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Bayesian data analysis: Difference between revisions
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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 == | ||