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:
formula_1
The essential concept is to transform the matrix, A, into a conditional probability and the variables, formula_2 and formula_3 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 formula_4, we can write:
formula_5
where "m" and "n" are the numbers of elements in formula_2 and formula_3 respectively formula_8 is the matrix to be solved (the linear or linearised forward model) and formula_9 is the covariance matrix of the vector formula_3. This can be similarly done for formula_2:
formula_12
Here formula_13 is taken to be the so-called "a-priori" distribution: formula_14 denotes the a-priori values for formula_15 while formula_16 is its covariance matrix.
The nice thing about the Gaussian distributions is that only two parameters are needed to describe them and so the whole problem can be converted once again to matrices. Assuming that formula_17 takes the following form:
formula_18
formula_19 may be neglected since, for a given value of formula_2, it is simply a constant scaling term. Now it is possible to solve for both the expectation value of formula_2, formula_22, and for its covariance matrix by equating formula_17 and formula_24. This produces the following equations:
formula_25
formula_26
Because we are using Gaussians, the expected value is equivalent to the maximum likely value, and so this is also a form of maximum likelihood estimation.
Typically with optimal estimation, in addition to the vector of retrieved quantities, one extra matrix is returned along with the covariance matrix. This is sometimes called the resolution matrix or the averaging kernel and is calculated as follows:
formula_27
This tells us, for a given element of the retrieved vector, how much of the other elements of the vector are mixed in. In the case of a retrieval of profile information, it typical indicates the altitude resolution for a given altitude. For instance if the resolution vectors for all the altitudes contain non-zero elements (to a numerical tolerance) in their four nearest neighbours, then the altitude resolution is only one fourth that of the actual grid size.