Another important application of the decision-making framework of this section is parameter estimation [89,268]. In this case, nature selects a parameter, , and represents a parameter space. Through one or more independent trials, some observations are obtained. Each observation should ideally be a direct measurement of , but imperfections in the measurement process distort the observation. Usually, , and in many cases, . The robot action is to guess the parameter that was chosen by nature. Hence, . In most applications, all of the spaces are continuous subsets of . The cost function is designed to increase as the error, , becomes larger.
Suppose that a Bayesian approach is taken. The prior probability density is given as uniform over an interval . An observation is received, but it is noisy. The noise can be modeled as a second action of nature, as described in Section 9.2.3. This leads to a density . Suppose that the noise is modeled with a Gaussian, which results in
The optimal parameter estimate based on is obtained by selecting to minimize
If a sequence, , , , of independent observations is obtained, then (9.39) is replaced by
Steven M LaValle 2012-04-20