**Y**et another question on X validated reminded me of a discussion I had once with Jay Kadane when visiting Carnegie Mellon in Pittsburgh. Namely the fundamentally ill-posed nature of conjugate priors. Indeed, when considering the definition of a conjugate family as being a parameterised family Þ of distributions over the parameter space Θ stable under transform to the posterior distribution, this property is completely dependent (if there is such a notion as completely dependent!) on the dominating measure adopted on the parameter space Θ. Adopted is the word as there is no default, reference, natural, &tc. measure that promotes one specific measure on Θ as being *the* dominating measure. This is a well-known difficulty that also sticks out in most “objective Bayes” problems, as well as with maximum entropy priors. This means for instance that, while the Gamma distributions constitute a conjugate family for a Poisson likelihood, so do the truncated Gamma distributions. And so do the distributions which density (against a Lebesgue measure over an arbitrary subset of (0,∞)) is the product of a Gamma density by an arbitrary function of θ. I readily acknowledge that the standard conjugate priors as introduced in every Bayesian textbook are standard because they facilitate (to a certain extent) posterior computations. But, just like there exist an infinity of MaxEnt priors associated with an infinity of dominating measures, there exist an infinity of conjugate families, once more associated with an infinity of dominating measures. And the fundamental reason is that the sampling model (which induces the shape of the conjugate family) does not provide a measure on the parameter space Θ.

## Archive for Pittsburgh

## dominating measure

Posted in Books, pictures, Statistics, Travel, University life with tags Bayesian textbook, Carnegie Mellon University, conjugate priors, cross validated, dominating measure, Jay Kadane, Pittsburgh, posterior distribution on March 21, 2019 by xi'an## RSS tribute

Posted in Statistics, University life with tags David Spiegelhalter, Joyce Fienberg, Pittsburgh, Royal Statistical Society, shooting, StatsLife, Steve Fienberg on November 4, 2018 by xi'an## remembering Joyce Fienberg through Steve’s words

Posted in Statistics with tags Joyce Fienberg, Paris, Pittsburgh, Statistical Science, Steve Fienberg, University of Pittsburgh, University of Toronto on October 28, 2018 by xi'anI just learned the horrific news that Joyce Fienberg was one of the eleven people murdered yesterday morning at the Tree of Life synagogue. I had been vaguely afraid this could be the case since hearing about the shooting there, just because it was not far from the University of Pittsburgh, and CMU, but then a friend emailed me she indeed was one of the victims. When her husband Steve was on sabbatical in Paris, we met a few times for memorable dinners. I think the last time I saw her was a few years ago in a Paris hotel where Joyce, Steve and I had breakfast together to take advantage of one of their short trips to Paris. In remembrance of this wonderful woman who got assassinated by an anti-Semitic extremist, here is how Steve described their encounter in his Statistical Science interview:

I

had met my wife Joyce at the University of Toronto when we were both undergraduates. I was actually working in the fall of 1963 in the registrar’s office, and on the first day the office opened to enroll people, Joyce came through. And one of the benefits about working in the registrar’s office, besides earning some spending money, was meeting all these beautiful women students passing through. That first day I made a note to ask Joyce out on a date. The next day she came through again, this time bringing through another young woman who turned out to be the daughter of friends of her parents. And I thought this was a little suspicious, but auspicious in the sense that maybe I would succeed in getting a date when I asked her. And the next day, she came through again! This time with her cousin! Then I knew that this was really going to work out. And it did. We got engaged at the end of the summer of 1964 after I graduated, but we weren’t married when I went away to graduate school. In fact, yesterday I was talking to one of the students at the University of Connecticut who was a little concerned about graduate school; it was wearing her down, and I told her I almost left after the first semester because I wasn’t sure if I was going to make a go of it, in part because I was lonely. But I did survive, and Joyce came at the end of the first year; we got married right after classes ended, and we’ve been together ever since.

## Unusual timing shows how random mass murder can be (or even less)

Posted in Books, R, Statistics, Travel with tags birthday problem, coincidence, Conan Doyle, mass murders, Pittsburgh, Richard von Mises, The Sign of Four, Toronto, USA Today on November 29, 2013 by xi'an**T**his post follows the original one on the headline of the USA Today I read during my flight to Toronto last month. I remind you that the unusual pattern was about observing four U.S. mass murders happening within four days, “for the first time in at least seven years”. Which means that the difference between the four dates is at most 3, not 4!

**I** asked my friend Anirban Das Gupta from Purdue University are the exact value of this probability and the first thing he pointed out was that I used a different meaning of “within 4”. He then went into an elaborate calculation to find an upper bound on this probability, upper bound that was way above my Monte Carlo approximation and my rough calculation of last post. I rechecked my R code and found it was not achieving the right approximation since one date was within 3 days of three other days, at least… I thus rewrote the following R code

T=10^6 four=rep(0,T) for (t in 1:T){ day=sort(sample(1:365,30,rep=TRUE)) #30 random days day=c(day,day[day>363]-365) #account for toric difference tem=outer(day,day,"-") four[t]=(max(apply(((tem>-1)&(tem<4)),1,sum)>3)) } mean(four)

*[checked it was ok for two dates within 1 day, resulting in the birthday problem probability]* and found 0.070214, which is much larger than the earlier value and shows it takes an average 14 years for the “unlikely” event to happen! And the chances that it happens within seven years is 40%.

**A**nother coincidence relates to this evaluation, namely the fact that two elderly couples in France committed couple suicide within three days, last week. I however could not find the figures for the number of couple suicides per year. Maybe because it is extremely rare. Or undetected…

## Pittsburgh snapshot

Posted in pictures, Running, Travel, University life with tags Carnegie Mellon University, Pennsylvania, Pittsburgh, sunrise, University of Pittsburgh on November 8, 2013 by xi'an## Unusual timing shows how random mass murder can be (or not)

Posted in Books, R, Statistics, Travel with tags birthday problem, coincidence, Conan Doyle, mass murders, Pittsburgh, Richard von Mises, The Sign of Four, Toronto, USA Today on November 4, 2013 by xi'an**T**his was one headline in the USA Today I picked from the hotel lobby on my way to Pittsburgh airport and then Toronto this morning. The unusual pattern was about observing four U.S. mass murders happening within four days, “for the first time in at least seven years”. The article did not explain why this was unusual. And reported one mass murder expert’s opinion instead of a statistician’s…

**N**ow, there are about 30 mass murders in the U.S. each year (!), so the probability of finding at least four of those 30 events within 4 days of one another should be related to von Mises‘ birthday problem. For instance, Abramson and Moser derived in 1970 that the probability that at least two people (among n) have birthday within k days of one another (for an m days year) is

but I did not find an extension to the case of the four (to borrow from Conan Doyle!)… A quick approximation would be to turn the problem into a birthday problem with 364/4=91 days and count the probability that four share the same birthday

which is surprisingly large. So I checked with a R code in the plane:

T=10^5 four=rep(0,T) for (t in 1:T){ day=sample(1:365,30,rep=TRUE) four[t]=(max(apply((abs(outer(day,day,"-"))<4),1,sum))>4)} mean(four)

and found 0.0278, which means the above approximation is far from terrible! I think it may actually be “exact” in the sense that observing *exactly* four murders within four days of one another is given by this probability. The cases of five, six, &tc. murders are omitted but they are also highly negligible. And from this number, we can see that there is a 18% probability that the case of the four occurs within seven years. Not so unlikely, then.