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.

(9.35) |

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

in which the mean is and the standard deviation is .

The optimal parameter estimate based on is obtained by selecting to minimize

(9.37) |

in which

(9.38) |

by Bayes' rule. The term does not depend on , and it can therefore be ignored in the optimization. Using the prior density, outside of ; hence, the domain of integration can be restricted to . The value of is also a constant that can be ignored in the optimization. Using (9.36), this means that is selected to optimize

which can be expressed in terms of the standard error function, (the integral from 0 to a constant, of a Gaussian density over an interval).

If a sequence, , , , of independent observations is obtained, then (9.39) is replaced by

Steven M LaValle 2020-08-14