Bayesian data analysis - Revision history

Admin at 10:41, 26 January 2023

2023-01-26T10:41:28Z

← Older revision		Revision as of 12:41, 26 January 2023
Line 1:		Line 1:
	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>

Admin: Updated reference links

2018-04-03T13:58:54Z

Updated reference links

← Older revision		Revision as of 15:58, 3 April 2018
Line 39:		Line 39:
	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\|~~J. Svensson et al, ''Current tomography for axisymmetric plasmas'',~~ Plasma Phys. Control. Fusion '''50''' (2008) 085002]]</ref>		<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 ===
Line 64:		Line 64:

	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>~~[http://dx.doi.org/10.1088/0741-3335/44/8/306~~ R. Fischer, C. Wendland, A. Dinklage, et al, '' Thomson scattering analysis with the Bayesian probability theory'', Plasma Phys. Control. Fusion '''44''' (2002) 1501]</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>~~[http://dx.doi.org/10.1088/0741-3335/45/7/304~~ R. Fischer, A. Dinklage, and E. Pasch, ''Bayesian modelling of fusion diagnostics'', Plasma Phys. Control. Fusion '''45''' (2003) 1095-1111]</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>~~[http://link.aip.org/link/?RSINAK/75/4237/1~~ R. Fischer, A. Dinklage, ''Integrated data analysis of fusion diagnostics by means of the Bayesian probability theory'', Rev. Sci. Instrum. '''75''' (2004) 4237]</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>~~[http://www.new.ans.org/pubs/journals/fst/a_575~~ A. Dinklage, R. Fischer, and J. Svensson, ''Topics and Methods for Data Validation by Means of Bayesian Probability Theory'', Fusion Sci. Technol. '''46''' (2004) 355]</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>~~[http://dx.doi.org/10.1109/WISP.2007.4447579~~ J. Svensson, A. Werner, ''Large Scale Bayesian Data Analysis for Nuclear Fusion Experiments'', IEEE International Symposium on Intelligent Signal Processing (2007) 1]</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>~~[http://www.new.ans.org/pubs/journals/fst/a_10892~~ R. Fischer, C.J. Fuchs, B. Kurzan, et al., ''Integrated Data Analysis of Profile Diagnostics at ASDEX Upgrade'', Fusion Sci. Technol. '''58''' (2010) 675]</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>~~[http://link.aip.org/link/doi/10.1063/1.3608551~~ B.Ph. van Milligen, T. Estrada, E. Ascasíbar, et al, ''Integrated data analysis at TJ-II: the density profile'', Rev. Sci. Instrum. '''82''' (2011) 073503]</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 ==

Admin: /* Parametric formulation */

2011-12-10T10:47:04Z

Parametric formulation

← Older revision		Revision as of 12:47, 10 December 2011
Line 39:		Line 39:
	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\|J. Svensson et al, ''Current tomography for axisymmetric plasmas'', Plasma Phys. Control. Fusion '''50''' (2008) 085002</ref>		<ref>[[doi:10.1088/0741-3335/50/8/085002\|J. Svensson et al, ''Current tomography for axisymmetric plasmas'', Plasma Phys. Control. Fusion '''50''' (2008) 085002]]</ref>

	=== Maximization ===		=== Maximization ===

Admin: /* Parametric formulation */

2011-12-10T10:46:06Z

Parametric formulation

← Older revision		Revision as of 12:46, 10 December 2011
Line 36:		Line 36:
	:<math>p(\alpha\|D) = \frac{p(D\|\alpha)p(\alpha)}{p(D)}</math>		:<math>p(\alpha\|D) = \frac{p(D\|\alpha)p(\alpha)}{p(D)}</math>
	where ''D'' represents the available data.		where ''D'' represents the available data.
	The likelihood ''p(D\|α)'' ~~speficies~~ the probability of a specific measurement outcome ''D'' for a given choice of parameters ''α''.		The likelihood ''p(D\|α)'' specifies the probability of a specific measurement outcome ''D'' for a given choice of parameters ''α''.
	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.
			<ref>[[doi:10.1088/0741-3335/50/8/085002\|J. Svensson et al, ''Current tomography for axisymmetric plasmas'', Plasma Phys. Control. Fusion '''50''' (2008) 085002</ref>

	=== Maximization ===		=== Maximization ===

Admin at 09:23, 10 December 2011

2011-12-10T09:23:33Z

← Older revision		Revision as of 11:23, 10 December 2011
Line 2:		Line 2:
	<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 ''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>
	from which follows ''Bayes' theorem'':		from which follows ''Bayes' theorem'':

Admin: /* Forward modelling */

2011-12-10T09:13:33Z

Forward modelling

← Older revision		Revision as of 11:13, 10 December 2011
Line 20:		Line 20:
	Mathematically, this forward model (mapping system parameters to measurements) is often much easier to evaluate than the reverse mapping (from measurements to system parameters), as the latter is often the inverse of a projection, which is therefore typically ill-determined.		Mathematically, this forward model (mapping system parameters to measurements) is often much easier to evaluate than the reverse mapping (from measurements to system parameters), as the latter is often the inverse of a projection, which is therefore typically ill-determined.
	On the other hand, evaluating the forward model requires detailed knowledge of the physical system and the complete measurement process.		On the other hand, evaluating the forward model requires detailed knowledge of the physical system and the complete measurement process.

			=== The Likelihood ===

			The forward model is used to predict the measurements ''y'', based on the physical state ''x'' of the system. The Likelihood ''p(y\|x)'' specifies the most probable measurement outcome, which corresponds to the maximum of the distribution ''p(y\|x)'', as well as its uncertainty, given by the width of the distribution.

			In a typical case, assume that the model is such that the measurement outcomes are distributed like a Gaussian around a most probable value ''y<sub>0</sub>'', with an error ''Δ y''. Then the likelihood will be
			:<math>p(y\|x) = \frac{1}{\sqrt{2\pi \Delta y^2}}\exp \left ( -\frac{(y-y_0)^2}{2\Delta y^2}\right )</math>
			Note that the negative logarithm of the likelihood is proportional to the [[:Wikipedia:Chi-squared distribution\|''χ<sup>2</sup>'']] of the fit of the data ''y'' to the model ''y<sub>0</sub>'', and maximizing the likelihood will minimize ''χ<sup>2</sup>'', thus establishing the link between the Bayesian approach and the common [[:Wikipedia:Least squares\|least squares]] fit. However, the Bayesian approach is more general than the standard least squares fit, as it can handle any type of probability distribution.

	== Parametric formulation ==		== Parametric formulation ==

Admin: /* Comparison with Function Parametrization */

2011-12-09T20:01:02Z

Comparison with Function Parametrization

← Older revision		Revision as of 22:01, 9 December 2011
Line 45:		Line 45:

	[[Function parametrization]] (FP) is another statistical technique for recovering system parameters from diverse measurements.		[[Function parametrization]] (FP) is another statistical technique for recovering system parameters from diverse measurements.
	~~Like~~ FP, Bayesian data analysis ~~requires~~ having a ''forward model'' to predict the measurement readings for any given state of the physical system, and the state of the physical system and the measurement process is ''parametrized''. However		Both FP and Bayesian data analysis require having a ''forward model'' to predict the measurement readings for any given state of the physical system, and the state of the physical system and the measurement process is ''parametrized''. However
	* instead of computing an estimate of the inverse of the forward model (as with FP), Bayesian analysis finds the best model state corresponding to a specific measurement by a maximization procedure (maximization of the likelihood);		* instead of computing an estimate of the inverse of the forward model (as with FP), Bayesian analysis finds the best model state corresponding to a specific measurement by a maximization procedure (maximization of the likelihood);
	* the handling of error propagation is more sophisticated within Bayesian analysis, allowing non-Gaussian error distributions and absolutely general and complex parameter interdependencies; and		* the handling of error propagation is more sophisticated within Bayesian analysis, allowing non-Gaussian error distributions and absolutely general and complex parameter interdependencies; and

Admin: /* Parametric formulation */

2011-12-09T19:56:36Z

Parametric formulation

← Older revision		Revision as of 21:56, 9 December 2011
Line 23:		Line 23:
	== Parametric formulation ==		== Parametric formulation ==

	The description of the system state is usually done by defining a parametric representation, e.g., by defining ~~spatial~~ density or temperature profiles as a function of space via a vector of ''N'' parameters, α: ''n(r,α)'' or ''T(r,α)''.		The description of the system state is usually done by defining a parametric representation, e.g., by defining density or temperature profiles as a function of space via a vector of ''N'' parameters, α: ''n(r,α)'' or ''T(r,α)''.
	The parameters obey a prior distribution ''p(α)'', expressing physical or other constraints.		The parameters obey a prior distribution ''p(α)'', expressing physical or other constraints.
	Applying Bayes theorem one obtains		Applying Bayes theorem one obtains

Admin at 19:55, 9 December 2011

2011-12-09T19:55:43Z

@@ Line 20: / Line 20: @@
 Mathematically, this forward model (mapping system parameters to measurements) is often much easier to evaluate than the reverse mapping (from measurements to system parameters), as the latter is often the inverse of a projection, which is therefore typically ill-determined.
 On the other hand, evaluating the forward model requires detailed knowledge of the physical system and the complete measurement process.
 == Comparison with Function Parametrization ==

Admin at 15:53, 9 December 2011

2011-12-09T15:53:19Z

@@ Line 1: / Line 1: @@
-The goal of [[:Wikipedia:Bayesian inference|Bayesian]]
+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>P. Gregory, ''Bayesian Logical Data Analysis for the Physical Sciences'', Cambridge University Press, Cambridge (2005) ISBN 052184150X</ref>
-or 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.
+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 ''conditional'' probability of observing ''x'', given ''y''. Then, by definition
 <ref>[http://dx.doi.org/10.1088/0741-3335/44/8/306 R. Fischer, C. Wendland, A. Dinklage, et al, '' Thomson scattering analysis with the Bayesian probability theory'', Plasma Phys. Control. Fusion '''44''' (2002) 1501]</ref>
 <ref>[http://dx.doi.org/10.1088/0741-3335/45/7/304 R. Fischer, A. Dinklage, and E. Pasch, ''Bayesian modelling of fusion diagnostics'', Plasma Phys. Control. Fusion '''45''' (2003) 1095-1111]</ref>
@@ Line 10: / Line 40: @@
 <ref>[http://www.new.ans.org/pubs/journals/fst/a_10892 R. Fischer, C.J. Fuchs, B. Kurzan, et al., ''Integrated Data Analysis of Profile Diagnostics at ASDEX Upgrade'', Fusion Sci. Technol. '''58''' (2010) 675]</ref>
 <ref>[http://link.aip.org/link/doi/10.1063/1.3608551 B.Ph. van Milligen, T. Estrada, E. Ascasíbar, et al, ''Integrated data analysis at TJ-II: the density profile'', Rev. Sci. Instrum. '''82''' (2011) 073503]</ref>
 == See also ==