“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always...“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always of the same gender as the member making the selection. This model for evolution is isomorphic to the classic Pólya’s urn. The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, then replacing it and adding a new ball of the same color. It is well known that for Pólya’s urn, the proportion of red balls in the urn is a martingale. It follows that for a consanguinity, the proportion of the membership that is male is a martingale. Furthermore, being bounded, this martingale converges to a limit. For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is also a well-known second-degree martingale from which the variance of the limiting distribution can be deduced. What the author discovered, in the process of solving his own examination problem, is that a similar martingale exists also for Pólya’s urn, even though in this case the number of red balls is the sum of random variables that are not independent. This new martingale can be used to calculate the variance of the limiting distribution. Traditionally, the probability that r red balls will be drawn from Pólya’s urn in n trials is derived by a rather tricky argument involving conditional probability. This article uses an obvious but overlooked simpler approach. Pólya’s formula for the probability that m male members will be chosen in n weeks is derived, without any mention of conditional probability, by an elementary counting argument, and its limit is shown to be a beta distribution.展开更多
The Surveillance, Epidemiology and End Results (SEER) cancer database contains survival data forUS individuals diagnosed with cancer. Semiparametric Bayesian methods are computationallyexpensive to fit for such large ...The Surveillance, Epidemiology and End Results (SEER) cancer database contains survival data forUS individuals diagnosed with cancer. Semiparametric Bayesian methods are computationallyexpensive to fit for such large data-sets. This paper develops a cost-effective Markov chain MonteCarlo strategy for censored outcomes to fit a semiparametric bayesian analysis of SEER data ofNew Mexico. We use an accelerated failure time model, with Dirichlet process random effectsfor inter-subject variation, and intrinsic conditionally autoregressive random effects for spatialcorrelations. The results offer insights into differences in breast cancer mortality rates betweenethnic groups, tumor grade and spatial effect of counties.展开更多
文摘“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always of the same gender as the member making the selection. This model for evolution is isomorphic to the classic Pólya’s urn. The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, then replacing it and adding a new ball of the same color. It is well known that for Pólya’s urn, the proportion of red balls in the urn is a martingale. It follows that for a consanguinity, the proportion of the membership that is male is a martingale. Furthermore, being bounded, this martingale converges to a limit. For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is also a well-known second-degree martingale from which the variance of the limiting distribution can be deduced. What the author discovered, in the process of solving his own examination problem, is that a similar martingale exists also for Pólya’s urn, even though in this case the number of red balls is the sum of random variables that are not independent. This new martingale can be used to calculate the variance of the limiting distribution. Traditionally, the probability that r red balls will be drawn from Pólya’s urn in n trials is derived by a rather tricky argument involving conditional probability. This article uses an obvious but overlooked simpler approach. Pólya’s formula for the probability that m male members will be chosen in n weeks is derived, without any mention of conditional probability, by an elementary counting argument, and its limit is shown to be a beta distribution.
基金National Science Foundation[grant number DMS-1461948].
文摘The Surveillance, Epidemiology and End Results (SEER) cancer database contains survival data forUS individuals diagnosed with cancer. Semiparametric Bayesian methods are computationallyexpensive to fit for such large data-sets. This paper develops a cost-effective Markov chain MonteCarlo strategy for censored outcomes to fit a semiparametric bayesian analysis of SEER data ofNew Mexico. We use an accelerated failure time model, with Dirichlet process random effectsfor inter-subject variation, and intrinsic conditionally autoregressive random effects for spatialcorrelations. The results offer insights into differences in breast cancer mortality rates betweenethnic groups, tumor grade and spatial effect of counties.
