In applied statistics, optimal estimation is a regularized matrix inverse method based on Bayes' theorem. It is used very commonly in the geosciences, particularly for atmospheric sounding. A matrix inverse problem looks like this:
A x → = y → {\displaystyle \mathbf {A} {\vec {x}}={\vec {y}}}
The essential concept is to transform the matrix, A, into a conditional probability and the variables, x → {\displaystyle {\vec {x}}} and y → {\displaystyle {\vec {y}}} into probability distributions by assuming Gaussian statistics and using empirically determined covariance matrices.
Derivation Typically, one expects the statistics of most measurements to be Gaussian. So for example for P ( y → | x → ) {\displaystyle P({\vec {y}}|{\vec {x}})} , we can write:
P ( y → | x → ) = 1 ( 2 π ) m n / 2 | S y | exp [ − 1 2 ( A x → − y → ) T S y − 1 ( A x → − y → ) ] {\displaystyle P({\vec {y}}|{\vec {x}})={\frac {1}{(2\pi )^{mn/2}|{\boldsymbol {S_{y}}}|}}\exp \left[-{\frac {1}{2}}({\boldsymbol {A}}{\vec {x}}-{\vec {y}})^{T}{\boldsymbol {S_{y}}}^{-1}({\boldsymbol {A}}{\vec {x}}-{\vec {y}})\right]}
where m and n are the numbers of elements in x → {\displaystyle {\vec {x}}} and y → {\displaystyle {\vec {y}}} respectively A {\displaystyle {\boldsymbol {A}}} is the matrix to be solved (the linear or linearised forward model) and S y {\displaystyle {\boldsymbol {S_{y}}}} is the covariance matrix of the vector y → {\displaystyle {\vec {y}}} . This can be similarly done for x → {\displaystyle {\vec {x}}} :
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