The Dream Must March On

“I am happy to join with you today in what will go down in history as the greatest demonstration for freedom in the history of our nation”. These were the opening words of what I think is a masterpiece of modern rhetoric. These words were spoken 48 years ago by Martin Luther King in his I have a Dream speech on the steps of the Lincoln Memorial.

King and the vast assembly African Americans had come the nation’s capital to cash a check, a check that would give “us upon demand the riches of freedom and the security of justice”. But in his oratory, replete with powerful imagery and delivered with unsurpassed oratory, “the check had come back marked insufficient funds”. But King was not deterred. And he proclaimed, “We refuse to believe that there are insufficient funds in the great vaults of opportunity of this nation”.

King had also come to the nation’s capital to remind America of the “fierce urgency of now”.

The Martin Luther King Jr. Memorial was to be dedicated on the National Mall on Sunday. But thanks to hurricane Irene, the ceremony was postponed.


But just like his life and what he stood for, the memory of King still draws fierce controversy.

Commenting on the King Memorial on the National Mall in the New York Times, Cornel West, philosopher and professor at Princeton noted that the “events constituted major milestones in the turbulent history of race and democracy.”

According to Andrew Young, one of King’s loyal lieutenants, former Mayor of Atlanta and former US Ambassador to the United Nations, “as America’s first black president, Barack Obama is doing the legacy of Martin Luther King proud. Cornel West disagrees. Cornel believes that “the age of Obama has fallen tragically short of fulfilling King’s prophetic legacy”.

Andrew Young believes “King would be most proud of the statue’s sheer size. King spent all his life as 5’ 7’’ so his ego would make him admire it”. That is vintage witty Andrew Young. But Cornel West thinks, “King weeps from his grave. He never confused substance with symbolism. He never conflated a flesh and blood sacrifice with a stone and mortar edifice.”

Cornel West reminds us that King’s dream of a more democratic America had become, in his words, “a nightmare,” owing to the persistence of “racism, poverty, militarism and materialism.”

According to Andrew Young, “We know you are enslaved when you are in a democracy without the right to vote but when you are in a free enterprise system, in capitalism, without equal access to capital, we still have problems. Equal access to capital means equal access also to education and employment opportunity, so we still have a long way to go on Martin Luther King’s dream.”

But here is what King would say today, just like he did 48 years ago on the steps of the Lincoln Memorial “ I say to you today, my friends, so even though we face the difficulties of today and tomorrow, I still have a dream. It is a dream deeply rooted in the American dream.”

And against all odds and tribulations, the dream must live on.

Fixing the Crisis in Education

Sol Garfunkel and David Mumford are excellent scholars and educators of mathematics. In a profoundly wise and engaging article published in the New York Times on August 24 2011 they discuss how to fix math education in the US.

In the article the authors decry the deplorable performance of US students in various international tests. The national response to remedy low student achievement was the enactment of the No Child Left Behind Law, which requires public school students to pass standardized math by 2014. The law punishes schools if students do not pass. President Obama has also been drumming up support for his signature education reform program, Race To The TOP.
The article I think inadvertently raises fundamental practical and philosophical questions about education. They observe that the worry of poor grades or under achievement in math by US students is founded on the assumption that there is such a thing as a universal body of mathematical knowledge and skills that students must demonstrate competent mastery of.

The authors note that math curriculum is highly abstract and largely unhelpful in preparing students for different careers. This I can say with confidence is true for curriculum in all disciplines. Education as delivered through the school curriculum is largely irrelevant to the world of work and the challenges of ordinary day-to-day living.

The authors ask for instance how often most adults encounter situations in which they must apply a quadratic equation to solve a real life problem. But they observe that all adults need to understand how mortgages are priced. As investing adults, we all need to understand how to read a company’s balance sheet.

At the core of these questions is one major concern or worry. How can we make education relevant to the learner? Or more fundamentally, what kind of education will be relevant for work and daily living?

Many employers rate that skills and knowledge of most high school or college graduates as fair or poor. And I have met and interviewed many.

There is in a sense a crisis in education. It is a crisis of relevance as well as substantive useful knowledge and skills necessary to strive in a globalized and complex economy.
The authors give a brilliant example of how we could make math relevant and fun to learn. “Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages”.

The article concludes “It is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now”. I rephrase this by saying, it is through the real-life application of knowledge that understanding, and discovery has flourished and connects to innovation.

We all have a real choice. But time is not on our side. We must make education relevant and prepare the next generation to take their rightful place in society.

Native Informants No More,

I have heard African scholars whinge about the asymmetry of research partnerships with colleagues from the universities in North America and Europe. For the most part researchers from developing countries do not have the resources, financial and infrastructure, necessary to undertake advanced multi-year science projects.

Invariably, universities in the North have the cash and infrastructure to dedicate to research. But often the field site for research happens to be in Africa. This is especially true for research in areas such as tropical ecology and biology. What tends to happen is that African scholars are invited by counterparts from the North as partners in a research project that is conceived and funded to serve not their interests or address priority research questions that are urgent or immediately relevant to a local problem.

African scholars in this arrangement tend to play the role of "native informants". For the most part they facilitate access to field sites, arrange introductions with government and local officials. So for practical reasons, African scholars merely serve a PR role. If they are lucky they may get a graduate student on the project and also get their name close to the end of the list of authors in a publication.

The reason for this bad situation is because public spending on research in Africa is infinitesimal or nonexistent. Hence invitation to "collaborate" in a project by researchers from the North is by far an irresistible proposition.

But this is to change.


The US National Science Foundation (NSF) and the Agency for International Development have opened a funding stream for scientists in the developing world. The Partnerships for Enhanced Engagement in Research (PEER) will enable collaborations with scientists who are funded by the NSF; the US National Academies will help to administer the initiative. Applicants need a letter of support from their US-based partners. The first request for proposals will be released in August, and the first round of funding will be awarded later this year. Six PEER pilot projects — focused on areas such as hydrology, biodiversity and seismology — are already being financed in Tanzania, Bangladesh and elsewhere.

--Nature 475, 415 (01 July 2011) doi:10.1038/nj7356-415b

Published online 20 July 2011

Essential Tools for Post Normal Science

We are in an era where the old dichotomy of knowledge and ignorance, and of facts and values is being transcended.

Science is no longer seen as the inevitable advance in certainty of our knowledge and control of the natural world. Science must now be judged by its capacity to anticipate uncertainty and emergence.

The new science must be based on assumptions of unpredictability, imperfect data and incomplete understanding. Uncertainty must not be banished but cultivated.

The new science must deploy an orientation to posing prior and posterior probabilities. The new science liberate fact to the power of new evidence.

The resurgence of Baye's Theorem offers an invaluable tool for Post Normal Science.

And if you are not thinking like a Bayesian, perhaps you should.

John Allen Paulos' review of Sharon McGrayne's book on Bayesian Theory is elegant and should cause us to be more hopeful about grappling in a world where reductionist analytic worldview is yielding to systemic, recursive and humanistic approaches.

Here is the review.

THE THEORY THAT WOULD NOT DIE

How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines and Emerged Triumphant From Two Centuries of Controversy

By Sharon Bertsch McGrayne

320 pp. Yale University Press. $27.50.

Book Review By JOHN ALLEN PAULOS
Published: August 07, 2011, New York Times

Sharon Bertsch McGrayne introduces Bayes's theorem in her new book with a remark by John Maynard Keynes: "When the facts change, I change my opinion. What do you do, sir?"

Bayes's theorem, named after the 18th-century Presbyterian minister Thomas Bayes, addresses this selfsame essential task: How should we modify our beliefs in the light of additional information? Do we cling to old assumptions long after they've become untenable, or abandon them too readily at the first whisper of doubt? Bayesian reasoning promises to bring our views gradually into line with reality and so has become an invaluable tool for scientists of all sorts and, indeed, for anyone who wants, putting it grandiloquently, to sync up with the universe. If you are not thinking like a Bayesian, perhaps you should be.

At its core, Bayes's theorem depends upon an ingenious turnabout: If you want to assess the strength of your hypothesis given the evidence, you must also assess the strength of the evidence given your hypothesis. In the face of uncertainty, a Bayesian asks three questions: How confident am I in the truth of my initial belief? On the assumption that my original belief is true, how confident am I that the new evidence is accurate? And whether or not my original belief is true, how confident am I that the new evidence is accurate? One proto-Bayesian, David Hume, underlined the importance of considering evidentiary probability properly when he questioned the authority of religious hearsay: one shouldn't trust the supposed evidence for a miracle, he argued, unless it would be even more miraculous if the report were untrue.

The theorem has a long and surprisingly convoluted history, and McGrayne chronicles it in detail. It was Bayes's friend Richard Price, an amateur mathematician, who developed Bayes's ideas and probably deserves the glory that would have resulted from a Bayes-Price theorem. After Price, however, Bayes's theorem lapsed into obscurity until the illustrious French mathematician Pierre Simon Laplace extended and applied it in clever, nontrivial ways in the early 19th century. Thereafter it went in and out of fashion, was applied in one field after another only to be later condemned for being vague, subjective or unscientific, and became a bone of contention between rival camps of mathematicians before enjoying a revival in recent years.

The theorem itself can be stated simply. Beginning with a provisional hypothesis about the world (there are, of course, no other kinds), we assign to it an initial probability called the prior probability or simply the prior. After actively collecting or happening upon some potentially relevant evidence, we use Bayes's theorem to recalculate the probability of the hypothesis in light of the new evidence. This revised probability is called the posterior probability or simply the posterior. Specifically Bayes's theorem states (trumpets sound here) that the posterior probability of a hypothesis is equal to the product of (a) the prior probability of the hypothesis and (b) the conditional probability of the evidence given the hypothesis, divided by (c) the probability of the new evidence.

Consider a concrete example. Assume that you're presented with three coins, two of them fair and the other a counterfeit that always lands heads. If you randomly pick one of the three coins, the probability that it's the counterfeit is 1 in 3. This is the prior probability of the hypothesis that the coin is counterfeit. Now after picking the coin, you flip it three times and observe that it lands heads each time. Seeing this new evidence that your chosen coin has landed heads three times in a row, you want to know the revised posterior probability that it is the counterfeit. The answer to this question, found using Bayes's theorem (calculation mercifully omitted), is 4 in 5. You thus revise your probability estimate of the coin's being counterfeit upward from 1 in 3 to 4 in 5.

A serious problem arises, however, when you apply Bayes's theorem to real life: it's often unclear what initial probability to assign to a hypothesis. Our intuitions are embedded in countless narratives and arguments, and so new evidence can be filtered and factored into the Bayes probability revision machine in many idiosyncratic and incommensurable ways. The question is how to assign prior probabilities and evaluate evidence in situations much more complicated than the tossing of coins, situations like global warming or autism. In the latter case, for example, some might have assigned a high prior probability to the hypothesis that the thimerosal in vaccines causes autism. But then came new evidence - studies showing that permanent removal of the compound from these vaccines did not lead to a decline in autism. The conditional probability of this evidence given the thimerosal hypothesis is tiny at best and thus a convincing reason to drastically lower the posterior probability of the hypothesis. Of course, people wedded to their priors can always try to rescue them from the evidence by introducing all sorts of dodges. Witness die-hard birthers and truthers, for example.

McGrayne devotes much of her book to Bayes's theorem's many remarkable contributions to history: she discusses how it was used to search for nuclear weapons, devise actuarial tables, demonstrate that a document seemingly incriminating Colonel Dreyfus was most likely a forgery, improve low-resolution computer images, judge the authorship of the disputed Federalist papers and determine the false positive rate of mammograms. She also tells the story of Alan Turing and others whose pivotal crypto-analytic work unscrambling German codes may have helped shorten World War II.

Statistics is an imperialist discipline that can be applied to almost any area of science or life, and this litany of applications is intended to be the unifying thread that sews the book into a coherent whole. It does so, but at the cost of giving it a list-like, formulaic feel. More successful are McGrayne's vivifying sketches of the statisticians who devoted themselves to Bayesian polemics and counterpolemics. As McGrayne amply shows, orthodox Bayesians have long been opposed, sometimes vehemently, by so-called frequentists, who have objected to their tolerance for subjectivity. The nub of the differences between them is that for Bayesians the prior can be a subjective expression of the degree of belief in a hypothesis, even one about a unique event or one that has as yet never occurred. For frequentists the prior must have a more objective foundation; ideally that is the relative frequency of events in repeatable, well-defined experiments. McGrayne's statisticians exhibit many differences, and she cites the quip that you can nevertheless always tell them apart by their posteriors, a good word on which to end.

John Allen Paulos, a professor of mathematics at Temple University, is the author of several books, including "Innumeracy" and, most recently, "Irreligion."
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