corresponding to the risks under and respectively. An example is shown on the right. Admissibility An admissible decision rule is one that is not dominated by any other decision rule, i.e. there is no decision rule that has equal risk as or lower risk than it for all parameters and strictly lower risk than it for some parameter. In a finite decision problem, the risk point of an admissible decision rule has either lower x-coordinates or y-coordinates than all other risk points or, more formally, it is the set of rules with risk points of the form such that . Thus the left side of the lower boundary of the risk set is the set of admissible decision rules. Minimax A minimax Bayes rule is one that minimises the supremum risk among all decision rules in . Sometimes, a randomised decision rule may perform better than all other nonrandomised decision rules in this regard. In a finite decision problem with two possible parameters, the minimax rule can be found by considering the family of squares . The value of for the smallest of such squares that touches is the minimax risk and the corresponding point or points on the risk set is the minimax rule. If the risk set intersects the line , then the admissible decision rule lying on the line is minimax. If or holds for every point in the risk set, then the minimax rule can either be an extreme point (i.e. a nonrandomised decision rule) or a line connecting two extreme points (nonrandomised decision rules). Bayes A randomised Bayes rule is one that has infimum Bayes risk among all decision rules. In the special case where the parameter space has two elements, the line , where and denote the prior probabilities of and respectively, is a family of points with Bayes risk . The minimum Bayes risk for the decision problem is therefore the smallest such that the line touches the risk set. This line may either touch only one extreme point of the risk set, i.e. correspond to a nonrandomised decision rule, or overlap with an entire side of the risk set, i.e. correspond to two nonrandomised decision rules and randomised decision rules combining the two. This is illustrated by the three situations below: As different priors result in different slopes, the set of all rules that are Bayes with respect to some prior are