A Technical Explanation of Technical Explanation

A Technical Explanation of Technical Explanation

Ð'©2005 by Eliezer Yudkowsky.

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This essay is meant for a reader who has attained a firm grasp of Bayes' Theorem. An introduction to Bayes' Theorem may be found at An Intuitive Explanation of Bayesian Reasoning. You should easily recognize, and intuitively understand, the concepts "prior probability", "posterior probability", "likelihood ratio", and "odds ratio". This essay is intended as a sequel to the Intuitive Explanation, but you might skip that introduction if you are already thoroughly Bayesian. Where the Intuitive Explanation focused on providing a firm grasp of Bayesian basics, the Technical Explanation builds, on a Bayesian foundation, theses about human rationality and philosophy of science.

The Intuitive Explanation of Bayesian Reasoning promised that mastery of addition, multiplication, and division would be sufficient background, with no subtraction required. To this the Technical Explanation of Technical Explanation adds logarithms. The math is simple, but necessary, and it appears first in the order of exposition. Some pictures may not be drawn with words alone.

As Jaynes (1996) emphasizes, the theorems of Bayesian probability theory are just that, mathematical theorems which follow inevitably from Bayesian axioms. One might naively think that there would be no controversy about mathematical theorems. But when do the theorems apply? How do we use the theorems in real-world problems? The Intuitive Explanation tries to avoid controversy, but the Technical Explanation willfully walks into the whirling helicopter blades. Bluntly, the reasoning in the Technical Explanation does not represent the unanimous consensus of Earth's entire planetary community of Bayesian researchers. At least, not yet.

The Technical Explanation of Technical Explanation is so named because it begins with this question:

What is the difference between a technical understanding and a verbal understanding?

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A fable:

Once upon a time, there was a teacher who cared for a group of physics students. One day she called them into her class, and showed them a wide, square plate of metal, next to a hot radiator. The students each put their hand on the plate, and found the side next to the radiator cool, and the distant side warm. And the teacher said, write down your guess why this happens. Some students guessed convection of air currents, and others guessed strange patterns of metals in the plate, and not one put down 'This seems to me impossible', and the answer was that before the students entered the room, the teacher turned the plate around.

(Taken from Verhagen 2001.)

There are many morals to this fable, and I have told it with different morals in different contexts. I usually take the moral that your strength as a rationalist is measured by your ability to be more confused by fiction than by reality. If you are equally good at explaining any story, you have zero knowledge. Occasionally I have heard a story that sounds confusing, and reflexively suppressed my feeling of confusion and accepted the story, and then later learned that the original story was untrue. Each time this happens to me, I vow anew to focus consciously on my fleeting feelings of bewilderment.

But in this case, the moral is that the apocryphal students failed to understand what constituted a scientific explanation. If the students measured the heat of the plate at different points and different times, they would soon see a pattern in the numbers. If the students knew the diffusion equation for heat, they might calculate that the plate equilibrated with the radiator and environment two minutes and fifteen seconds ago, turned around, and now approaches equilibrium again. Instead the students wrote down words on paper, and thought they were doing physics. I should rather compare it to the random guessing of Greek philosophers, such as Heraclitus who said "All is Fire", and fancied it his theory of everything.

As a child I read books of popular physics, and fancied myself knowledgeable; I knew sound was waves of air, light was waves of electromagnetism, matter was waves of complex probability amplitudes. When I grew up I read the Feynman Lectures on Physics, and discovered a gem called 'the wave equation'. I thought about that equation, on and off for three days, until I saw to my satisfaction it was dumbfoundingly simple. And when I understood, I realized that during all the time I had believed the honest assurance of physicists that sound and light and matter were waves, I had not the vaguest idea what 'wave' meant to a physicist.

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So that is the difference between a technical understanding and a verbal understanding.

Do you believe that? If so, you should have applied the knowledge, and said: "But why didn't you give a technical explanation instead of a verbal explanation?"

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In "An Intuitive Explanation of Bayesian Reasoning" I tried to provide visual and physical metaphors for Bayesian probability; for example, evidence is a weight, a pressure upon belief, that slides prior probabilities to posterior probabilities.

Now we add a new metaphor, which is also the mathematical terminology: Visualize probability density or probability mass - probability as a lump of clay that you must distribute over possible outcomes.

Let's say there's a little light that can flash red, blue, or green each time you press a button. The light flashes one and only one color on each press of the button; the possibilities are mutually exclusive. You're trying to predict the color of the next flash. On each try, you have a weight of clay, the probability mass, that you have to distribute over the possibilities red, green, and blue. You might put a fourth of your clay on the "green" possibility, a fourth of your clay on the "blue" possibility, and half your clay on