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Bayesian data analysis: Difference between revisions
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Bayesian data analysis is based on [[:Wikipedia:Bayesian inference|Bayesian inference]]. | Bayesian data analysis is based on [[:Wikipedia:Bayesian inference|Bayesian inference]]. | ||
<ref>D.S. Sivia, ''Data Analysis: A Bayesian Tutorial'', Oxford University Press, USA (1996) ISBN 0198518897</ref> | <ref>D.S. Sivia, ''Data Analysis: A Bayesian Tutorial'', Oxford University Press, USA (1996) {{ISBN|0198518897}}</ref> | ||
<ref>P. Gregory, ''Bayesian Logical Data Analysis for the Physical Sciences'', Cambridge University Press, Cambridge (2005) ISBN 052184150X</ref> | <ref>P. Gregory, ''Bayesian Logical Data Analysis for the Physical Sciences'', Cambridge University Press, Cambridge (2005) {{ISBN|052184150X}}</ref> | ||
Briefly, this approach is based on the following straightforward property of probability distributions. Let ''p(x,y)'' be the joint probability of observing ''x'' and ''y'' simultaneously. Let ''p(x|y)'' be the [[:Wikipedia:conditional probability|conditional probability]] of observing ''x'', given ''y''. Then, by definition | Briefly, this approach is based on the following straightforward property of probability distributions. Let ''p(x,y)'' be the joint probability of observing ''x'' and ''y'' simultaneously. Let ''p(x|y)'' be the [[:Wikipedia:conditional probability|conditional probability]] of observing ''x'', given ''y''. Then, by definition | ||
:<math>p(x|y)p(y) = p(x,y) = p(y|x)p(x)\,</math> | :<math>p(x|y)p(y) = p(x,y) = p(y|x)p(x)\,</math> | ||
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The advantage of the parametric representation is that the abstract 'system state' is reduced to a finite set of parameters, greatly facilitating numerical analysis. | The advantage of the parametric representation is that the abstract 'system state' is reduced to a finite set of parameters, greatly facilitating numerical analysis. | ||
This parametrization may involve, e.g., smooth (orthogonal) expansion functions such as [[:Wikipedia:Fourier-Bessel_series|Fourier-Bessel functions]], or discretely defined functionals on a grid. | This parametrization may involve, e.g., smooth (orthogonal) expansion functions such as [[:Wikipedia:Fourier-Bessel_series|Fourier-Bessel functions]], or discretely defined functionals on a grid. | ||
<ref>[[doi:10.1088/0741-3335/50/8/085002| | <ref>J. Svensson et al, ''Current tomography for axisymmetric plasmas'', [[doi:10.1088/0741-3335/50/8/085002|Plasma Phys. Control. Fusion '''50''' (2008) 085002]]</ref> | ||
=== Maximization === | === Maximization === | ||
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The goal of Integrated Data Analysis (IDA) is to combine the information from a set of diagnostics providing complementary information in order to recover the best possible reconstruction of the actual state of the system subjected to measurement. This goal overlaps with the goal of Bayesian data analysis, but IDA applies Bayesian inference in a relatively loose manner to allow incorporating information obtained with traditional or non-Bayesian methods. | The goal of Integrated Data Analysis (IDA) is to combine the information from a set of diagnostics providing complementary information in order to recover the best possible reconstruction of the actual state of the system subjected to measurement. This goal overlaps with the goal of Bayesian data analysis, but IDA applies Bayesian inference in a relatively loose manner to allow incorporating information obtained with traditional or non-Bayesian methods. | ||
<ref> | <ref>R. Fischer, C. Wendland, A. Dinklage, et al, '' Thomson scattering analysis with the Bayesian probability theory'', [[doi:10.1088/0741-3335/44/8/306|Plasma Phys. Control. Fusion '''44''' (2002) 1501]]</ref> | ||
<ref> | <ref>R. Fischer, A. Dinklage, and E. Pasch, ''Bayesian modelling of fusion diagnostics'', [[doi:10.1088/0741-3335/45/7/304|Plasma Phys. Control. Fusion '''45''' (2003) 1095-1111]]</ref> | ||
<ref> | <ref>R. Fischer, A. Dinklage, ''Integrated data analysis of fusion diagnostics by means of the Bayesian probability theory'', [[doi:10.1063/1.1787607|Rev. Sci. Instrum. '''75''' (2004) 4237]]</ref> | ||
<ref> | <ref>A. Dinklage, R. Fischer, and J. Svensson, ''Topics and Methods for Data Validation by Means of Bayesian Probability Theory'', [http://www.new.ans.org/pubs/journals/fst/a_575 Fusion Sci. Technol. '''46''' (2004) 355]</ref> | ||
<ref> | <ref>J. Svensson, A. Werner, ''Large Scale Bayesian Data Analysis for Nuclear Fusion Experiments'', [[doi:10.1109/WISP.2007.4447579|IEEE International Symposium on Intelligent Signal Processing (2007) 1]]</ref> | ||
<ref> | <ref>R. Fischer, C.J. Fuchs, B. Kurzan, et al., ''Integrated Data Analysis of Profile Diagnostics at ASDEX Upgrade'', [http://www.new.ans.org/pubs/journals/fst/a_10892 Fusion Sci. Technol. '''58''' (2010) 675]</ref> | ||
<ref> | <ref>B.Ph. van Milligen, T. Estrada, E. Ascasíbar, et al, ''Integrated data analysis at TJ-II: the density profile'', [[doi:10.1063/1.3608551|Rev. Sci. Instrum. '''82''' (2011) 073503]]</ref> | ||
== See also == | == See also == | ||