Information Is Surprise (2015)

Just to be clear. I don't disagree. Albeit information theory is a grand name, "communication" theory would not strike at the general applicability.

Yes, it is from Hamming's "The Art of Doing Science and Engineering".

From page 89:

> Information Theory was created by C.E.Shannon in the late 1940s. The management of Bell Telephone Labs wanted him to call it “'Communication Theory” as that is a far more accurate name, but for obvious publicity reasons “Information Theory” has a much greater impact—this Shannon chose and so it is known to this day. The title suggests the theory deals with information—and therefore it must be important since we are entering more and more deeply into the information age.

Then later on on page 90:

> First, we have not defined “information”, we merely gave a formula for measuring the amount. Second, the measure depends on surprise, and while it does match, to a reasonable degree, the situation with machines, say the telephone system, radio, television, computers, and such, it simply does not represent the normal human attitude towards information. Third, it is a relative measure, it depends on the state of your knowledge. If you are looking at a stream of “random numbers” from a random source then you think each number comes as a surprise, but if you know the formula for computing the “random numbers” then the next number contains no surprise at all, hence contains no information! Thus, while the definition Shannon made for information is appropriate in many respects for machines, it does not seem to fit the human use of the word. This is the reason it should have been called “Communication Theory”, and not “Information Theory”. It is too late to undo the definition (which produced so much of its initial popularity, and still makes people think it handles “information”) so we have to live with it, but you should clearly realize how much it distorts the common view of information and deals with something else, which Shannon took to be surprise.

> This is a point which needs to be examined whenever any definition is offered. How far does the proposed definition, for example Shannon’s definition of information, agree with the original concepts you had, and how far does it differ? Almost no definition is exactly congruent with your earlier intuitive concept, but in the long run it is the definition which determines the meaning of the concept—hence the formalization of something via sharp definitions always produces some distortion.

> I'm sure that story has additional meaning (and less surprise) to folk in Europe

Indeed, information is a receiver dependent quantity, as you say, and is formalized as such in any information theory textbook. Really, the information in a message about variable X is how much the receiver's subjective probability distribution about X is constrained by the message.

In practice, creating a complete model of the system i.e. enumerating all the possible variables and their prior probabilities is impossible. However, this doesn't mean the entire field is useless. In fact, the work of Shannon and others helps us understand the limits of what is possible to model and gain information about, and by how much. What those limits mean requires grappling with philosophical questions far above my paygrade.

I don't see any reason why information theory has to be limited to encoded information in bits. It deals generally with a quantity p(x), with no qualification for what p(x) is. You could, for example, tokenize words and figure out how complex the language used is, given some measure over the distribution of words in the english language. Similarly, you could go higher and work with characters, and figure out the surprise factor of using bears, but again you would need some model or measure of the distribution of character types in english literature. Figuring out the entropy of a coin toss is fairly trivial, but figuring out the entropy of, say, a story book, requires coming up with a model of the distribution over story books, which is the hard part. In that regard, information theory is quite powerful and gives us the tools to do all of that, it's just that the models are highly non-trivial, or otherwise missing.

I have a feeling you may be conflating some concepts in information theory, and possibly redefining information in multiple places.

Goldilocks and Red Riding Hood require a codec to decipher. Otherwise they are literally just strings of bits. A simple codec can take the bitstring and encode it as characters. So that you, may use your codec (your eyes) to send an encoding to another codec (your brain) to decipher and finally intepret the meaning. Codecs are kind of a tricky thing, in that they can compress a lot of information into a single bit, but to set up that compression requires a lot of ‘work’ outside the scope of a given transmission window.

To answer your question as to which story contains more information, you need to first ask ‘how sophisticated is my codec?’ I can tell you from watching children grow up, the same story can have vastly different information conveyed depending on when the story is consumed.

>How much information is in the story of Little Red Riding Hood? You could measure the bits encoded in the text but is that really measuring the amount of information communicated?

Yes, in several important ways.

It's not about putting a number to the "subjective experience" of reading/appreciating the story however.