When I wrote last weekend’s piece about how interesting a basketball tournament was, I let some terms slide without definition, mostly so I could explain what ideas I wanted to use and how they should relate. My love, for example, read the article and looked up and asked what exactly I meant by “interesting”, in the attempt to measure how interesting a set of games might be, even if the reasoning that brought me to a 63-game tournament having an interest level of 63 seemed to satisfy.
When I spoke about something being interesting, what I had meant was that it’s something whose outcome I would like to know. In mathematical terms this “something whose outcome I would like to know” is often termed an `experiment’ to be performed or, even better, a `message’ that presumably I wil receive; and the outcome is the “information” of that experiment or message. And information is, in this context, something you do not know but would like to.
So the information content of a foregone conclusion is low, or at least very low, because you already know what the result is going to be, or are pretty close to knowing. The information content of something you can’t predict is high, because you would like to know it but there’s no (accurately) guessing what it might be.
This seems like a straightforward idea of what information should mean, and it’s a very fruitful one; the field of “information theory” and a great deal of modern communication theory is based on them. This doesn’t mean there aren’t some curious philosophical implications, though; for example, technically speaking, this seems to imply that anything you already know is by definition not information, and therefore learning something destroys the information it had. This seems impish, at least. Claude Shannon, who’s largely responsible for information theory as we now know it, was renowned for jokes; I recall a Time Life science-series book mentioning how he had built a complex-looking contraption which, turned on, would churn to life, make a hand poke out of its innards, and turn itself off, which makes me smile to imagine. Still, this definition of information is a useful one, so maybe I’m imagining a prank where there’s not one intended.
And something I hadn’t brought up, but which was hanging awkwardly loose, last time was: granted that the outcome of a single game might have an interest level, or an information content, of 1 unit, what’s the unit? If we have units of mass and length and temperature and spiciness of chili sauce, don’t we have a unit of how informative something is?
We have. If we measure how interesting something is — how much information there is in its result — using base-two logarithms the way we did last time, then the unit of information is a bit. That is the same bit that somehow goes into bytes, which go on your computer into kilobytes and megabytes and gigabytes, and onto your hard drive or USB stick as somehow slightly fewer gigabytes than the label on the box says. A bit is, in this sense, the amount of information it takes to distinguish between two equally likely outcomes. Whether that’s a piece of information in a computer’s memory, where a 0 or a 1 is a priori equally likely, or whether it’s the outcome of a basketball game between two evenly matched teams, it’s the same quantity of information to have.
So a March Madness-style tournament has an information content of 63 bits, if all you’re interested in is which teams win. You could communicate the outcome of the whole string of matches by indicating whether the “home” team wins or loses for each of the 63 distinct games. You could do it with 63 flashes of light, or a string of dots and dashes on a telegraph, or checked boxes on a largely empty piece of graphing paper, coins arranged tails-up or heads-up, or chunks of memory on a USB stick. We’re quantifying how much of the message is independent of the medium.
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