ring theory an approximate identity is defined in a similar way, except that the ring is given the discrete topology so that a ae sub sub for some . A module over a ring with approximate identity is called non degenerate if for every m in the module there is some with m me sub sub . See a ... more details
Orphan date November 2011 In mathematics approximate limits are a generalization of the ordinary limit of a function limit for real valued functions of several real variables. A function f on math mathbb R k math has an approximate limit y at a point x if there exists a set F that has Lebesgue density theorem density one at the point such that if x sub n sub is a sequence in F that converges towards x then f x sub n sub converges towards y . Properties The approximate limit of a function, if it exists, is unique. If f has an ordinary limit at x then it also has an approximate limit with the same value. We denote the approximate limit of f at x sub 0 sub by math lim limits x rightarrow x 0 operatorname ap f x 0 . math Many of the properties of the ordinary limit are also true for the approximate limit. Thus if a is a scalar and f and g are functions the following equations are true if values on the right hand side are well defined that is the approximate limits exist and in the last equation the approximate limit of g is non zero. math begin align lim x rightarrow x 0 operatorname ap a cdot f x 0 & a cdot lim x rightarrow x 0 operatorname ap f x 0 lim x rightarrow x 0 operatorname ap f x 0 g x 0 & lim x rightarrow x 0 operatorname ap f x 0 lim x rightarrow x 0 operatorname ap g x 0 lim x rightarrow x 0 operatorname ap f x 0 g x 0 & lim x rightarrow x 0 operatorname ap f x 0 lim x ... x 0 operatorname ap g x 0 end align math Approximate continuity and differentiability If math lim ... 0 h f x 0 h math has an approximate limit as h approaches zero we say that f has an approximate derivative at x sub 0 sub . It turns out that approximate differentiability implies approximate continuity ... to the approximate derivative. There is no generalization of the chain rule that is true in general ... A a012870.htm title Approximate limit last Tolstov first G.P accessdate 2010 04 25 work Encyclopaedia of Mathematics publisher Springer DEFAULTSORT Approximate Limit Category Real analysis Category ... more details
orphan date December 2011 In statistics , an approximate entropy ApEn is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time series data. ref name Pincus1991 cite journal last1 Pinus first1 S. M. last2 Gladstone first2 I. M. last3 Ehrenkranz first3 R. A. title A REGULARITY STATISTIC FOR MEDICAL DATA ANALYSIS journal Journal of Clinical Monitoring and Computing year 1991 volume 7 issue 4 pages 335 345 DOI 10.1007 BF01619355 ref For example, there are two series of data series 1 10,20,10,20,10,20,10,20,10,20,10,20... , which alternates 10 and 20. series 2 10,10,20,10,20,20,20,10,10,20,10,20,20... , which has either a value of 10 or 20, randomly chose, each with probability 1 2. Moment statistics , such as mean and variance , will not distinguish between these two series. Nor will rank order statistics distinguish between these series. Yet series 1 is perfectly regular knowing one term has the value of 20 enables one to predict with certainty that the next term will have the value of 10. Series 2 is randomly valued knowing one term has the value of 20 gives no insight into what value the next term will have. Regularity was originally measured by exact regularity statistics, which has mainly centered around various entropy measures. ref name Pincus1991 ref However, accurate entropy calculation requires vast amounts of data, and the results will be greatly influenced by system noise ref name Pincus21991 cite journal last1 Pinus first1 S. M. title Approximate entropy as a measure of system complexity journal Proceedings of the National Academy of Sciences year 1991 volume 88 issue 6 pages 2297 2301 PMID 11607165 ref , therefore it is not practical to apply these methods to experimental data. ApEn was developed by Steve M. Pincus ... math Define approximate entropy math mathrm ApEn math as math mathrm ApEn Phi m r Phi m 1 r . math for math ... J.S. last2 Moorman first2 J.R. title Physiological time series analysis using approximate entropy ... more details
of templates. A Bayesian network , Bayes network , belief network or directed acyclic graphical ..., a Bayesian network could represent the probabilistic relationships between diseases and symptoms ... diseases. Formally, Bayesian networks are directed acyclic graph s whose nodes represent random variables in the Bayesian probability Bayesian sense they may be observable quantities, latent variable ... algorithms exist that perform inference and machine learning learning in Bayesian networks. Bayesian networks that model sequences of variables e.g. speech recognition speech signals or peptide sequence protein sequences are called dynamic Bayesian network s. Generalizations of Bayesian networks ... equivalent definitions of a Bayesian network. For all the following, let G V , E be a directed ... . Factorization definition X is a Bayesian network with respect to G if its joint probability density ... X is a Bayesian network with respect to G if it satisfies the local Markov property each variable ... of non descendants because the graph is acyclic . Developing Bayesian networks To develop a Bayesian ... of these conditional distributions, then X is a Bayesian network with respect to G . ref Neapolitan, R.E., Learning Bayesian Networks , Prentice Hall, Upper Saddle River, NJ, 2004 ref Markov ... other parents of its children. X is a Bayesian network with respect to G if every node is conditionally ... and Pearl, Judea title Identifying independence in Bayesian Networks journal Networks year 1990 ... are d separated. If u and v are not d separated, they are called d connected. X is a Bayesian network ... v from all other nodes. Causal networks Although Bayesian networks are often used to represent causality ... that X sub v sub is causally dependent on X sub u sub . This is demonstrated by the fact that Bayesian ... requirements. A causal network is a Bayesian network with an explicit requirement that the relationships ... 400px thumb right A simple Bayesian network. Suppose that there are two events which could cause grass ... more details
and dynamical systems theory approximateBayesiancomputation ABC are also becoming increasingly popular ...More footnotes date April 2009 Bayesian statistics In statistics , Bayesian inference is a method of statistical ... changes due to Evidence evidence . Bayesian inference is justified by the philosophy of Bayesian probability Bayesian probability , which asserts that degrees of belief may be represented by Probability ... ref Stanford encyclopedia of philosophy Bayesian Epistemology http plato.stanford.edu entries epistemology bayesian ref ref Gillies, Donald 2000 Philosophical Theories of Probability Routledge ... prior and the updated degree of belief the Posterior probability posterior . Bayesian inference has ... employ Bayesian inference see Bayesian cognitive science . Bayesian inference is used in science and engineering for model selection, see Bayes factor . Philosophical background Main Bayesian probability The philosophy of Bayesian probability considers the degrees of belief in a set of propositions .... This is the essence of Bayesian inference math P P E frac P E P P E cdot P P math math P ... function likelihood . Method General formulation File Bayesian inference event space.svg thumb Diagram illustrating event space math Omega math in general formulation of Bayesian inference. Although ..., Andrew Carlin, John B. Stern, Hal S. Rubin, Donald B. 2003 . Bayesian Data Analysis, Second Edition ... be viewed as more fully respecting the Bayesian philosophy, as any value with non zero posterior ... a prediction File Bayesian inference archaeology example.jpg thumb Example results for archaeology ... dablink See Bayesian model selection Applications Computer applications Bayesian inference has applications in artificial intelligence and expert system s. Bayesian inference techniques have been ... an ever growing connection between Bayesian methods and simulation based Monte Carlo method Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while ... more details
Bayesian filtering may refer to Bayesian spam filtering , a method to detect spam. Recursive Bayesian estimation , a method to estimate the state of a system evolving in time. Bayes theorem Disambig ... more details
Bayesian brain is a term that is used to refer to the ability of the nervous system to operate in situations of uncertainty in a fashion that is close to the optimal prescribed by Bayesian statistics. This term ... by neural computation neural processing of sensory information using methods approximating those of Bayesian probability . ref Kenji Doya Editor , Shin Ishii Editor , Alexandre Pouget Editor , Rajesh P. N. Rao Editor 2007 , Bayesian Brain Probabilistic Approaches to Neural Coding, The MIT Press 1 edition Jan 1 2007 ref ref Knill David,Pouget Alexandre 2004 , The Bayesian brain the role of uncertainty in neural coding and computation,TRENDS in Neurosciences Vol.27 No.12 December 2004 ... , experimental psychology and Bayesian statistics . As early as the 1860s, with the work of Hermann ... Westheimer, G. 2008 Was Helmholtz a Bayesian? Perception 39, 642 50 ref The basic idea is that the nervous ..., R. M. 1995 . The Helmholtz machine. Neural Computation, 7, 889 904. ref ref Dayan, P. and Hinton .... Gallinari editors ICANN 95, 483 490 ref Bayesian probability, has been developed by a large field ..., Bayesian Methods General Background, in Maximum Entropy and Bayesian Methods in Applied Statistics ... for using Bayesian Probability to model mental processes. ref Jaynes, E. T., 1988, How Does the Brain Do Plausible Reasoning? , in http books.google.com books?id UjixarjDFH0C Maximum Entropy and Bayesian ... realized early on that the Bayesian statistical framework holds the potential to lead to insights into the function of the nervous system. A wide range of approaches exist that link Bayesian ideas ... with Bayesian statistics. This approach, with its emphasis on behavioral outcomes as the ultimate ... using Bayesian decision theory. Examples are the work of Landy ref Tassinari H, Hudson TE & Landy ... RN 2003 . http www.opticsinfobase.org abstract.cfm?URI josaa 20 7 1391 Bayesian integration of visual ... of Vision, 5 2 , 103 15. ref ref Knill DC 2007 . http journalofvision.org 7 8 13 Learning Bayesian ... more details
Bayesian econometrics is a branch of econometrics which applies Bayesianism Bayesian principles to economic modelling. Bayesianism is based on a degree of belief probability interpretations interpretation of probability , as opposed to a relative frequency interpretation. The Bayesian principle relies on Bayes theorem which states that the Bayesian probability probability of B conditional on A is the ratio of joint probability of A and B divided by probability of B. Bayesian econometricians assume that coefficients in the model have prior distribution s. This approach was first propagated by Arnold Zellner . Citation needed date August 2010 References Tony Lancaster 2004 An Introduction to Modern Bayesian Econometrics , Blackwell Publishing. ISBN 1405117206 Gary Koop, Dale J. Poirier, Justin L. Tobias 2007 Bayesian Econometric Methods , Cambridge University Press. ISBN 0521855713 Zellner, A. 1996 An Introduction to Bayesian Inference in Econometrics , Wiley. ISBN 0471169374 reprint of 1971 edition DEFAULTSORT Bayesian Econometrics Category Econometrics Category Bayesian statistics econometrics stub ... more details
Bayesian statistics Bayesian probability is one of the different Probability interpretations interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian ... of a hypothesis , the Bayesian probabilist specifies some prior probability, which is then updated ... retrieved 2011 08 06 ref The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation. Bayesian probability interprets the concept of probability as a probability ... ghxaib Jaynes, E.T. Bayesian Methods General Background. In Maximum Entropy and Bayesian Methods in Applied ... . The term Bayesian refers to the 18th century mathematician and theologian Thomas Bayes 1702&ndash 1761 , who provided the first mathematical treatment of a non trivial problem of Bayesian inference ... what is now called Bayesian probability. ref Stigler, Stephen M. 1986 The history of statistics. , Harvard University press. pg 97 98, pg 131. ref Broadly speaking, there are two views on Bayesian ... view , the rules of Bayesian statistics can be justified by Cox s theorem requirements of rationality ... learning methods are based on objectivist Bayesian principles. ref name ReferenceA Bishop, C.M. Pattern Recognition and Machine Learning. Springer, 2007 ref In the Bayesian view, a probability is assigned ... hypothesis test tested without being assigned a probability. Bayesian methodology In general, Bayesian ... parameter s. In most cases, the computation is Intractability complexity intractable , but good ... , so that the frequentist probability of a frequentist hypothesis is either one or zero. In Bayesian ... Bayesian probabilities Broadly speaking, there are two views on Bayesian probability that interpret ... of Bayesian statistics, which can be justified by Cox s theorem requirements of rationality and consistency ... variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability. History Main History of statistics Bayesian statistics The term Bayesian refers ... more details
Bayesian statistics Bayesian statistics is that subset of the entire field of statistics in which the evidence about the true state of the world is expressed in terms of degrees of belief or, more specifically, Bayesian probability Bayesian probabilities . Such an interpretation is only one of a number of Probability interpretations interpretations of probability and there are many other statistical techniques that are not based on degrees of belief . Outline The general set of statistical techniques can be divided into a number of activities, many of which have special Bayesian versions. Statistical inference Main Bayesian inference Bayesian inference is an approach to statistical inference , that is distinct from the more traditional frequentist inference . It is specifically based on the use of Bayesian probability Bayesian probabilities to summarise evidence. Statistical modelling The formulation of statistical model s for use in Bayesian statistics has the additional feature, not present with other types of statistical techniques, of requiring the formulation of a set of prior distribution s for any unknown parameters. Such prior distributions are as much part of the statistical model as the part that expresses the probability distribution of observations given the model parameters. The specification of a set of prior distributions for a problem may involve hyperparameter s and hyperprior distribution s. Design of experiments main Bayesian design of experiments The usual considerations in the design of experiments are extended in the case of Bayesian design of experiments to include the influence of prior beliefs. Importantly, the application of sequential analysis techniques ... example of the Bayesian design of experiments aimed at such efficiency is the multi armed bandit problem ..., etc. The use of certain modern computational techniques for Bayesian inference, specifically .... unreferenced date July 2011 Category Bayesian statistics statistics stub ... more details
Bayesian Vector Autoregression BVAR is a term which indicates that bayesian inference Bayesian methods are used to estimate a vector autoregression VAR . In that respect, the difference with standard VAR models lies on the fact that the model parameters are treated as random variable random variables , and prior probability prior probabilities are assigned to them. The parameter space of VARs proliferates with the number of dependent variables and the number of lags. At the same time, macro economic datasets involve monthly, quarterly or annual observations and, thus are only of moderate size. Bayesian methods have attracted attention because full and empirical bayes estimator Bayes estimators help provide shrinkage over unrestricted least squares estimates ref Koop and Korobilis 2010 ref . A typical example is the shrinkage prior proposed by Robert Litterman ref Litterman 1979, 1984 ref , and subsequently developed by other researchers at University of Minnesota ref Doan, Litterman and Sims 1984 , Sims 1989 ref , which came to stay in the BVAR literature as the Minnesota prior . Recent research have shown that vector auto regression with Bayesian shrinkage is an appropriate tool for large scale dynamic models. ref Banbura, Giannone and Reichlin 2010 ref Notes reflist References cite journal first1 T. last1 Banbura first2 R. last2 Giannone first3 C. last3 Reichlin title Large Bayesian vector auto regressions journal Journal of Applied Econometrics volume 25 1 pages 71&ndash 92 year 2010 url http ideas.repec.org a jae japmet v25y2010i1p71 92.html cite journal first1 T. last1 Doan first2 R. last2 Litterman first3 C. last3 Sims title Forecasting and conditional projection using realistic prior distributions journal Econometric Reviews volume 3 pages 1&ndash 100 year 1984 cite journal first1 G. last1 Koop first2 D. last2 Korobilis title Bayesian multivariate time series methods ... research DP DP14.pdf pdf year 1989 See also Bayesian econometrics Category Econometrics ... more details
Cleanup date June 2009 Unreferenced date December 2009 A Bayesian average is a method of estimating the mean of a population consistent with Bayesian probability Bayesian interpretation , where instead of estimating the mean strictly from the available data set, other existing information related to that data set may also be incorporated into the calculation in order to minimize the impact of large deviations, or to assert a default value when the data set is small. For example, in a calculation of an average review score of a book where only two reviews are available, both giving scores of 10, a normal average score would be 10. However, as only two reviews are available, 10 may not represent the true average had more reviews been available. The review site may instead calculate a Bayesian average of this score by adding the average review score of all books in the store to the calculation. For example, by adding five scores of 7 each, the Bayesian average becomes 7.86 instead of 10, which the review site would hope that it will better represent the quality of the book. Note that the additional information incorporated into the mean calculation does not have to be the true prior mean of the larger population, but rather a value subjectively determined by the person calculating the average as relevant and serving the purpose of the calculation. Therefore, the quality of the Bayesian average in term of representing the data set is dependent on the judgment of the person doing the calculation. Calculation Calculating the Bayesian average uses the prior mean m and a constant C . C is assigned a value that is proportional to the typical data set size. The value is larger .... Example The goal is to calculate the Bayesian average of the heights of various occupations of adult ..., for an average height of 201 cm. border 1 rules all Group N Group mean Bayesian average Basketball ... cm 191.4 cm See also Additive smoothing Category Bayesian statistics statistics stub ... more details
Bayesian efficiency addresses an appropriate economic definition of Pareto efficiency where there is incomplete information . ref name implementation Palfrey, Thomas R. Srivastava, Sanjay Postlewaite, A. 1993 http books.google.com books?id lZTls JJSxgC&pg PA14&dq there is no other Bayesian incentive compatible allocation rule that is&sig ZXeayrAFXGaZM4iwZGwRrgA qow PPA13,M1 Bayesian Implementation. Pg. 13 14. ISBN 3718653141 ref Under Pareto efficiency, an allocation of a resource is Pareto efficient if there is no other allocation of that resource that makes no one worse off while making some agents strictly better off. ref name implementation A limitation with the concept of Pareto efficiency is that it assumes that knowledge about other market participants is available to all participants in that every player knows the payoffs and strategies available to other players so as to have incomplete information. ref name implementation Often, the players have types that are hidden from the other player. ref name implementation The lack of complete information raises a question of when the efficiency calculation should be made. ref name implementation Should the efficiency check be made at the ex ante stage before the agent sees their types, at the interim stage after the agent sees their types, or at the ex post stage where the agent will have complete information about their types? Another issue is incentive. ref name implementation If a resource allocation rule is efficient but there is no incentive to abide by that rule or accept that rule, then the revelation principle asserts that there is no mechanism by which this allocation rule can be realized. ref name implementation Bayesian efficiency overcomes problems of the Pareto efficiency by accounting for incomplete information, by addressing the timing of the evaluation ex ante efficient, interim efficient, or post ante efficient , and by adding an incentive qualifier so that the allocation rule is incentive compatible ... more details
Unreferenced date December 2009 In game theory , a Bayesian game is one in which information about characteristics ... C. Harsanyi s framework, a Bayesian game can be modelled by introducing Nature as a player in a game ... distribution across each player s type space . Harsanyi s approach to modelling a Bayesian game ... for whom the type is specified is that type. In a Bayesian game, the incompleteness of information .... Such games are called Bayesian because of the probabilistic analysis inherent in the game. Players .... Specification of games The Normal form game normal form representation of a non Bayesian game with perfect ... for every player. In a Bayesian game, it is necessary to specify the strategy spaces, type spaces ... t i ,t i in C i, forall t i . math A Bayesian Equilibrium of the game G is defined to be a possibly ... S N, hat u u rangle math . So for any finite game G, Bayesian Equilibria always exists. A signalling example Signaling games Signalling games constitute an example of Bayesian games. In such a game ... wages to skilled workers and low wages to unskilled. Bayesian Nash equilibrium In a non Bayesian game ... yield a higher payoff, given all the strategies played by the other players. In a Bayesian game where ... or risk loving, the assumption is that players are expected utility expected utility maximizing . A Bayesian ... on players beliefs. This makes Bayesian Nash equilibrium an incomplete tool with which to analyse dynamic games of incomplete information. Perfect Bayesian equilibrium Bayesian Nash equilibrium ... by the Bayesian Nash solution concept or subgame perfection, one can apply the Perfect Bayesian ... in non singleton information sets to be dealt with more satisfactorily. So far in discussing Bayesian ... in Bayesian games can be approached more rigorously in PBE. A belief system is an assignment of probabilities ... set. Definition A perfect Bayesian equilibrium is a strategy profile and a belief system such that the strategies ... to them. An example Image Extensive form game 2.svg 250px left thumb A Bayesian game with imperfect ... more details
Bayesian poisoning is a technique used by e mail Spam electronic spammers to attempt to degrade the effectiveness of spam filter s that rely on Bayesian spam filtering . Bayesian filtering relies on Bayesian probability to determine whether an incoming mail is spam or is not spam. The spammer hopes that the addition of random or even carefully selected words that are unlikely to appear in a spam message will cause the spam filter to believe the message to be legitimate a statistical type II error . Spammers also hope to cause the spam filter to have a higher false positive rate by turning previously innocent words into spammy words in the Bayesian database statistical type I error because a user who trains their spam filter on a poisoned message will be indicating to the filter that the words added by the spammer are a good indication of spam. Empirical results Graham Cumming and Brighenti At the Spam Conference held at MIT in 2006 John Graham Cumming and Stefano Brighenti presented two possible attacks on POPFile s Bayesian engine. ref http www.jgc.org SpamConference011604.pps ref One was unsuccessful and the other worked, but was impractical. In doing this they identified two types of poisoning attack passive where words are added without any feedback to the spammer and active where the spammer gets feedback after the spam has been received . The passive method of adding random ... to determine whether the spam was received. If it was, another Bayesian system was trained using the same ... in the emails they were using most Bayesian spam filters make extensive use of header information ... false positives by turning previously innocent words into spammy words in the Bayesian database ... words to spam were ineffective against a na ve Bayesian filter. In fact, they showed, as John ... was effective against a na ve Bayesian filter, and enabled spam to slip through. They went on to detail ... sb200602 poison Does Bayesian Poisoning Exist? registration required Category Spam filtering Category ... more details
The study of the physics of computation relates to understanding the fundamental physical limits of computer s. This field has led to the investigation of how thermodynamics limits information processing, the understanding of Chaos theory chaos and dynamical systems , and a rapidly growing effort to invent new quantum computer s. See also list of publications in physics Physics of computation important publications in physics of computation See also Digital physics Computation Theory of computation Reversible computation Hypercomputation Physical information Limits to computation Bremermann s limit References Lloyd, S., 2000, Ultimate physical limits of computation, Nature journal Nature , 406 1047 1054. Category Computational physics physics stub ... more details
Indeterminancy in computation may refer to Quantum indeterminacy in quantum computer s Nondeterministic finite automata Nondeterministic algorithm In concurrency Indeterminacy in concurrent computation Unbounded nondeterminism disambig ... more details
Morphological computation may refer to Morphological computation robotics Computational linguistics disambig Long comment to avoid being listed on short pages ... more details
Unreferenced date December 2009 A computation tree is a representation for the computation steps of a non deterministic Turing machine on a specified input. A computation tree graph theory tree is a rooted tree of nodes and edges. Each node in the tree represents a single computational state, while each edge represents a transition to the next possible computation. The number of nodes of the tree is the size of the tree and the length of the path from the root to a given node is the depth of the node. The largest depth of an output node is the depth of the tree. The output nodes of the tree are called leaves. In a computation tree each output node is labeled Yes or No. If a tree, T, with an input space X, if math x in X math and the path for x ends in node labeled yes, then the input x is accepted. Else it is rejected. The depth of the computation tree for a given input is the computation time for the Turing machine on that input. One of the primary methods of showing that a computational problem L is complete complexity complete for a given complexity class C is to show that the computation tree of any algorithm in C can be directly analyzed in terms of L . DEFAULTSORT Computation Tree Category Computational complexity theory ... more details
In computer science , interactive computation is a mathematical model for computation that involves communication with the external world during the computation. This is in contrast to the traditional understanding of computation which assumes a simple interface between a computing agent and its environment, consisting in asking a question input and generating an answer output . The famous Church Turing thesis attempts to define computation and computability in terms of Turing machines . However the Turing machine model only provides an answer to the question of what computability of functions means and, with interactive tasks not always being reducible to functions, it fails to capture our broader intuition of computation and computability. While this fact was admitted by Alan Turing himself, it was not until recently that the theoretical computer science community realized the necessity to define adequate mathematical models of interactive computation. Among the currently studied mathematical models of computation that attempt to capture interaction are http www.csc.villanova.edu japaridz Japaridze s hard and easy play machines elaborated within the framework of computability logic , http www.cse.uconn.edu dqg Goldin s persistent Turing machines, and http research.microsoft.com gurevich Gurevich s abstract state machines. Peter Wegner has additionally done a great deal of work on this area of computer science. See also Human based computation Computability logic Game semantics Interactive programming Quasi empiricism in mathematics Quasi empiricism References and external web sources Interactive Computation The New Paradigm ISBN 354034666X. Edited by D.Goldin, S.Smolka and P.Wegner. ... dqg D.Q.Goldin , Persistent Turing Machines as a model of interactive computation . Lecture Notes ... Machines, Transition Systems, and Interaction . J. Information and Computation 194 2 2004 , pp. ... . Theoretical Computer Science 192 1998 , pp. 315 351. Category Theory of computation Category ... more details
Italic title Mathematics of Computation ref http www.ams.org mcom aboutmcom.html Mathematics of Computation Journal overview , retrieved April 2007 ref is a quarterly mathematics journal focused on computational mathematics that is published by the American Mathematical Society . It was established in 1943. The articles in all volumes older than five years are available electronically free of charge. ref http www.ams.org jourcgi jrnl toolbar nav mcom all Mathematics of Computation Archive ref References reflist Category Mathematics journals Category Quarterly journals sci journal stub ... more details
of computation that is more restricted than the set of operations that you could use in practice and therefore ... Models of Computation An Introduction to Computability Theory publisher Springer year 2009 series Undergraduate Topics in Computer Science isbn 978 1 84882 433 1 DEFAULTSORT Model Of Computation Category Models of computation Category Theory of computation Comp sci stub bg es Modelo ... more details
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