> (p is independent)
But this does not hold for most real-life data streams.
But this does not hold for most real-life data streams.
Learning advice: You shouldn't read line by line stopping to think after every sentence. Skim each chapter first. What is the main point? You may also want look at the next chapters to see where they go before finishing the chapter.
It is actually a very reasonable assumption for almost all kinds of data -- given that suitable compression is applied. Data, which is well-compressed, is essentially uniformly random.
When p('a') = p and p('b') = 1 - p, it means that the probability of 'a' and 'b' do not depend on anything. The probabilities must sum to 1, since it is the certain event, so p('a') + p('b') = p('a' + 'b') = p + 1 - p = 1. That is when we assume that only the symbols 'a' and 'b' are possible.
If there was a relationship between 'a' and 'b', say that 'b' always occurs after every 'aaa', then the when we receive 'aaa', we know that the next symbol is 'b' -- always.
So in this case the probability of 'b' has a relationship, a condition on the history, which would be written as p('b' | hist = 'aaa') = 1. A much more useful framework for this is a Markov process the a history/memory of 3. A graph for such a process can be seen here: https://mermaidjs.github.io/mermaid-live-editor/#/view/eyJjb...
Each node is a state and the edges represent the possible outputs. The rul for such a graph is that the probabilities of the "leaving" edges must sum to 1 -- we must of course always leave the current state we are in. Notice that it takes a sequence of 'aaa' to enter node "D", afterwhich we _must_ output a 'b'. Using some matrix formulations it is possible to calculate the probabilities of 'a' and 'b' (the stationary distribution I think it is called).
And to return to the first point, why is independence a reasonable assumption. In the Markov process, in node D, we known that we must always output a 'b'. In terms of information theory, if we receive 'aaa' then the 'b' is given and provides no new information. There we can perfectly predict it and we could also remove it (compress the data), without _losing_ information.
'abaabbaaab'
contains the same information as
'abaabbaaa'
since we _know_ that there must be a 'b'.
I hope that explains why independence is reasonable.
> It is actually a very reasonable assumption for almost all kinds of data -- given that suitable compression is applied. Data, which is well-compressed, is essentially uniformly random.
What kinds of data are an exception? Your explanation seems to cover everything
Second, you might (will) not be able to completely compress the data. A picture might be worth a thousand words, but they still take out a megabyte or so on disk. That makes for about 1000 bytes per word ;) So the entropy/information of a picture might be very small ("A dog jumping into water"), but we have no chance of truly understanding a general source (reality) and expressing its full machinery.
Think about the difference between JPEG and PNG (or GZIP and a JavaScript minifier). They are designed for completely different assumptions about the source and even receiver. JPEG assumes that the most important part of an image is the human-visual understand; PNG is lossless, but assumes high inter-pixel dependence. GZIP assumes general bytes (I think); JS minification assumes that a there is a more fundamental representation of the source without noise (formatting, comments, reasonable names, dead functions).