The Wolf of Wall Street

Statistics is a much maligned field. Its practitioners are seen as dull, gray cardboard cutouts. Its methods are tedious and complex. Its results can be stunning. 

I studied the topic in college, but unfortunately was enrapt of a young female during that time, and so had little ability to focus. Much of it went by me, and I felt fortunate to pull a C out of the courses. The young female? If my fellow classmates that I saw at my fiftieth class reunion in 2018 are any hint, she is probably married and as big around as a water tank. Something to do with aging and estrogen, I am told. 

In real life I saw the field applied almost magically with election results and a process called “exit polling.” Many people might note that the entire population of our country can be polled by using a sampling of 3,000 people, maybe even fewer. The quality of results depends on the selection of people to be polled. But when dealing with people who have just now voted, many variables are eliminated. People can lie in polling, over-enthused about their own importance. They might not even vote. Exit polling reduces the field of data to people who have actually voted, and by rigorous questioning, places them in various categories that resemble to electorate as a whole.  

The idea is to construct a microcosm of the larger population in the smaller one, a tiny mirror image. If done correctly, and if bias is eliminated in questioning, the sample can be deadly accurate. In fact, it was often said during the days of exit polls that the results were more accurate than the elections themselves, which can have many mechanical shortcomings. 

But statistics never yield concrete answers. They only state probability – that is, if a poll finds that Elmer Fudd is ahead of Bugs Bunny, it will give a range of probability, as in “Fudd leads Bunny by seven percentage points with a margin of error of one percentage point either way,” or that the lead is 6-8 points. The likelihood that the result will fall within that range is called a “confidence interval,”  and the level of confidence is stated as a standard deviation (don’t go there), usually with a professional poll in the area of 97% or so. So there is a 97% chance that on election day Fudd will beat Bunny by 6-8 points. But suppose Elmer leads by only 1/2 point with a 1 point margin or error. Then the result might fall between Bugs winning by a point or Elmer winning by 1.5%. The results are said to “cross zero”, so no winner is named in the poll. 

Notice that nothing is ever definite with statistics. But it was a rare thing prior to the 2000 election for exit polls to be wrong. What happened in 2000? Bush v Gore, and HAVA, or the Help America Vote Act, which introduced electronic voting throughout the country. Exit polls went south on us, and ceased to be reliable. In fact, in the subsequent years when exit polls were still done, they were massaged afterwards to adjust them to the “actual” vote count. Then they stopped doing them entirely because they were not “reliable.” 

These days I do not trust any election outcome. As our friend Miles Mathis noted recently, popularity polls concerning Joe Biden are fake, and his real level of popularity is probably less than 10%. But no matter, as votes are not counted, and elections are anybody’s guess (Last sentence there is me, not MM.) 

Beneath the fold is an excerpt from a book I read many years ago, The Metaphysical Club: The Story of Ideas in America, by Louis Menand (2001). I was still quite naive when I read it, and was totally taken by it, and still admire the work and its author. It is mostly about four men, Charles Sanders Pierce (pronounced “perz“), Oliver Wendell Holmes, William James and John Dewey. Yesterday I took time to dictate a section of the book about The Witch of Wall Street, Hetty Robinson. It’s a 3,000-word excerpt, I warn you, but for those who dive into it, I assure you it will hold your interest.  Keep in mind that Benjamin and Charles Pierce plied their work before statistics was a formal science taught in colleges, so that their techniques are nothing short of pure and original genius. (If you come upon typos, please let me know in the comments. I used Nuance Dragon to dictate, and then spent as much time fixing typos, but I am sure I missed some.)

[PS: Methodology can be confusing. What the Pierce’s have done is to quantify how unlikely it is that each signature in the (forged) will is exactly like the signature on the original. The extremely high number, one in five to the 30th power, is the result of multiplication of unlikelihoods. For instance, the odds of rolling a one with a die are one in six. The odds of rolling snake-eyes is 1/6 x 1/6, or 1/36. The odds of rolling snake-eyes twice in a row are 1 in 1,296. Etc.]

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Edward Robinson died on June 14, 1865, in New York City, leaving an estate worth almost $6 million; less than three weeks later, Sylvia Ann Howland died in New Bedford at the age of 59. Her estate was worth $2,145,029; she was said to have been, at the time of her death, the richest person in New Bedford. She had never married, she suffered from a spinal affliction, she was obliged to have herself carried around on a litter. Her death left a single heir-at-law to the entire Howland fortune: her niece (Edward Robinson’s daughter), Hetty Robinson.

Hetty Robinson was now 30 years old. She was a handsome woman, but she had, as it turned out, only one interest in life. By her father’s will, she had received $910,000 outright and the income from most of the remaining $5 million. Sylvia Ann Howland’s will, written in 1863, ordered that roughly half of her estate be given in bequests to various individuals and corporations and the rest, $1,132,929, be placed in trust, with the income to go to Hetty during her lifetime and the principal to be distributed to the lineal descendants of Sylvia Ann’s grandfather, Gideon Howland, after Hetty’s death. The will name Thomas Mandel, the former Howland partner, as executor.

Hetty Robinson was thus left with nearly one million of her own plus an income for life on some six million more. But she did not feel well used, and on December 2, 1865, she sued. She filed a bill of complaint in federal court (she was a resident of New York) against the executor and trustees of her aunt’s estate, claiming that, according to an earlier will, the entire estate, minus $100,000 in unspecified bequests, should have gone to her outright. Hetty produced a copy of this earlier will, which had been signed on January 11, 1862, by Sylvia Ann Howland and three witnesses; and she produced two copies of an additional page as well, signed only by Sylvia Ann Howland, revoking “all wills made by me before or after this one.” “I give this will to my niece,” it said, “to shew, if absolutely necessary to have it, to appear against another well found after my death.” This piece of paper became known during the trial as the “Second page.”

Hetty Robinson swore that she had personally drawn up the 1862 will, including the “Second page” with her aunt Anna taken the duplicate, also signed, back to New York City. She and her aunt had concealed the “Second page” from the three witnesses to the will proper, she explained, in order to keep a secret, which was that the will was one-half of the “mutual will.” Sylvia Ann Howland, Hetty told the court, and became estranged from her brother-in-law, Edward Robinson, and she wanted to make sure that none of the Howland fortune ended up in his hands. So, she made Hetty promise to write her own will, excluding her father from any money she might inherit; in exchange, she agreed to leave all her property, excluding $100,000 in “presents to my friends and relations,” to Hetty, with no strings attached. The two women agreed that neither would alter her own will without notifying the other (an understanding memorialized on the “Second page”); and since Hetty Robinson had not, she claimed, been notified of her aunt’s later will, dated 1863 – the will and probate – that will must be invalid. Hettie was therefore asking the court to award her aunt’s entire estate to herself. She would take care of the $100,000 in bequests, she said. She said she knew the beneficiaries her aunt had in mind.…

There were three signatures in evidence: the one on the 1862 will, which have been witnessed by three people in which both sides agree was genuine, and the signatures on the two copies of the “Second page,” which the estate charged were forgeries. Those signatures did look remarkably alike, and also remarkably like the signature on the will. They even appeared in the same place on the page – the same distance from the margins – as the witness signature. Had Hetty drawn up the “Second page” herself, without her aunt’s knowledge, and then traced her aunt’s signature?

No expense was spared (there was, after all, plenty of money available) to decide this question. Photographs of the three signatures were taken and blown up; photographs were taken of the signatures of other people and introduced into evidence for comparison. Chemists analyze the ink and engravers evaluated handwriting. Bankers and brokers were brought in from Boston and New York to testify based on their extensive experience with signatures. The presidents of commercial colleges offered their views. It was, after all, an era in which penmanship was still a professional skill. People made a living from handwriting….

… Hetty Robinson’s lawyers’ expert witnesses were Louis Agassiz and Oliver Wendell Holmes, Sr., men introduced to the court as persons skilled in the use of the microscope. Holmes testified that after inspecting the signatures in question under a microscope, he had found nothing to indicate that different inks had been used on the different documents, or that the signatures on the “Second page” had been traced. Agassiz … was a good deal more expansive; he treated the lawyers to a disposition on the microscopic interactions of ink and paper fiber which all the parties claim to have found fascinating. But he came to the same conclusion as his Harvard colleague: examined under a compound microscope, the fibers of the paper revealed no evidence of lead from a pencil (which might have been used to do the tracing), and the distribution of the ink did not suggest unusual movements of the hand when the signatures were written. There was no reason, he concluded, to believe that the signatures on the “Second page” had been forged.

… To offset the testimony of Holmes and Agassiz, the defendants needed their own giant of science from Harvard, and the man they selected was Benjamin Pierce [pronounced perz]. He brought [his son] Charles along with them. The Pierce’s testified on June 5 and 6th, 1867.

Charles was deposed first. He explained that his father had given him 44 samples of Sylvia Ann Howland’s signature (not including the ones in dispute), and had identified 30 separate “positions” for him to compare. These “positions” were places in the signature where the formation of a letter required a downward stroke of the pen. There were two such places on each letter S, two on each y one on each l, and so on. The forty-four signatures had been enlarged and printed on oil paper, and Charles’s assignment was to superimpose each signature on the other 43, one at a time, and count the number of down strokes that coincided. (A coincidence was counted when to down strokes were started at the same point on the letter in question in both signatures.) Benjamin had already determined that the down strokes in the disputed signature (the signature on the “Second page”) coincided with the down strokes in the genuine signature (the one on the will itself) at all 30 positions. What he was attempting to determine was the likelihood that the disputed signature was produced independently of the signature on the will proper – the likelihood that the degree of coincidence happened by chance.

Two of the reproductions were flawed, so Charles ended up comparing 42 signatures. It required 861 comparisons; but since 30 separate “positions” had to be compared in each case he had to tabulate the results of a total of 25,830 comparisons. (People who did this sort of work in the 19^th century were not called “computers” for nothing.) Charles found 5,325 cases of coincidence – 5,325/25,830 possible cases in which the start of a downstroke in one signature coincided with the start of the same downstroke on the same letter in another signature. In other words, one out of every five of Sylvia Ann Howland’s downstroke positions overlapped. To put it technically, the relative frequency of coincidence in the position of Sylvia Ann Howland’s downstrokes was one-fifth.  (The Pierce’s had hypothesized, of course, that each downstroke was an independent event – that is, that the existence of a coincidence on the first downstroke in two signatures does not affect the probability that the second down strokes will also coincide, and so on.) Charles telegraphed this information to his father, who was at the Coast Survey offices in Washington.

Benjamin Pierce was not supplied with two pieces of information: the total number of signature comparisons (861) and the relative frequency of coincidence of the downstroke’s (one-fifth). He proceeded to calculate the number of comparisons in which – if the coincidences were occurring by chance – just one of the 30 downstrokes should overlap, the number in which two should overlap, and so on, up to the number of cases in which all thirty could be expected to overlap. He ended up with the following table.

Number of cases of less than three or more than twelve expected coincidences: 41.

Off in Cambridge, Charles was making the same tabulation by actually counting the number of cases in which one downstroke, two down strokes, three down strokes, and so on coincided these were his results:

Cases of less than three and more than twelve actual coincidences: 35. (Charles counted 15 cases of two coincidences, and 20 cases of more than 12.) In other words, Charles’s results, produced by counting actual matches, approximated Benjamin’s predictions, produced mathematically remarkably closely.

When he was deposed on the day after his son, Benjamin Pierce was asked what conclusion he drew from these results. He had an impressive answer prepared. The chances that Sylvia Ann Howland could have produced two signatures in which all 30 downstroke’s coincided was, he said, “one in five to the 30^th power, or more exactly it is one in… two thousand six hundred and sixty-six millions of millions of millions of times, or 2,666,000,000,000,000,000,000.” Such a number, he advised the court,

“transcends human experience. So vast an improbability is practically an impossibility. Such evanescent shadows of probability cannot belong to actual life. They are measurably less than those least things which the law cares not for.

“The coincidence which is presented in this case cannot therefore be reasonably regarded as having occurred in the ordinary course of signing a name. Under a solid sense of responsibility involved in the assertion, I declare that the coincidence which has here occurred must have had its origin in an intention to produce it.… [I]t is utterly repugnant to sound reason to attribute this coincidence to any cause but design.

He hadn’t even bothered, he added, to factor in the likelihood of any two signatures being exactly the same distance from the margins of the paper, as was the case with all three of the signatures in question. If he had, he estimated that this would have increased the improbability of mere coincidence by at least a factor of 10, and probably by a factor of 100.

The lawyers for Hetty Robinson treated the whole demonstration as mathematical voodoo, and they had some fun during the oral arguments that fall, when the testimony was presented to the Circuit Court, ridiculing Benjamin Pierce’s portentousness. “[A] most extraordinary piece of evidence drawn from the shades of the Academy,” Hetty’s lead attorney, Sidney Bartlett, told the judges. “It is a pleasure to read it, may it please your Honors, to me, illustrating as it does the fervor and breadth with which science, only grant its postulates, can express itself. Given it its postulates, and nothing can be more beautiful to read, but it is the most baseless statement that ever came from a learned man.”

Many people who were not parties to the lawsuit had a similar reaction. There was, they felt, an air of the parlor trick about the Pierce’s performance – the father in Washington predicting with uncanny accuracy the figures the son in Cambridge would get when he tallied up the comparisons. In the Pierce’s fantastically infinitesimal number, one fifth to the 30^th power, representing the odds that the similarities in the disputed signatures had occurred by chance, seemed a hyperbolic flourish. People didn’t like the idea that something it was humanly possible to do could be declared inconceivable by statistics; it was an inverse case of the irritation people feel when they are told that their own behavior is statistically normal. It was as though some boundaries have been transgressed, and though Sylvia Ann Howland and somehow been denied the faculty of free will. “It is always a person’s intention to make the signature similar to others as nearly as possible every time,” complained a letter writer in The Nation a few weeks after the Pierces’ testimony was presented in court. “The elements of will and desire unfit it for judgment by such laws. Figures can be prostituted to prove almost anything, and were it not for Prof. Pierce’s high position, one might be led to think his evidence nothing more than a special plea. In the tone of his testimony is arrogant and positive, as if he were charging the judges.”

But of course, Benjamin Pierce did not believe that “elements of will and desire” made a thing unfit for mathematical reasoning, and the procedure he had followed with the signatures in the Howland will case was exactly the procedure he would articulate in Linear Associative Algebra three years later. He started with an idea: that hidden within the randomness of the collection of different Sylvia Ann Howland signatures was a certain kind of statistical order, an idea that might be tested by comparing pairs of signatures in a way he and his son devised. The empirical test of this idea yielded a hypothesis: that the relative frequency of coincidence in the downstroke since Sylvia Ann Howland’s signatures was one fifth. Pierce’s next step was to deduce the logical consequence of his hypothesis, which was that the number of coincidences would be distributed among the 861 signature comparisons in a particular way. He then compared the results he had arrived at by calculation (the distribution that ought to exist if one fifth is the frequency in which coincidences occur by chance) with this son’s observations (the actual distribution in the samples at hand), and the fit confirmed the hypothesis.

If Charles had already determined that the relative frequency of coincidence was 1/5, why did he and his father have to calculate the way the coincidences were distributed? Because if the actual distribution had differed significantly from the predicted distribution, then either the coincidences were not occurring by chance or the sample of 42 signatures was not a random sample. By establishing that the actual distribution matched the predicted one, Pierce’s had verified that one fifth was indeed the relative frequency with which coincidences appeared by chance. Final step was simply a matter of raising that fraction to the power represented by the total number of coincidences raised for all 30 positions to match – each pair of downstrokes having a one in five chance of coinciding – in order to arrive at a numerical expression of the probability that the thirty-for-thirty rate of coincidence in the disputed signatures had occurred by chance, or, to put it another way, the numerical expression of a how often Sylvia Ann Howland could be expected to duplicate her signatures in unintentionally but exactly: one in every 2,666,000,000,000,000,000,000 attempts.…

… In the end, all the expensive evidence about the signatures turned out to be irrelevant. In its verdict, handed down in 1868, the Circuit Court held that Hetty Robinson’s testimony on her own behalf during the trial had violated a federal statute prohibiting parties to a suit over a will from giving testimony unless called by the other side or commanded to testify by the court. Since Hetty was the only witness supporting the contention that her  and her aunt’s wills had been mutual, “the court is of the opinion that the contract is not proved. The estate won. The Howland case was decided on a technicality.  

The plaintiff had already skipped town. In 1867, while her case was still underway, Hetty Robinson had married Edward Green, a wealthy Vermont businessman. Seeing the shadow of a criminal fraud charge across her path if the signatures in dispute were proved to be forgeries, she took the precaution of moving to London, where she and her husband lived for eight years, and where they had two children. After their return, the Greens moved to New York City and the money that had come out of the New Bedford whaling industry was put to work on Wall Street. Hetty Green became a moneylender, a phenomenally successful one – eventually, a character in the popular imagination. As one might imagine, she proved to be a ruthless businesswoman and a great miser. She became known as the Witch of Wall Street, and when she died, in 1916, at the age of 82, her fortune was estimated to be between $100 and $200 million. The New York Times, in its obituary, called her the richest woman in the country.

The Metaphysical Club: The Story of Ideas in America

pp 165-176