丸山 祐造

This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which claimed that the half-Cauchy prior is a sensible default prior for a scale parameter in hierarchical models. For estimation of a normal mean vector under the quadratic loss, they showed that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In terms of the shrinkage coefficient, the half-Cauchy prior has a U-shape and can be interpreted as a continuous spike and slab prior. In this paper, we consider a general class of priors with U-shapes and theoretically establish sufficient conditions for the minimaxity of the corresponding (generalized) Bayes estimators. We also develop an algorithm for posterior sampling and present numerical results.

We propose a new method of perturbing a major variable by adding noise such<br /> that results of regression analysis are unaffected. The extent of the<br /> perturbation can be controlled using a single parameter, which eases an actual<br /> perturbation application. On the basis of results of a numerical experiment, we<br /> recommend an appropriate value of the parameter that can achieve both<br /> sufficient perturbation to mask original values and sufficient coherence<br /> between perturbed and original data.

Moran&#039;s I statistic, a popular measure of spatial autocorrelation, is<br /> revisited. The exact range of Moran&#039;s I is given as a function of spatial<br /> weights matrix. We demonstrate that some spatial weights matrices lead the<br /> absolute value of upper (lower) bound larger than 1 and that others lead the<br /> lower bound larger than -0.5. Thus Moran&#039;s I is unlike Pearson&#039;s correlation<br /> coefficient. It is also pointed out that some spatial weights matrices do not<br /> allow Moran&#039;s I to take positive values regardless of observations. An<br /> alternative measure with exact range [-1,1] is proposed through a monotone<br /> transformation of Moran&#039;s I.

A new class of minimax Stein-type shrinkage estimators of a multivariate<br /> normal mean is studied where the shrinkage factor is based on an l_p norm. The<br /> proposed estimators allow some but not all coordinates to be estimated by 0<br /> thereby allow sparsity as well as minimaxity.

We study probit regression from a Bayesian perspective and give an<br /> alternative form for the posterior distribution when the prior distribution for<br /> the regression parameters is the uniform distribution. This new form allows<br /> simple Monte Carlo simulation of the posterior as opposed to MCMC simulation<br /> studied in much of the literature and may therefore be more efficient<br /> computationally. We also provide alternative explicit expression for the first<br /> and second moments. Additionally we provide analogous results for Gaussian<br /> priors.

For the balanced ANOVA setup, we propose a new closed form Bayes factor<br /> without integral representation, which is however based on fully Bayes method,<br /> with reasonable model selection consistency for two asymptotic situations<br /> (either number of levels of the factor or number of replication in each level<br /> goes to infinity). Exact analytical calculation of the marginal density under a<br /> special choice of the priors enables such a Bayes factor.