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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. | ||
== 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 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)'' speficies 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. | |||
=== 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 == | == Comparison with Function Parametrization == | ||