“It gets easier further on.”
Chapter Nine
George Tiao and the Bayes Book
George Tiao received his MBA from New York University in 1958, and then he came to Wisconsin with the intention of getting a Ph.D. in international finance. Eventually he pursued econometrics and was one of the students in the first course I taught. He was very interested in Bayes, and while he was a student, he and I wrote an article about Bayesian methods for Biometrika.1 His thesis was on the robustness of linear models from a Bayesian viewpoint.2
When Bayes, your conclusions need not be based on the assumptions that the errors were normally and identically distributed and independent. You could for example explore:
It was clear from the beginning that George would make an important contribution to our growing department, so when he completed the Ph.D. in 1962, I offered him an assistant professorship with a joint appointment with the Business School. In 1965–1966, when I was invited to spend a year at the Harvard Business School, I arranged for him to come too, which provided an opportunity for us to work together on a book about Bayesian inference.3
George searched housing for our respective families, and he found two places that would do. One was the old, dark house with many rooms that I mentioned earlier, where we lived. George and his family stayed in a more reasonably priced home that was not far away. We set up an office in one of the bedrooms where we could lay out our papers, books, and scribbled notes. By the end of the year, the book was well on its way.
George Tiao's logical mind is illustrated in the following story. While in Cambridge, we discovered that the Harvard campus has strict rules about parking. Once, George and I had to bring a large and heavy tape recorder from our car to the business school, so we tried to park as close as we could. We found the perfect spot, but it was illegal to park there. A sign warned that the first-time offender would receive an orange ticket, the second-time offender would get a red ticket, and a third offense revoked all parking privileges any where on the Harvard campus. When I told George that we couldn't park there, he said, “Yes we can—twice.”
We returned to Madison with plans to work on the book whenever possible, but the pace of academic life made it difficult to have sustained periods of collaboration. In 1968, however, George and I wrote a paper on Bayes and outliers that generated a good deal of interest.4 Finally, we got our chance to finish the book in 1970–1971, while we were both on leave at the University of Essex.
In 1963, a new university was started at Essex, in southern England, and in 1966 George Barnard left Imperial College to become chair of the mathematics department there. In 1970–1971, he invited me to come to Essex for the academic year. At the time I had three Ph.D. students (Larry Haugh, Hiro Kanemasu, and John MacGregor) and I was partway through the Bayes research with George. George Tiao decided to go to Essex as well, and he had two Ph.D. students (William Cleveland and David Pack). George warmly welcomed the large entourage that went in the end: George Tiao and I went with our respective families, the five graduate students, and some of their families as well. Bill Cleveland's wife gave birth to their first child there, and Larry Haugh went with his wife, Jane, and one-year-old daughter. Larry recalls that the Americans in our group sometimes had to “rough it” in England: “Jane and I lived in a small village, Wivenhoe, which was just an English country path (across farm fields) … away from the Essex campus. … Living in the English village was quite the experience, …wearing sweaters so the thermostat could be lowered, living thru [sic] the postal strike and various rail strikes.”5 None of us knew that six years later, Jimmy Carter would be urging Americans to keep their thermostats down.
When we arrived in Essex, at George Barnard's suggestion, we met each week for a seminar studying Fisher's earliest papers. Each of us studied one of the articles and then presented it to the others. When Fisher first went to Rothamsted in 1919, he had been asked to review the data from the long-term Broadbalk soil fertility experiment, which had begun in the mid-nineteenth century and continues to the present day. A large area of land planted in wheat had been divided into long strips, each of which had been treated with the same nutrients (N, P, K, etc.) since 1843. Differences in the resulting wheat crops were plain to see, but it had been Fisher's task to determine whether further information could be gleaned from such experiments. Fisher's response to this and many other problems can be found in a series of papers called “Studies in Crop Variation.”
What we learned from these papers exceeded all expectations. There were many things we had not associated with Fisher and many that we believed were of a much later date. For example, there was an analysis of residuals (including how they are autocorrelated), the use of distributed lag models, and the multivariate distribution of regression coefficients. Fisher later referred to these studies as “raking over the muck heap,” for these investigations made him realize that often desired information was just not available from data such as was supplied by the Broadbalk trials. This led to his inventing the design of experiments6 that made it possible to study efficiently and simultaneously those specific effects that were of interest to the experimenter. The design of experiments later became a research area of great interest to me and my students.
Winter weather in England can be cold and dispiriting so it was not a surprise when two of the students, Hiro Kanemasu and David Pack asked if there was anywhere they might go over Christmas where it was warm. I suggested Spain, and they went there, but before they were due to come back, they ran out of money. We sent funds via American Express, but because of some complicated snafu, the money didn't arrive. We sent some more, but they never received that either. They later told us that they got so hungry toward the end that they got into trouble sitting in restaurants and eating the sugar.
Hiro had not finished his thesis when he went to work for the World Bank, so periodically he sent me new sections handwritten on yellow sheets from Washington. I got one batch that was covered with tire marks extending in all directions over the pages. He tended to be forgetful, and one day he had left a large chunk of his thesis on top of his car. The pages had blown all over Pennsylvania Avenue, and Hiro had spent a large part of the morning running in and out of the traffic recovering them page by page. Although at that stage his thesis was hard to read, it was good stuff. Titled Topics in Model Building, it was successfully completed in 1973 and Hiro received his Ph.D.
Smog had plagued Los Angeles since the early 1940s. Scientists had been measuring the concentrations of a number of pollutants for many years, so they had a tremendous amount of data. Beginning in 1973, George Tiao and I worked closely with them and particularly with an excellent chemist, Walter J. Hamming. By the time we met him, Hamming was the Chief Air Pollution Analyst for the Los Angeles County Air Pollution Control District. To deal with pollution, the state of California had introduced various laws, for example, outlawing open burning in incinerators and controlling certain industrial emissions. Our friend Hamming, however, was certain that almost all the pollution was what came out of the back end of an automobile, although at that time nobody wanted to believe it.
In 1966, California adopted the nation's first emission standards for hydrocarbons and carbon monoxide. George and I developed intervention analysis initially to look for a significant change in the pollutants at the time of the enactment of this law, and we found it. It became indisputably clear that Hamming was right.
The theoretical aspects of this kind of “intervention analysis” were discussed in a paper that George Tiao and I had published.7 This concerned the general problem of estimating a change in level that might have occurred at a known point in a nonstationary time series. For the Los Angeles data the change in level may be estimated by a linear combination of the data for which the weight function consisted of two exponentials back to back. The first had positive weights, and the second, had negative weights. This was very sensible because what happened just before and just after the event got the largest weights and were obviously of most importance.8
George taught at Madison from 1962 to 1982, when he went to the University of Chicago. Subsequently he has spent a great deal of time working in Taiwan, where he is held in very high regard. Through the years, our friendship has remained strong. When I turned 80 in 1999, George organized and hosted a wonderful birthday party at the Fortnightly restaurant in Chicago. Old friends and colleagues came from afar. It seemed only appropriate to sing them “There's No Theorem Like Bayes' Theorem.”
1 G.E.P. Box and G.C. Tiao, ‘A Further Look at Robustness Via Bayes’ Theorem. Biometrika, Vol. 49, 1961, pp. 419–432.
2 G. C. Tiao, Bayesian Assessment of Statistical Assumptions, Ph.D. Economics, University of Wisconsin, 1962.
3 Bayesian Inference in Statistical Analysis, John Wiley and Sons, New York, 1973.
4 G.E.P. Box and G.C. Tiao, “A Bayesian approach to Some Outlier Problems,” Biometrika, Vol. 55 (1), 1968, pp. 119–129.
5 Personal communication from Larry Haugh, March 30, 2012.
6 R.A. Fisher, “The Arrangement of Field Experiments,” The Journal of the Ministry of Agriculture, Vol. 33, 1926, pp. 503–513 was the first article to discuss the design of experiments.
7 G.E.P. Box and G.C. Tiao, “Intervention Analysis with Applications to Economic and Environmental Problems,” Journal of the American Statistical Association, Vol. 70, No. 349, 1975, pp. 70–79.
8 G.C. Tiao, G.E.P. Box, and W.J. Hamming, “A Statistical Analysis of the Los Angeles Ambient Carbon Monoxide Data 1955–1972,” Journal of the Air Pollution Control Association, Vol. 25, No. 11, Nov. 1975, pp. 1129–1136.