The Craft of Probabilistic Modelling

Trendy phrases generate confusion. This is because everyone thinks it fun to use them, but not everyone uses them in the same way. “The art of modelling” is no exception. I shall try to demonstrate this by releasing a few skeletons from my own cupboard.

At City College in New York City, I took one course in mathematical statistics and three courses in what today we term noncalculus statistics. The former was given by the department of mathematics in which I was enrolled to fulfill the requirement for a major subject. Professor Selby Robinson was my instructor, as he was for generations of students. The noncalculus statistics courses were not listed in any specific department and, in fact, were labeled Unattached 15.1, 15.2, and 15.3. They were offered by Professor John Firestone, who also provided statistical instruction to countless students over the years. Among my contemporaries at City College, (1936–40) were a number of individuals such as Kenneth J. Arrow, Herman Chernoff and Milton Sobel who made their mark in statistics and allied subjects in later years.

There are a great many ways by which a scholar may come to choose his field of study. The story-book account where a man sees some great human need and devotes his life to providing for it is probably rarely true, and could hardly be true for a statistician. Who would believe a man who claimed to have been seized by a passionate desire to relieve human suffering through the provision of good methods of summarizing data? Some pure mathematicians believe that statisticians are failed mathematicians, and there is a grain of truth in this. However this is basically a superficial judgement, for science calls for different combinations of abilities and men naturally move to those fields best suited to their nature. Not an uncommon progression would be from the more abstract to the more concrete, because it is when we are young that we are most idealistic and only later that we begin to perceive how incredibly difficult it is to achieve anything worthwhile in a very theoretical way. To be a good statistician certainly requires some mathematical ability, but it also requires other qualities.

Probability, expressed as odds, dominated my childhood because my father and his brothers were inveterate (some professional) gamblers. Their father was driven off his farm in the western part of the state of Victoria, Australia, by a drought in the 1890s, and bought a hotel in the coastal town of Portland. So began some 50 years of Watson association with hotels and horses, surely an indication of their Ulster origins. Had they been content to be great horsemen (as indicated by the daring feats described to me, and some witnessed by me, as a child) all might have been well, but they had many racehorses which never seemed to win. My father thought of himself as a Scot (Scots did settle in Ulster) and despised the Irish. When young he won prizes for playing the bagpipes and for Highland dancing and later became quite infatuated with golf. Though he was Australia’s champion rifle shot for some years, he would have given away all his shooting medals for a low golfing handicap. He first wanted me to be a doctor, but changed this to dentist because he felt this would give me more assured time on the golf course, which would make up for the slight drop in social status. To ensure this rosy future, I was bribed to be at the top of my class. He hated to pay up when I managed to do this only by my unmanly skills in drawing and painting.

I have always been intrigued by numbers, symbols, formulae, and foreign alphabets. Perhaps because they seem to hold some promise of cabbalistic knowledge or insight not otherwise readily obtainable? Anyway, I took to elementary arithmetic early, and needed little encouragement to fall in with my mother’s idea that I should look rather foolish if, on going to school at the age of five, I did not know all the multiplication tables up to 12 × 12! After that, school seemed rather dull, though I do remember being fascinated by a boy whom I can still visualize and whose distinguishing characteristics were that his ears were always very green (due to lack of washing) and his “sum cards” involved division (which I could not do) with weird “÷” signs.

I was born in Leeuwarden, a provincial city located in a cattle-breeding and agricultural area in the north of the Netherlands. This is a rural part of the country with its own culture and language, a tough-minded rather stubborn and independent population, often well versed in the scriptures. Here my family had lived for about three centuries, making a living as small salesmen, their culture rooted in Jewish tradition. My father was a businessman, who had to solve week in, week out, the problem of earning a living. The records of his weekly activities were among my first experiences of problem-solving. At his workshop he had several mechanical devices, like jackscrews and tackles; when accompanying him on his business visits to machine mills, I noticed the driving wheels and axles which powered the machines. Such mechanical devices fascinated me, and with my extensive Meccano set I tried to imitate these mysterious constructions.

Mathematical modeling, like painting or photography, is an art, requiring proper balance between composition and the ability to convey a message. A good mathematical model, aiming to present an idealistic image of a real-life situation, should be accurate as well as selective in its description, and should use mathematical tools worthy of the problem.

I was born at Calicut, Kerala State, India and attended high school and (two-year) intermediate college there. I wanted to study mathematics, but in those days there was supposed to be no future for arts and science graduates, so I applied for admission to an engineering college. As it turned out, I failed to get this admission, and so I joined the Loyola College of Arts and Science, Madras, where I studied for a bachelor’s degree (with honours) in mathematics.

I was born on 21 August 1924 in Maglód, a little town 16 miles from Budapest. From a very early age I was fascinated by numbers. I must have been about four when a neighbor, Mrs Kéry, told my mother that she had sold her pig for 40 pengös. At that age I was more familiar with fillérs than pengös, and I asked my mother how many fillérs made up a pengö. She told me, 100. After thinking this over, I spoke up: “Mrs Kéry has 4000 fillérs.” From that time my reputation as a mathematician was established—at least in Maglód. Everyone who knew me wanted to try out my mathematical skills, and I was delighted to oblige.

When I was a small boy, I was fascinated by the flowers my father loved and raised, and imitating him I grew various plants in the garden at home. The main reason why I was attracted by the flowers was their beauty. In addition, I was fascinated by the mystery of development; I wondered how a beautiful tulip, for instance, could emerge from a mere bulb. My interest in plants continued through elementary school. Then, in the second year of middle school, I met an interesting teacher who was a devoted naturalist. He encouraged my interest in plants, and gradually I became absorbed in botany; as a result, I made up my mind to become a botanist.

I grew up in New York City in the Depression. Both my parents were immigrants. My father, Jonas I. Keilson, had arrived in his late teens from Lithuania and had built up a small business as middleman between families of modest means and large merchants in lower Manhattan selling suits, overcoats, and furniture. My father provided contacts, expertise, and credit, and his business flourished. My mother, Sarah Eimer, had come as a baby from Austria. I was the third of three boys. The first, Philip, died in infancy in an accident. The second, Sidney, became an accountant and businessman. I was born on 19 November 1924. A girl, Marcia, came some nine years later and is a clinical psychologist.

Modesty, together with an awareness of the quizzical reader, inclines me to write an article less autobiographical than has generously been invited. However, it is true, I realize, that the autobiographical form serves well as a thread upon which to string one’s thoughts, observations and prejudices.

What is an “applied probabilist” or a “probabilistic modeller”? How does one become one? There used to be a cartoon that hung on the walls of many engineering schools. On it was the face of Alfred E. Newman. Above the face were the words. “For years ago I couldn’t even spel ingineer”, and under it, “Now, I are one”. Despite the caricature there is a not-so-obvious message: one is hard pressed to explain whatever it is one is, and how one has become it. Often whatever one is, is not quite what one sees oneself as, and this in turn is not what others see.

It is a common observation that the intrinsic time of human experience is a highly nonlinear function of clock time. Anyone whose earliest efforts at scientific computation date back some 30 years or more must now feel separated from them by a very long span of time. The computational environment of the 1950s already belongs to a past that will be unimaginable to the generation for which the personal computer is a childhood toy. It may therefore be appropriate to start this essay with a few personal recollections typical of that era.

When I was 12 years old, my parents emigrated from England to New Zealand, and this, perhaps, was the single most important event in my early life. The more relaxed atmosphere in the New Zealand schools, and the opportunity this gave for reading and study outside the pressure of competition, led, I am sure, to a greater success than I could have achieved in the tenser, more status-conscious environment of England. It also seemed to pave the way for the travel opportunities which are one of the great privileges of being an academic. From that time on, without any tremendous effort on my part, opportunities for travel have arisen which have allowed me to remain based in New Zealand, while living and working for periods in Australia, Britain, Russia, India, Japan and elsewhere. These opportunities have come about not through being an explorer, a journalist, an interpreter even, but through being a mathematician, and that at a modest level. Whatever dreams of adventure I may have had as a child, I never thought that such opportunities could come, of all things, from a career in mathematics. Be a mathematician and see the world? It may sound an unusual slogan, but in fact there can be few disciplines, if any, where mutual understanding is so independent of race, politics, culture or religion, and (with a few exceptions, perhaps) recognition of important work is so freely acknowledged on an international basis. Be this as it may, from those early days onwards, while other interests have come and gone, mathematics and travel have remained the dominant concerns of my working like.

The first 20 years of my life were spent in the intensely religious and ritualistic atmosphere of a south Indian Brahmin family confined to the towns of Thanjavur and Madras and the paddy fields of the village of Thalanayar in Thanjavur district. Emphasis was laid at home on the philosophy and teachings of the Vaishnavite saints Ramanuja and Desika, the Hindu epics Ramayana and Mahabharata, the devotional songs of Tamil saints known as the Alwars and finally the Bhagavad Gita. To this day, I believe in their basic idea that personal and global peace are possible only through renunciation and action with detachment.

I was born on 12 August 1936 in Piteşti, then a small town of some 20,000 inhabitants situated on the river Argeş, about 120 km north-west of Bucharest. It has now become an important industrial centre with more than 150,000 inhabitants. During my childhood, Piteşti enjoyed the reputation of being a clean and attractive town with very good secondary schools modelled after the French lycées. My parents were teachers there, and as a child I thought I would also become a teacher.

Since the highlights of our professional life are what makes it interesting and rewarding, I have decided to describe in this account the events surrounding the derivation of the so-called “Ewens sampling formula” of population genetics, which has formed a high point in my career as an applied probabilist. Before doing this I will briefly describe my background and early training in population genetics theory.

When I was asked to contribute to this volume I was both flattered and depressed: flattered, to find myself in the company of many applied probabilists who were famous names in the field before I ever entered it; depressed, because I feel too young to be asked to write even this brief autobiographical note with all the intimations of a completed career that such an activity implies.