I had been talking about how much information there is in the outcome of basketball games, or tournaments, or the like. I wanted to fill in at least one technical term, to match some of the others I’d given.
In this information-theory context, an experiment is just anything that could have different outcomes. A team can win or can lose or can tie in a game; that makes the game an experiment. The outcomes are the team wins, or loses, or ties. A team can get a particular score in the game; that makes that game a different experiment. The possible outcomes are the team scores zero points, or one point, or two points, or so on up to whatever the greatest possible score is.
If you know the probability p of each of the different outcomes, and since this is a mathematics thing we suppose that you do, then we have what I was calling the information content of the outcome of the experiment. That’s a number, measured in bits, and given by the formula
The sigma summation symbol means to evaluate the expression to the right of it for every value of some index j. The pj means the probability of outcome number j. And the logarithm may be that of any base, although if we use base two then we have an information content measured in bits. Those are the same bits as are in the bytes that make up the megabytes and gigabytes in your computer. You can see this number as an estimate of how many well-chosen yes-or-no questions you’d have to ask to pick the actual result out of all the possible ones.
I’d called this the information content of the experiment’s outcome. That’s an idiosyncratic term, chosen because I wanted to hide what it’s normally called. The normal name for this is the “entropy”.
To be more precise, it’s known as the “Shannon entropy”, after Claude Shannon, pioneer of the modern theory of information. However, the equation defining it looks the same as one that defines the entropy of statistical mechanics, that thing everyone knows is always increasing and somehow connected with stuff breaking down. Well, almost the same. The statistical mechanics one multiplies the sum by a constant number called the Boltzmann constant, after Ludwig Boltzmann, who did so much to put statistical mechanics in its present and very useful form. We aren’t thrown by that. The statistical mechanics entropy describes energy that is in a system but that can’t be used. It’s almost background noise, present but nothing of interest.
Is this Shannon entropy the same entropy as in statistical mechanics? This gets into some abstract grounds. If two things are described by the same formula, are they the same kind of thing? Maybe they are, although it’s hard to see what kind of thing might be shared by “how interesting the score of a basketball game is” and “how much unavailable energy there is in an engine”.
The legend has it that when Shannon was working out his information theory he needed a name for this quantity. John von Neumann, the mathematician and pioneer of computer science, suggested, “You should call it entropy. In the first place, a mathematical development very much like yours already exists in Boltzmann’s statistical mechanics, and in the second place, no one understands entropy very well, so in any discussion you will be in a position of advantage.” There are variations of the quote, but they have the same structure and punch line. The anecdote appears to trace back to an April 1961 seminar at MIT given by one Myron Tribus, who claimed to have heard the story from Shannon. I am not sure whether it is literally true, but it does express a feeling about how people understand entropy that is true.
Well, these entropies have the same form. And they’re given the same name, give or take a modifier of “Shannon” or “statistical” or some other qualifier. They’re even often given the same symbol; normally a capital S or maybe an H is used as the quantity of entropy. (H tends to be more common for the Shannon entropy, but your equation would be understood either way.)
I’m not comfortable saying they’re the same thing, though. After all, we use the same formula to calculate a batting average and to work out the average time of a commute. But we don’t think those are the same thing, at least not more generally than “they’re both averages”. These entropies measure different kinds of things. They have different units that just can’t be sensibly converted from one to another. And the statistical mechanics entropy has many definitions that not just don’t have parallels for information, but wouldn’t even make sense for information. I would call these entropies siblings, with strikingly similar profiles, but not more than that.
But let me point out something about the Shannon entropy. It is low when an outcome is predictable. If the outcome is unpredictable, presumably knowing the outcome will be interesting, because there is no guessing what it might be. This is where the entropy is maximized. But an absolutely random outcome also has a high entropy. And that’s boring. There’s no reason for the outcome to be one option instead of another. Somehow, as looked at by the measure of entropy, the most interesting of outcomes and the most meaningless of outcomes blur together. There is something wondrous and strange in that.
18 thoughts on “A Little More Talk About What We Talk About When We Talk About How Interesting What We Talk About Is”
Clever title to go with an interesting post, Joseph :)
Thank you. I hope you found it interesting.
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There is so much entropy in my life that I just didn’t know there were two different kinds.
It’s worse than that: there’s many kinds of entropy out there. There’s even a kind of entropy that describes how large black holes are.
Shannon Entropy is so interesting … The last paragraph of your post is eloquent… Thanks for teaching us about the The sigma summation in which the pj means the probability of outcome number j.
Best wishes to you. Aquileana :star:
Thank you; I’m glad you enjoyed.
I always enjoy trying to follow along with your math posts, and throwing some mathmatician anecdotes in there seasons it to perfection.
Thank you. I’m fortunate with mathematician anecdotes that so many of them have this charming off-kilter logic. They almost naturally have the structure of a simple vaudeville joke.
I totally agree on your way of introducing the entropy ‘siblings’. Actually, I had once wondered why you call the ‘information entropy’ ‘entropy’ just because of similar mathematical definitions.
Again Feynman comes to my mind: In his physics lectures he said that very rarely did work in engineering contribute to theoretical foundations in science: One time Carnot did it – describing his ideal cycle and introducing thermodynamical entropy – and the other thing Feynman mentioned was Shannon’s information theory.
It’s curious to me how this p-times-log-p form turns up in things that don’t seem related. I do wonder if there’s a common phenomenon we need to understand that we haven’t quite pinned down yet and that makes for a logical unification of the different kinds of entropy.
I hadn’t noticed that Feynman quote before, but he’s surely right about Carnot and Shannon. They did much to give clear central models and definitions to fields that were forming, and put out problems so compelling that they shaped the fields.
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Omg the TITLE of this! Lol :D I’m getting motion sickness as I speak.
Yeah, I was a little afraid of that. But it’s just so wonderful to say. And more fun to diagram.
I hope the text came out all right.