## Analytical framework for predicting performance

Navigation is an estimation problem; a set of unknown parameters, location and attitude, are estimated from a set of observations. The CRLB is a standard tool for determining the uncertainty in the estimate based on uncertainty in the observations and a model relating the observed and estimated quantities.

Consider the estimation of an unknown parameter vector x from a set of observations z with known probability density pz(z\x). An estimator extracts the information from these observations to derive and estimate of the parameters based on the measurements, x{z). The uncertainty in this estimate is a direct consequence of how much information is available from the measurements. When it exists, the CRLB gives the lower bound on the variance of any valid unbiased estimator (Bar-Shalom, Li, & Kirubarajan, 2001). The Fisher information, Iz{x) is the information about the parameters, x contained in the observations, z.

Where £[ ] is the expectation operator. The CRLB, A(i(z)), is the inverse of the Fisher information, i.e.,

The CRLB is the minimum uncertainty achievable by an unknown optimal estimator. An estimator that approaches this existence of the lower bound is efficient, but the bound does not guarantee that an efficient estimator exists or that one can be found. Another consequence of this principle is that an efficient estimator extracts all the available information from the observations. Efficiency amounts to the extracted information being equal to the existing information.

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