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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]]. | |||
<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> | |||
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> | |||
from which follows ''Bayes' theorem'': | |||
:<math>p(x|y) = \frac{p(y|x)p(x)}{p(y)}</math> | |||
& | == Interpretation of Bayes' Theorem == | ||
& | |||
& | The interpretation of this expression in the framework of data interpretation is as follows. | ||
Given the initial knowledge of a system, quantified by the ''prior'' probability ''p(x)'' of system states ''x'', one makes an observation ''y'' with probability (or degree of confidence) ''p(y)'', adding to the prior knowledge. | |||
* instead of computing an estimate of the inverse of the forward model (as with FP), | The ''posterior'' probability ''p(x|y)'' quantifies this enhanced knowledge of the system, ''given'' the observation ''y''. | ||
* the handling of error propagation is more sophisticated within | Thus, the Bayesian inference rule allows one to (a) gradually improve the knowledge of a system by adding observations, and (b) easily combining information from diverse sources by formulating the degree of knowledge of the system and the observations in terms of probability distributions. | ||
The quantity ''p(y|x)'' is a fundamental quantity linking the prior and posterior distributions, called the ''likelihood'', and expresses the probability of observing ''y'', given the prior knowledge ''x''. | |||
== Forward modelling == | |||
The likelihood is evaluated using a ''forward model'' of the experiment, returning the value of simulated measurements while ''assuming'' a given physical state ''x'' of the experimental system. | |||
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. | |||
=== 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 == | |||
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. | |||
Applying Bayes theorem one obtains | |||
:<math>p(\alpha|D) = \frac{p(D|\alpha)p(\alpha)}{p(D)}</math> | |||
where ''D'' represents the available data. | |||
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. | |||
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>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 === | |||
The maximization of the posterior probability as a function of the parameters ''α'' yields the most likely value of the parameters, given the data ''D'', which is the basic answer to the data interpretation problem. | |||
=== Marginalization === | |||
The width of the posterior distribution yields the error in the parameters. | |||
To obtain the error in a given parameter ''α<sub>i</sub>'', the posterior distribution is ''marginalized'' by integrating over the remaining ''N-1'' parameters: | |||
:<math>p(\alpha_i|D) = \int{p(\alpha|D) d\alpha_1 \cdots d\alpha_{i-1}d\alpha_{i+1}\cdots d\alpha_N}</math> | |||
The width of this one-dimensional distribution is found using standard procedures. | |||
== Comparison with Function Parametrization == | |||
[[Function parametrization]] (FP) is another statistical technique for recovering system parameters from diverse measurements. | |||
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); | |||
* the handling of error propagation is more sophisticated within Bayesian analysis, allowing non-Gaussian error distributions and absolutely general and complex parameter interdependencies; and | |||
* additionally, it provides a systematic way to include prior knowledge into the analysis. | * additionally, it provides a systematic way to include prior knowledge into the analysis. | ||
Typically, the maximization process is CPU intensive, so that Bayesian analysis is not usually suited for real-time data analysis (unlike FP). | |||
== Integrated Data Analysis == | |||
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>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>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>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>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>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>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>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 == | ||
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== References == | == References == | ||
<references /> | |||