the same as the set of admissible rules. Note that no situation is possible where a nonrandomised Bayes rule does not exist but a randomised Bayes rule does. The existence of a randomised Bayes rule implies the existence of a nonrandomised Bayes rule. This is also true in the general case, even with infinite parameter space, infinite Bayes risk, and regardless of whether the infimum Bayes risk can be attained. This supports the intuitive notion that the statistician need not utilise randomisation to arrive at statistical decisions. In practice As randomised Bayes rules always have nonrandomised alternatives, they are unnecessary in Bayesian statistics. However, in frequentist statistics, randomised rules are theoretically necessary under certain situations, and were thought to be useful in practice when they were first invented: Egon Pearson forecast that they 'will not meet with strong objection'. However, few statisticians actually implement them nowadays. Randomised test Randomized tests should not be confused with permutation tests. In the usual formulation of the likelihood ratio test, the null hypothesis is rejected whenever the likelihood ratio is smaller than some constant , and accepted otherwise. However, this is sometimes problematic when is discrete under the null hypothesis, when is possible. A solution is to define a test function , whose value is the probability at which the null hypothesis is accepted: This can be interpreted as flipping a biased coin with a probability of returning heads whenever and rejecting the null hypothesis if a heads turns up. A generalised form of the Neyman–Pearson lemma states that this test has maximum power among all tests at the same significance level , that such a test must exist for any significance level , and that the test is unique under normal situations. As an example, consider the case where the underlying distribution is Bernoulli with probability , and we would like to test the null hypothesis against the alternative hypothesis . It is natural to choose some such that , and reject the null whenever , where is the test statistic. However, to take into account cases where , we define the test function: where is chosen such that . Randomised confidence intervals An analogous problem arises in the construction of confidence intervals. For instance, the Clopper-Pearson interval is always conservative because of the discrete nature of the binomial distribution. An alternative is to find the upper and lower confidence limits and by