Appropriate and Inappropriate Estimation Techniques

\1ode (also called \1AP\ estimation. mean esumation and median estimation are examined here to determine when they can be 'afely used to derive (posterior) cost mmimwng estimates. !These are all Bayes procedures. using the mode. mean. or median of the posterior distnbu· lion.) [t is found that modal estimation only returns cost m1mmizmg estimates when the cost function is 0L. [f the cost funcuon is a function of distance then mean estimation only returns cost mmimizing estimates when the cost function 1s squared distance from the true value and median estimation only returns cost minimizing estimates when the cost function 1s the distance from the true value. Results are presented on the good· ness of modal estimation with non 0· L cost functions. In low-level vision small-scale image phenomena are often assigned labels such as "on a boundary" or "moving at .3 pixels per frame". \-lost current approaches to low-level vision return estimates of the feature labeling [Ballard82] [Andrews77b]. Any such technique operates implicitly by deriving an estimate from a posterior distribution. Given that the correct posterior distribution is derived an estimation tech­ nique, such as returning the label with highest proba­ bility. returns the most useful estimates when the user has an appropriate cost function. A cost function is appropriate for an estimation technique if the estima­ tion technique returns the expected cost minimizing estimate. The question addressed in this paper is: "What are the appropnate cost functions for different estimation techmques and how much extra cost is incurred by using an estimation technique with an inappropriate cost function?" This work is motivated by and lies within the field of decision theory [Berger80b].