David Blackwell's paper "Comparison of Experiments" introduces a method to compare two sampling procedures or experiments, defined by Bohnenblust, Shapley, and Sherman. The key concept is that one experiment \(a\) is more informative than another experiment \(\beta\) (written \(a \succ \beta\)) if any risk attainable with \(\beta\) can also be achieved with \(a\). If \(a\) is a sufficient statistic for a procedure equivalent to \(\beta\), then \(a \succ \beta\). The paper explores various properties of this relationship, such as the independence of outcomes and the convexity of loss vectors. The paper defines an experiment as a set of probability measures on a Borel field, and a decision problem involves choosing actions based on observed outcomes. The loss from an action is defined as the expected loss from the observed outcome. The set of attainable loss vectors is denoted by \(R(a, A)\), where \(A\) is the set of possible actions. The paper proves that if \(a \succ \beta\), then \(R(a, A) \supseteq R(\beta, A)\) for any closed convex set \(A\). The paper also discusses the equivalence of certain conditions for \(a \supset \beta\), including inequalities involving convex functions and cumulative distribution functions. It introduces the concept of a standard experiment, which is an experiment whose outcome is a point in a probability simplex, and shows that every experiment is equivalent to a standard experiment. The paper further explores the sufficiency of experiments, where one experiment \(M\) is said to be sufficient for another \(m\) if there exists a function that transforms the distribution of \(m\) into that of \(M\). The paper proves that if \(M \supseteq m\), then \(M > m\), and provides conditions for the equivalence of \(\succ\) and \(\supset\) for \(N = 2\). Finally, the paper discusses the combination of experiments, showing that the standard experiment for the combination of two experiments is the same as the standard experiment for the individual experiments combined. It also provides an application to a problem in \(2 \times 2\) tables, demonstrating that selecting individuals based on their rarity in the general population is more informative than other procedures.David Blackwell's paper "Comparison of Experiments" introduces a method to compare two sampling procedures or experiments, defined by Bohnenblust, Shapley, and Sherman. The key concept is that one experiment \(a\) is more informative than another experiment \(\beta\) (written \(a \succ \beta\)) if any risk attainable with \(\beta\) can also be achieved with \(a\). If \(a\) is a sufficient statistic for a procedure equivalent to \(\beta\), then \(a \succ \beta\). The paper explores various properties of this relationship, such as the independence of outcomes and the convexity of loss vectors. The paper defines an experiment as a set of probability measures on a Borel field, and a decision problem involves choosing actions based on observed outcomes. The loss from an action is defined as the expected loss from the observed outcome. The set of attainable loss vectors is denoted by \(R(a, A)\), where \(A\) is the set of possible actions. The paper proves that if \(a \succ \beta\), then \(R(a, A) \supseteq R(\beta, A)\) for any closed convex set \(A\). The paper also discusses the equivalence of certain conditions for \(a \supset \beta\), including inequalities involving convex functions and cumulative distribution functions. It introduces the concept of a standard experiment, which is an experiment whose outcome is a point in a probability simplex, and shows that every experiment is equivalent to a standard experiment. The paper further explores the sufficiency of experiments, where one experiment \(M\) is said to be sufficient for another \(m\) if there exists a function that transforms the distribution of \(m\) into that of \(M\). The paper proves that if \(M \supseteq m\), then \(M > m\), and provides conditions for the equivalence of \(\succ\) and \(\supset\) for \(N = 2\). Finally, the paper discusses the combination of experiments, showing that the standard experiment for the combination of two experiments is the same as the standard experiment for the individual experiments combined. It also provides an application to a problem in \(2 \times 2\) tables, demonstrating that selecting individuals based on their rarity in the general population is more informative than other procedures.