Bayesian Reconstructions of $m, n$-Patterns

Abstract

The notion of $m, n$-pattern is introduced--namely, a division of the unit interval into at most $n$ cells (intervals or points), each having one of $m$ colors. Given an unknown $m, n$ pattern, it is desired to produce a reconstruction of the pattern using $r \geqq 1$ sample points (fixed or chosen at random) where the color is determined. The problem is studied from a decision-theoretic point of view. A way to obtain all the probability measures on the set of $m, n$-patterns is given. The notion of a Bayesian reconstruction rule (B.R.R.) is introduced. It is proved that when B.R.R.'s are considered, it is sufficient to use certain fixed sample points. A complete class of reconstruction rules is obtained. Finally an example of a B.R.R. is given for 2,2-patterns.