The article touches on two themes in which Larry Brown has been interested, namely foundations, and mathematical analysis of Bayesian decision theory. In the section on foundations, a new formulation of the problem of testing is given, and is theoretically explored. The formulation strikes a balance between false discoveries and missed discoveries. Basic higher order asymptotic theory for the α level that this formulation would imply is then worked out. Accuracy is investigated and examples are

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... given. In the section on Bayesian decision theory, first, the Brown identities are connected to a set of inequalities of elliptic boundary value problems. It is shown by four specific results that sometimes a result in that field can lead to new results in Bayesian decision theory, and sometimes a result in decision theory can give surprisingly useful information about a problem in that field. For instance, Brown identities can provide amazingly good estimates of best constants in the Nash inequalities over Sobolev spaces. The article ends with a result on the modern theory of high dimensional Gaussian mean estimation. By means of a triangulation of the Brown identity, the Rayleigh-Ritz variational formula of boundary value problems, and the famous Donsker-Varadhan result connecting the Rayleigh-Ritz formula to the absorption time of a Brownian motion into the boundary of a smooth bounded open domain, we show that the minimax risk of estimating the Gaussian mean can be approximated by chasing a Brownian motion to the boundary of the parameter space. This link should be tested by simulation. 1 2 A. DasGupta eries. The traditional Neyman-Pearson formulation and the current work in the popular area of multiple testing place an asymmetric emphasis on false discoveries. In the symmetric formulation of Section 2 of this article, it is argued, via appropriate asymptotic expansions, that the traditional α levels are quite possibly too large in some problems and that the level α should go down to zero at a suitable rate, depending on the particular problem, and depending on which particular alternatives are practically significant in that problem. Mathematically, this requires the specification of a density g on the alternative. The asymptotic expansions show the rate at which α should go to zero, and this rate depends on the smoothness of g at the boundary. The exact asymptotic expansions should not be taken literally. Asymptotic expansions often produce ugly coefficients, and here too they do. The expansions should be separated from the formulation and the outcome that α should go to zero at some suitable rate, which is not universal, but depends on the problem. Coming from other angles, this has been argued in some of the Bayesian literature, for example Berger and Sellke (1987) ; also see Casella and Berger (1987) . Section 2.4 offers some thoughts on where this general approach might next go. The final section, Section 3, goes back to where many of us, including Larry Brown himself, started. It is the mutual connection of various Bayesian techniques, tools, and theorems and risk based decision theoretic properties, such as minimaxity. Of course, admissibility is also mostly about Bayes, but it is not mentioned here. Diaconis and Holmes (1996) is a very good place to look for new possibilities and good connections. The main thesis of Section 3 is that there are aspects of the Brown identity (Brown (1971) ) that have unexplored connections to a very well developed area of analysis, namely inequalities of elliptic boundary value problems. Typically, such an inequality is of the form f (k) q,G ≤ K( f α p,G )( f (n) β r,G ) for some suitably large class of Sobolev type spaces of functions f on some subset G of an Euclidean space, and for flexible p, q, r, k, n, with the restriction that 0 ≤ k < n. The inequality is supposed to hold with a universal constant K = K (G, k, n, p, q, r). When the smallest possible universal constant has been found, one calls it the best constant for that inequality. Usually, best constants are very hard to find, and appear to be known analytically only in isolated cases. The link of all these to Bayesian decision theory is through the Brown identity for Bayes risks. The Brown identity showed that the Bayes risk in a multivariate normal mean problem is related to the Fisher information of the marginal, which is essentially the square of the L 2 norm of the gradient of the square root of the marginal density. Once this identity is at hand, the inequalities of boundary value problems, such as the ones mentioned above, lead to a large collection of connections, and these connections go both ways. Section 3 shows that sometimes a known result in boundary value problems can lead to a result in decision theory, with the Brown identity being the link; and sometimes, quantities in Bayesian decision theory lead to information about a question in boundary value problems, for example, how small can the best constant in an inequality possibly be? Four such specific connections are laid out in Section 3, by using a generalized Heisenberg's uncertainty inequality, an inequality of the HELP type, the Nash inequality, and an inequality of Landau (although commonly ascribed to Kolmogorov and Landau). The number of such connections would be essentially unlimited, with the Brown identity always being the link. Diaconis and Saloff-Coste (1996) have used Nash inequalities to study the speed of convergence of some finite Markov chains to stationarity. So, in some sense, there is a history of the Nash inequalities being linked to problems in mathematical statistics. Hopefully, someone will pursue these