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Apr 9, 2018 · Let $A$ and $B$ be two events and $p$ and $q$ be their corresponding probabilities. One of the properties of surprise function is: $$S(AB) = S(A) + S(B)$$ Which seems to be fine if the two events are independent.
- Information Is Surprise
- Producing Gibberish
- Calculating Surprise
- Average Surprise
- An Example
- About This Article
Shannon wanted to measure the amount of information you could transmit via various media. There are many ways of sending messages: you could produce smoke signals, use Morse code, the telephone, or (in today's world) send an email. To treat them all on equal terms, Shannon decided to forget about exactly how each of these methods transmits a messag...
Shannon appears to have had some fun playing around with this idea. On page 7 of A mathematicaltheory of communicationhe reproduced a string of words that werepicked at random, independently of each other and with a probabilityreflecting their frequency: REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NATURALHERE HE THE A IN CAME THE...
This line of thinking led Shannon to consider an idealised situation. Suppose there’s a random process which produces strings of symbols according to a certain probability distribution, which we know. Assume, for the moment, that each symbol is picked independently of the one before. We could simply define the surprise associated to a single symbol...
This sorts out the amount of surprise, which we've related to the amount of information, of a string of symbols produced by our machine. For reasons that will become clear later, we can also calculate the expected amount of surprise the machine produces per symbol. This is akin to an average, but takes into account that symbols with higher probabil...
Let’s look at an example. Suppose the machine can only produce two symbols, an and a and that it picks them with equal probability of It might do so by flipping a fair coin, hence the choice of the symbols. The entropy of this probability distribution is Choosing the base of the logarithm to be 2, this gives us a nice round value: Now suppose that ...
This article is part of our Information about information project, run in collaboration with FQXi. Click hereto find out about other ways of measuring information. Marianne Freiberger is co-editor of Plus. She would like to thank Scott Aaronson, a computer scientist at the Massachusetts Institute of Technology, for a very useful conversation about ...
The information content, also called the surprisal or self-information, of an event is a function which increases as the probability () of an event decreases.
These functions like the Cantor function and the continuous-but-not-differentiable function are all well and good, but contrived - the only place you ever see them is as counterexamples. Here is a function that has many uses in Number Theory, and still manages to have a strange property or two.
- A Brownian motion sample path . These are about the most bizarrely behaved continuous functions on $\mathbb{R}^+$ that you can think of. They are n...
- The empty function $\emptyset:\emptyset\to\emptyset$ is quite strange when you first meet it.
- The Busy Beaver function Let Σ be a finite alphabet, for instance {0, 1}; let M be the set of Turing machines with alphabet Σ, and let H ⊆ M be the...
- I like the Cantor function. A continuous, increasing function $f:[0,1]\rightarrow[0,1]$ with derivative $0$ almost everywhere. See wiki article here .
- The Conway base 13 function has to be the weirdest function I know. This function is continuous nowhere, yet it satisfies the intermediate value th...
- The Ackermann function $A(n,m)$ is defined on the natural numbers by a very simple recursion, but the values grow enormously, almost beyond concept...
- I suppose the strangest function in mainstream mathematics is Riemann zeta function http://en.wikipedia.org/wiki/Riemann_zeta_function $ \zeta(s) =...
- I'm still quite impressed about $f(x)=\mathrm e^x$ …
- Thomae's function , also called the "popcorn function". It's continuous at all irrationals and discontinuous at all rationals. Here a picture:
unexpected: He gave a quite surprising answer. It's hardly / scarcely /not surprising (that) you're putting on weight, considering how much you're eating. I have to say that it's surprising to find you agreeing with me for once. Synonym. amazing. Opposite. unsurprising. Fewer examples.
Mar 28, 2024 · “Not surprising” is an adjective phrase used to characterize an object, phenomenon, or action that is expected, while “not surprisingly” is an adverbial phrase that refers to an anticipated occurrence.
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Definition of surprising adjective in Oxford Advanced Learner's Dictionary. Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more.
