Memories of some self-study episodes

Comments from JL stirred memories of my own experience with self-learning. I remember trying to learn how to model biological phenomenon by picking up John Maynard Smith’s Mathematical Ideas in Biology from the library many years ago. One of the problems that I had a really tough time understanding was the following problem in probabilistic thinking:

Of three prisoners, Matthew, Mark and Luke, two are to be executed, but Matthew does not know which. He therefore asks the jailer ‘Since either Mark or Luke are certainly going to be executed, you will give me no information about my own chances if you give me the name of one man, either Mark or Luke, who is going to be executed’. Accepting this argument, the jailer truthfully replied ‘Mark will be executed’. Thereupon, Matthew felt happier, because before the jailer replied his own chances of execution were 2/3, but afterwards there were only two people, himself and Luke, who could be the one not to be executed, and so his chance of execution is only 1/2. Is Matthew right to feel happier?

This problem is also known as the “Serbelloni problem” and according to John Maynard Smith, it “nearly wrecked a conference in theoretical biology in 1966”. It seems that there is nothing wrong with Matthew’s intuition – one could reason that given the information that Mark would be executed, only two possibilities remain: (Mark, Matthew) or (Mark, Luke) would be executed. Since Matthew is in one of the two possible outcomes, his chance of dying is surely 1/2.

After checking the hint at the back of the book, I was puzzled that that was not the case. Maynard Smith merely said that an application of Bayes’ Theorem or “common sense” should solve the problem. I had some ideas about Bayes’ Theorem at that time, so following the hint, one arrives at the solution of 2/3, which is the same as the probability of dying prior to receiving any information from the guard. To be precise, let I be the information given by the guard and H be the hypothesis that Matthew will die. What is required is the conditional probability P(H|I). According to Bayes’ Theorem, this can be related to P(I|H) as follows:

P(H|I) = P(I|H)P(H) / P(I)

Let’s consider P(I|H). If Matthew is known to die, then the outcomes must be either (Matthew, Mark) or (Matthew, Luke). Since the guard cannot explicitly say whether Matthew will die, he either says Mark or Luke will die, with probability 1/2. P(H) is just 2/3 since it is equal to P(Matthew, Mark) + P(Matthew, Luke), both of which occurs with probability 1/3.

How about P(I)? We could express P(I) as P(I,H)+P(I,H’), where H’ is the complement of H, and further rewrite it as P(I|H)P(H) + P(I|H’)P(H’). The first term is the same as the numerator. For the second term, P(H’) = 1/3, and P(I|H’) = 1/2, since if we know that Matthew does not die, then the only outcome is (Mark, Luke), and the guard reveals that Mark dies with probability 1/2. Putting everything together, we get

P(H|I) = (1/2)(2/3) / { (1/2)(2/3) + (1/2)(1/3)} = 2/3.

Bayes’ Theorem works but may lead to mechanical application without stimulating more intuition about conditional thinking. With more experience, P(H|I) could be computed as follows. Given information from the guard, only two outcomes are possible: (Mark, Matthew), (Mark, Luke). The total probability shrinks from 1 to 2/3. Now between these two outcomes, the guard can only reveal Mark or Luke. Consider the case of the guard revealing Mark. Since there are two Marks to one Luke, the guard reveals Mark with probability 2/3, after which, only Matthew or Luke could die, each with probability 1/2. Now consider the other case, where the guard reveals Luke. This occurs with probability 1/3, after which, only Mark or Matthew could die. But there are two Marks to one Matthew, so the probability of Matthew dying is 1/3. Putting these together we have {(2/3)(1/2) + (1/3)(1/3)} / (2/3) = 2/3.

Things may be clearer with the help of the figure below. Maybe this is the “common sense” that Maynard Smith talked about.

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Quotable Conversation

Gaston Gonnet (founder of the MAPLE computer algebra system) in an interview with Thomas Haigh in 2005 (SIAM History of Numerical Analysis and Scientific Computing Project; see here for the complete interview). I have fond memories of MAPLE as as a tremendously helpful tool for checking operations involving special functions, something that I dabbled in quite some time ago.

Now that I have worked several years in bioinformatics, the work in bioinformatics can be summarized as: you have to be good at algorithms, and you have to be very good at probability and statistics. You are not working with completely deterministic objects. You are not working with mathematical formulas that go only one, you are not working with problems which have a unique and precise answer. You are working with nature that has gone into a process of evolution in a relatively random way. This randomness percolates everything that you do because this randomness is not only in nature, but in all the data that you acquire. You acquire data, and the data is not exact. It’s subject to error because of the nature of the data or the nature of the acquisition of the data.

What I tell all my students and my grad students when they come is to make sure that their background in algorithms and their background in probability and statistics are really strong. If they have a good background in algorithms and statistics, quite a bit of scientific computation helps. It helps if someone knows how to integrate a system of differential equations or finding a minimum in an efficient way. Those kinds of basic scientific computation abilities are also very helpful. But if you are good at those two and possibly that third one, you are going to be good a bioinformatician. There is no two ways about it. But you have to understand algorithms and statistics, and that’s maybe the crucial point.

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The academic landscape after autonomy?

Brian Martin has this to say about the sociology of the academia in Australia after funding to universities was drastically reduced in Australia back in the 70s.

…Some positions become vacant through retirements and resignations. Many of these are not filled. But some positions are filled, and even some new ones created. The competition for these positions is now incredibly intense. Indeed, a sizable fraction of tenured academics would be very lucky to obtain their own positions should they be openly advertised. There are some tutors for example, struggling one year at a time to keep their positions, whose teaching load and research productivity shames tenured academics on twice the salary. Universities have never been meritocracies, but the squeeze has made the resemblance even more remote…

Social science research didn’t mean anything to me until I discovered Martin‘s writings. 🙂 It’s also good to know that he came from a science background (physics)!

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Obstacles to academic integrity

This is another gem from Brian Martin. It’s probably too late for mandatory reading for faculty members, but would be of immense value to young people just finishing their degrees and pondering that enormous step into the academia.

An excerpt:

“Most people subscribe to high principles, but living up to them is another matter. Practical realities mean continual compromises. Integrity is about aligning behaviour and principles. The challenge in maintaining integrity is to decide when to stand firm by principles and when to allow compromise or deviation – especially when principles clash. This is seldom easy.”

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Status and envy in the academia

The academia is not a shelter for people uninterested in worldly gains! Brian Martin gets it so right here in this article.

To quote an excerpt:

“…One way to get beyond envy and the status race is to focus on intrinsic satisfaction. This is possible in teaching, for example in seeing students improve their understanding and performance over a period of time. It is also possible in research, when the topic is chosen for its intellectual or social importance rather than its utility as a vehicle for personal advance…”

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Delightful little gems from PLoS Computational Biology

Journals are not generally the first place one looks for good reads* that deal with the cultural aspects of doing science, but thanks to the emergence of high quality open access** (OA) journals in recent years, the situation has changed dramatically. The Ten Simple Rules series from PLoS Computational Biology is just awesome – all graduate students irrespective of discipline should find the articles highly useful.

Let’s start with Ten Simple Rules for Building and Maintaining a Scientific Reputation by Bourne & Barbour. This article provides much needed orientation for new academics who are just getting to know the challenges waiting for them at the academic landscape. For new graduate students, the Ten Simple Rules for Graduate Students by Gu & Bourne gives a morale boost, and reminds them why they chose this path. I expect that standards of communicating scientific results for graduate students should improve after reading Ten Simple Rules for a Good Poster Presentation and Ten Simple Rules for Making Good Oral Presentations.

Finally, this journal also has a great section on bioinformatics education that discusses curriculum issues and provides tutorial style articles to new methods. Reading journals has never been this fun!

* The American Statistician also has columns that deal with general aspects of graduate student life and also tutorials, but the focus is on statistics majors. It requires subscription for easy access.

** The author pays for publication and then the article becomes free to the world. Depending on the journal, this could cost somewhere between USD 500 to USD 5000 (USD 2250 for PLoS Computational Biology). This contrasts with traditional publishing where the publisher retains copyright of the article and charges readers for access. There have been some controversies surrounding OA journals because of concerns that the quality of papers may degenerate because the publishers would be tempted to accept as many papers as possible. However, I think the BMC and PLoS publishers have mostly avoided this problem because of their success in convincing leading scholars of their respective fields to sit on the editorial board. Other OA publishers have floundered because they didn’t.

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Quotable Quotes 9

Gian-Carlo Rota (1932-1999), a mathematician who made landmark contributions in combinatorics, and also a great teacher:

The lack of real contact between mathematics and biology is either a tragedy, a scandal or a challenge; it is hard to decide which.

P/S: Gian-Carlo Rota said this a long time ago back in the eighties. Unfortunately, his quote is still relevant today. If one does not do something about it, then it is a tragedy, because modern biology has become so data-rich that lack of mathematical appreciation will severely limit the possibility of progress. It is also a scandal because we will keep producing biology grads who lack confidence and the technical know-how to make any meaningful contribution in a rapidly changing field.

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