I'm one season behind in "Only Murders in the Building" and only watched the scene pictured above last night showing Steve Martin with a fish called President William McKinley. Could it be a coincidence that the Fanta-faced phalangist is also catching up on the show, thereby motivating his new found sympathy for McKinley's tariff policy, ... and, and His Orange-ness being flushed down the toilet later in the same episode?
Among the lesser injustices, there is the failure of capitalism to produce the odd item that in my humble opinion would improve life in general, or mine in particular. I've written earlier about the Bialetti Elettrodomestici Macchina Per Pasta. It would be nice if someone brought chervil to my local farmers market. Such market failures don't measure up to the failure of the Soviet state to produce enough copies of the poetry of Osip Mandelstam to satisfy demand.
Which brings me to Penelope Unbound, a novel by Mary Morrissy, published last year by Banshee Press an independent Irish press. The premiss of the novel is simple: What if Norah Barnacle had grown tired of waiting for James Joyce at the train station in Trieste and had been taken home by a good samaritan? From this idea, Morrissy weaves a funny, but very poignant tale that deserves a much wider audience. In particular the novel deserves a U.S. publisher. John Banville wrote an ecstatic review in the Guardian, but this doesn't seem to have been appreciated in America. Why?
Every 20 years or so I am reminded to reread Thorstein Veblen's essay, The Intellectual Pre-Eminence of Jews in Modern Europe, Political Science Quarterly (1919). With the war in Gaza raging and my belated reading of Michael Chabon's novel Moonglow nearly finished, this seemed to be another such moment. The Veblen essay is brilliant not only because it turns all the ambient eugenicist anti-semitic tripe of its time on its head, but because it offers a warning about the consequences to Europe of the Zionist project. Were this project to succeed and Jews were to find a comfortable homeland that allowed them to retreat into their religious heritage and lose their engagement and skepticism about science and society in the rest of the world the loss of their contribution would be a tragedy. Of course this dream is far from realization and for most Zionists opting out of world of science and culture was never the objective. Perhaps now it is time for me to read yet another Veblen opus, his Yale Phd dissertation in philosophy titled, "Ethical Grounds of a Doctrine of Retribution". Unfortunately, to the best of my googling this item appears to be lost in the ozone, just when we need it most.
1. All speech in and around universities should be encrypted.
2. All houses within a one kilometer radius of the home of a
school shooter should be bulldozed.
3. The following ten great ideas of statistics should be canceled:
a. Correlation because Karl Pearson was a eugenicist.
b. Regression because Francis Galton was a eugenicist
c. Likelihood because R.A. Fisher was a eugenicist
d. Likelihood Principle because George Barnard was a communist
e. 0-1 Law because Andrey Kolmogorov was a communist
f. Markov Processes because Andrey Markov was an atheist
g. Gini Coefficient because Corrado Gini was a fascist
h. Exchangeability because Bruno de Finetti was a fascist
i. Causality because Plato was a fascist
j. Time because Martin Heidegger was a fascist
~
~
In a 2021 post Cauchy priors I made a gigantic blunder in misinterpreting a twitter response to a question posed by Victor Chernozhukov about the mean E[X|X+Y] when X is standard Gaussian, and Y is independent and standard Cauchy. I reformulated this as: suppose Y|T ~ N(T,1) and T ~ Cauchy, what is E(T|Y=y)? This is a standard Bayesian problem with the idea of Cauchy priors going back to Jeffreys and explored more recently by Berger and others. My blunder was elementary and involved failing to remember that the normalizing factor for the conditional density was dependent on y. When this was fixed, I get the figure below. To accentuate the flat portion of the posterior mean I've reduced the scale of the Cauchy to be 0.1 rather than 1. The interpretation of figure is quite intuitive: when y is near zero and therefore in agreement with the prior, the posterior mean is aggressively shrunken toward zero. However, when |y| is far from zero, the prior says, "well, that could happen" and the posterior eventually looks indistinguishable from y.
There is a mildly amusing story associated with how I came to revisit this problem. I have been reading a recent JPE paper A/B Testing with Fat Tails that employs Student t priors with low degrees of freedom in an essential way. Having totally forgotten about the previous blog post, I proceeded to investigate how to compute this posterior mean, and not surprisingly my initial attempts faltered a bit, so I started to google around to see what was "out there" in webland. Early on I found a nice paper by Guy Nason that dealt with the case of Student on 3 dfs. It mentioned that there was a 1939 David Kendall paper that treated the Cauchy case. This must have been written when Kendall was still a grad student. It involves some quite exotic complex analysis, and among others cites a 1935 paper by Robert Oppenheimer! If I interpret Nason correctly, the Kendall paper produces a "closed form" expression for the marginal density of a Cauchy mixture of Gaussians. Kendall comments rather drolly that the expression isn't useful for computations because there was no tabulated version of the erfc function for complex arguments. This lack has been rectified in the intervening years, but my attempts in R, and then in Mathematica to check that this Kendall's expression integrated to one failed. Instead, the integral seemed to diverge slowly. I would be grateful for any and all suggestions about this, but I rather expect that it is all lost in the mists of time.
Meanwhile, fortunately, it is easy to cook up a numerical version of the posterior mean solution that I will append here:
# Berger problem
s <- 0.1
f <- function(t,x) dnorm(x-t, sd = 1) * dt(t/s,1)/s
k <- function(x) integrate(f, -Inf,Inf, x = x)$value
g <-function(t,x) t * f(t,x)
h <- function(x) integrate(g, -Inf,Inf, x = x)$value
x <- -100:100/10
m <- x
for(i in 1:length(x)) m[i] <- h(x[i])/k(x[i])
png("Cauchy.png")
plot(x, m, type = "l", xlab = "y", ylab = expression(E~theta|Y==y))
abline(c(0,1),col = 2)
abline(h = 0,col = 2)
dev.off()
I joined Twitter in 2009 in the futile hope that it would lead me to a Korean Taco truck on my first ever visit to LA. It didn't occur to me to tweet until 2021 when I decided that it was time to launch a quixotic attempt to get Bialetti to revive their legendary pasta machine. This failed miserably too, although the Guardian food columnist Racheal Roddy was very nice about it.
Since then I've tweeted a few times always in response to something someone else had written. This led me to wonder why I couldn't bring myself to originate a tweet. The answer to this query appeared to me yesterday in the form of a talk delivered by Frank Ramsey in 1925 that appears as the Epilogue in the collection of Ramsey's papers edited by R.B. Braithwaite titled: "The Foundations of Mathematics"
I've been reading about the Rasch model of item response in educational testing, in preparation for writing a brief section about it for the empirical Bayes book. Eventually, I recalled that Edgeworth had an amusing paper about this sort of thing, from which I quote the final paragraph.
To examiners at least it will be interesting to test the accuracy of the instrument with which they work. The statistical study may beguile the monotony of their task. The " charm severe of numbers " is celebrated by Wordsworth as
"Especially perceived when nature droops
And feeling is suppressed."
The poet is evidently describing in prophetic words words the condition of examiners, and prescribing their solace. More tropically another inspired bard has indicated the paregoric use of an interest in statistics. In one of the beautiful pictures with which Homer has adorned the shield of Achilles, the ploughman of the good old times, as he finishes each furrow, and turns to begin a new one, is presented with a refreshing cup of honey-sweet wine. So they who plough in the modern metaphorical sense, may, in the pauses of their labours, be refreshed with the cup of statistical science, which I have endeavoured to sweeten.
F.Y. Edgeworth (1890) The Element of Chance in Competitive Examinations, JRSS, 664-663.
Peter Hull posted on twitter this fragment from a paper by Don Rubin that perfectly encapsulates the W.H. Auden maxim: Poetry makes nothing happen.
http://www.asasrms.org/Proceedings/y1975/Bayesian%20Inference%20for%20Causality%20-%20The%20Importance%20of%20Randomization.pdf
" Good sense is dead, its child, science killed it to find out how it was made."
From the novel, Innocence by Penelope Fitzgerald, the phrase is attributed to Antonio Gramsci..
Innocence is a truly brilliant novel, with a sensibility somewhere between Jane Austen and Henry James. It is strange that someone, preferably Paolo Sorrentino, hasn't made a movie of this novel.
Edgeworth's 1920 paper "The Element of Chance in Competitive Examinations" mocks excessive reliance on "reasoning with the aid of the gens d'arme's hat -- from which as from a conjuror's, so much can be extracted." In this spirit Bruce Hansen's recent paper, "A Modern Gauss-Markov Theorem" argues that the econometrics slogan "OLS is BLUE" can be modified to "OLS is BUE", that is that we need not restrict attention to linear estimators, OLS can be considered minimum variance unbiased in a suitable class of more general regression models.
Since I'm thanked in the acknowledgments, I thought it might be prudent to make explicit a few reservations I have about Bruce's version of the GMT. Here then is my unexpurgated original comment on an earlier draft of the paper.
Bruce,
I hope that you won’t mind an unsolicited comment on your recent Gauss-Markov paper. I was wandering around somewhat aimlessly yesterday looking for recent work on model averaging for a refereeing task, and it attracted my attention. (Spoiler alert: I’ve always hated the GM Thm since it seemed to restrict attention to such a small class of estimators that its optimality claim was nearly vacuous.)
There is of course the (Rao?) result that ols is MVUE in the Gaussian linear model, but you want to say something much stronger, that it is MVUE in a much bigger class of models, but then the qualifiers become critical. I think that I understand where you are coming from, but I wonder whether you might be misleading the youth of econometrics by the way that you develop the argument. Your “for all F in calF_2” is quite strong. Of course median regression can be much more efficient than mean regression in iid error linear models and both are unbiased when the errors are symmetric. When errors are iid and not symmetric then median regression is biased, but only the intercept is biased, the slope parameters are still potentially much more efficient than the mean regression estimates would be. Here, I don’t mean to suggest that there is anything special about median regression — a plethora of other estimators would serve as well. There is merit, I concede, in the idea that “if you want to estimate a mean you should use the sample mean, etc” — I’ve heard this from Lars several times, but on the other side of the argument there is the infamous Bahadur-Savage result that the mean is never identified, in the sense that slight perturbations of the tails of the population distribution can make it bounce around arbitrarily. Of course, this depends upon what “slight” might mean. Your “for all F…” condition and unbiasedness for any linear contrast gets us back very close to requiring linearity, it seems.
The paper is fine, I just think that it might need a surgeon general’s warning of some sort.
Best
Roger
PS. The photo, taken recently at the Museum of the History of Paris (Carnavalet), depicts a metal, Napoleanic era, hat of the type that Edgeworth presumably had in mind.
PPS (added March 12 2022) There are now two papers circulating one by Steve Portnoy and the other by Benedikt Potscher showing that the unbiasedness condition of Hansen admits only linear estimators, so my "very close" in the note above could be strengthened a bit. It also occurred to me after writing the original message that the 1757 proposal of Boscovich defines an estimator that can have superior (to OLS) asymptotic MSE performance and is asymptotically unbiased for iid error linear models. Details are given here.
Terrific talk by Jean Tirole last night on "The Common Good after Covid" sponsored by the IFS. It covered a lot of ground in an hour. Perhaps my favorite bit -- apropos the lamentable failure to appreciate the arrival of new scientific evidence by the general public -- was this quote from Swann's Way: "The facts do not penetrate the world where our beliefs live." If this seems too pithy for Proust you can google to find a more elaborated version:
In the museum of the history of Lyon there is a beautiful example of a Jacquard loom used in weaving silk at the beginning of the 19th century. Designs were implemented on punch cards as shown in the photo below and produced the flamboyant pattern in the next photo. One can't help but envy the results especially by comparison with the paltry spew of Phillips curve coefficients emanating from my early experiences with Fortran and Hollerith cards.
Joliet Prison pictured above, built in 1858, and closed in 2002 has recently reopened for public tours.
With n teams of equal ability and one team that can beat the others with probability p > 1/2 how many teams are needed to assure the best team wins a round robin tournament in which each team plays all other teams once? Some details here.
Voronograms are a primitive form of nonparametric regression intended to explore edge detection methods. Given scattered data on the plane, the idea is to fit piecewise constant functions defined on the Voronoi tessellation. I was motivated to revive an 2004 R package for this by some recent conversations about total variation smoothing with Ryan Tibshirani and one of his students. The package is linked here. The idea was an off-shoot of work with Ivan Mizera on triograms which were intended to fit quantile surfaces using a roughness penalty that corresponded to total variation of the gradient of the fitted function on the Delone triangulation of a sample of scattered points. Triograms are piecewise linear with breaks in the gradient along the edges of the triangulation. In contrast Voronograms are piecewise constant with breaks in the function itself across edges of the Voronoi tessellation. Examples appear in the figure below which can be reproduced by the command demo(tseg) in the R package. As usual there is a lambda parameter that controls the strength of the penalization.
My DIY project over the Covid Containment has been putting together this Brezhnev Fone:
Victor Chernozhukov posed the following question on twitter a few days ago: "Suppose that X~ N(0,1), and Y ~ Cauchy, X and Y are independent. What is E[X | X+Y]?"
Thien An replied "isn't it the mean of the density proportional to exp(-x^2/2)/[1+(z-x)^2]?" and she included this nice plot
This recalled some rather ancient suggestions by Jim Berger about the utility of Cauchy priors.Reformulating Victor's question slightly, suppose that X ~ N(t,1) and you have prior t ~ C, Thien's nice plot can be interpreted as showing the posterior mean of t as a function of X. For small values of |X| there is moderate shrinkage back to 0, and as |X| grows the posterior mean does too. However, for large values of X, this tendency is reversed and eventually very large |X| values are ignored entirely. This might seem odd, why is the data being ignored?
I like to think about this in terms of the comedian Richard Pryor's famous question: "Who are you going to believe, me, or your lying eyes*." If the observed X is far from the center of the prior distribution, you decide that it is just an aberration that is totally consistent with your prior belief that t could be quite extreme since the prior has such heavy tails, but you stick with your prior belief that it is much more likely that t is near 0.
In contrast if the prior were Gaussian, say, t ~ N(0,1), then the posterior mean is determined by linear shrinkage, so E(t|X) is midway between 0 and X, and for large |X|, we end up with a posterior mean in a place where neither the prior or likelihood have any mass.
* In this case "me" should be interpreted as your prior, and X as what you see with your lying eyes.
You may have wondered, as I have, about the convexifying effect of peeling a potato, so it comes as a great relief that this is a well studied subject. The opening paragraph of Goodman (1981) is priceless:
A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?
A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?
A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?
I've added two new vinaigrettes to the growing list at http://www.econ.uiuc.edu/~roger/research/vinaigrettes/vinaigrette.html
One is a survey of currently available methods for estimation and inference for quantile regression, the other is about computational methods for univariate quantiles. Both incorporate some new methods in my quantreg package. As usual there is an element of samo-kritika to these notes, since they try to reflect several drawbacks of the current state of affairs, not just the successes.
