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Friday, July 20, 2007

Question Everything

It's no secret that I have a particular fondness for statistics. What I love most about it is its counterintuitive nature. It's the one branch of mathematics where the "right" answer may completely defy common sense, so that we have to trust our rationality over our gut feelings. According to a 2004 talk by Dick De Veaux entitled Math is like Music, Statistics is like Literature, stat sets itself apart by being the only math field which requires students to be subversive.
We haven’t evolved to be statisticians. Our students who think statistics is an unnatural subject are right. This isn’t how humans think naturally. But it is how humans think rationally. And it is how scientists think. This is the way we must think if we are to make progress in understanding how the world works and, for that matter, how we ourselves work.

While I think De Veaux makes a great point, I think his love for alliteration may have led him to a less than accurate title. In the talk itself, math (other than statistics) is compared to systems like music and chess, in that they are built on a relatively simple set of rules that create apparent complexity through combination. Chess is the better example, I think. It's rules are concrete and universal. It is played the same way in Central Park as Tienanmen Square. The "rules" of music, as pirate Captain Barbosa might say, are more like guidelines. The rules vary from culture to culture, and even when we limit the discussion to Western music, many composers have gained fame through the years by breaking the rules.

Nevertheless, the crux of the argument is true. Statistics, like literature, requires practitioners to temper algorithms with life experience. The best and most simple example of this is the Monty Hall problem. When it originally appeared in Marilyn vos Savant's Parade article, it sparked a firestorm of criticism from mathematicians. These experts of numeracy had fallen victim to their instincts, and in doing so, provided a counter-counter example of the confusing nature of probability. Let me explain.

We all know people that genuinely believe in luck. These are the people that kiss the dice before rolling past Park Place. These people are experiencing a glitch in their pattern recognition software. But there are also people who have studied just enough stat to be dangerous. They are the ones who know that a random die is as likely to come up heads as tails, and yet persist in believing that the probability of rolling heads is increased after a long string of tails. Here we are seeing the same glitch in software, even in people trained in the rules of the game. Often times, the better you know the rules, the more apt you are to misinform yourself, as in the case of vos Savant's critics. These learned men (and women, but mostly men,) lambasted the article for misrepresenting probability, and they were quite rude about it. They claimed that each of the three doors had an equal 1/3 chance of hiding the prize, and that switching doors was a poor tactic. They were, of course, wrong, and Marilyn was vindicated. The naive application of a standard algorithm produced an erroneous answer.

What I have always found particularly amusing is that the Monty Hall system is extremely simple. There are only three doors, after all. It is quite easy to arrive at the correct answer by counting out the set of possible outcomes. But instead, mathematicians sent hate mail to a woman who had correctly applied experience and skepticism to a problem rather than merely rote formulas. It is this skepticism that makes statistics so interesting to me. It requires you to question everything. You have to know how the numbers can lie to you, and you will never get by with just memorizing algorithms. You have to think, then rethink your conclusions, and sometimes scrap the whole thing and start from scratch. It truly is like writing a great work of fiction, only instead of creating fantasy it uncovers reality.


steve said...

Nice blog.
We need more people to help erradicate the phobia of Maths!

Anonymous said...

why bash algorithms?
i didn't know the truth myself
until i worked it out using
a "tree diagram" ...

though in hindsight, it's easier
to see what's going on if we
complicate the situation
(by adding more "doors").

pick a card, any card.
don't look at it.
you win if it's the four of hearts.
i go through the rest of the deck
one card at a time and show you
fifty of 'em, face up: no H4.

now: do you want to trade me
for the unseen card in my hand
or keep the one you've got?

seems pretty obvious now, right?