Optimal estimation

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:

${\displaystyle \mathbf {A} {\vec {x}}={\vec {y}}}$

The essential concept is to transform the matrix, A, into a conditional probability and the variables, ${\displaystyle {\vec {x}}}$ and ${\displaystyle {\vec {y}}}$ into probability distributions by assuming Gaussian statistics and using empirically-determined covariance matrices.