It is sometimes desired to update solutions to systems of equations or other problems as new information is to be appended. Also, a system that is too large to solve directly can often be managed by first solving a part of the system, and then updating the solution with the rest of the system. This updating procedure is often required to be both efficient and stable, and recomputing the solution from scratch may be too costly. Beside efficiency and stability, factors such as storage requirement, simplicity, and applicability are often important.
Updating the least squares solution to an over determined system of linear equations can be done in many ways. The method of Recursive Least Squares is simple and efficient, and works in most cases, although it is a bit sensitive to round-off errors. Updating the QR Decomposition, the Cholesky Factorization, or the Singular Value Decomposition is in general more stable, but these tasks are often a bit more complex than the Recursive Least Squares. Updating problems with constraints is possible when using Constrained Recursive Least Squares.
These methods all give a single solution as result, for example the least squares solution, but errors and uncertainty are not handled.
The Bayesian Inference offers a different type of updating. Here the answer is given in form of a probability distribution, with which it is possible to study the reasonableness of different solutions. This approach handles measuring errors, and information about the uncertainty in the answer is available.
Bayesian Inference can be applied to both linear and non-linear models, and one can incorporate additional information about the solution. When the noise and other errors are assumed Gaussian, the calculations are particularly simple and closely related to ordinary least squares problems. Some connections to non-Bayesian methods are pointed out, as well as a non-recursive property of the Bayesian Inference in linear problems with independent observation series.
Author: Lundman, Urban
Source: Luleå University of Technology
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