reheated vanilla Rao-Blackwellisation

Over the weekend, I came across a X validated question asking for clarification about our 2012 Vanilla Rao-Blackwellisation paper with Randal. Question written in a somewhat formal style that made our work difficult to recognise… At least for yours truly.

Interestingly this led another (major) contributor to X validation to work out an uncompleted illustration as attached, when the target distribution is (1-x)². It seems strange to me that the basics of the method proves such a difficulty to fathom, given that it is a simple integration of the (actual and virtual) uniforms…. The point of the OP that the improvement brought by Rao-Blackwellisation is only conditional on the accepted values is correct, though.

Natural statistical science [#1]

Calyampudi Radhakrishna Rao (1920-2023)

Just heard that C.R. Rao had passed away on Wednesday. Above is a 1941 picture I photographed while attending the jubilee of the Department of Statistics of the University of Calcuta. Showing R.A. Fisher and P.C. Mahalanobis surrounded by faculty and students from the Department. Including a very young Rao who would a few years later go to Cambridge and write a PhD thesis on ANOVA under the supervision of R.A. Fisher. While my own interactions with C.R. Rao have been quite limited, from attending a seminar dinner with him when he visited Purdue University in 1988, to writing a critical assessment of Pitman nearness that he reportedly disliked, to writing chapters in some of the handbooks he edited and a review paper on Rao-Blackwellisation for the International Statistical Review special issue for his 100th birthday (which almost coincided with mine, one day off), he stood as a giant of the field, having impacted statistics and beyond in many and profound ways. The Hindu published an obituary immediately after his death, while Current Science has a longer if older biography full of pictures and Significance a series of articles on “C.R. Rao’s Century”. However, I’d like to recall this quote of his’, acknowledging his mother for his work habits.

‘For instilling in me the quest for knowledge, I owe to my mother, A. Laxmikanthamma, who, in my younger days, woke me up every day at four in the morning and lit the oil lamp for me to study in the quiet hours of the morning when the mind is fresh.’

top off…

Monte Carlo swindles

While reading Boos and Hugues-Olivier’s 1998 American Statistician paper on the applications of Basu’s theorem I can across the notion of Monte Carlo swindles. Where a reduced variance can be achieved without the corresponding increase in Monte Carlo budget. For instance, approximating the variance of the median statistic Μ for a Normal location family can be sped up by considering that

by Basu’s theorem. However, when reading the originating 1973 paper by Gross (although the notion is presumably due to Tukey), the argument boils down to Rao-Blackwellisation (without the Rao-Blackwell theorem being mentioned). The related 1985 American Statistician paper by Johnstone and Velleman exploits a latent variable representation. It also makes the connection with the control variate approach, noticing the appeal of using the score function as a (standard) control and (unusual) swindle, since its expectation is zero. I am surprised at uncovering this notion only now… Possibly because the method only applies in special settings.

A side remark from the same 1998 paper, namely that the enticing decomposition

when X/Y and Y are independent, should be kept out of reach from my undergraduates at all costs, as they would quickly get rid of the assumption!!!