ONT Re: Inquiry Driven Systems
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| To see a World in a Grain of Sand
| And Heaven in a Wild Flower
| Hold Infinity in the palm of your hand
| And Eternity in an hour
|
| William Blake, 'Auguries of Innocence'
Of the other worlds that are visible in this,
our grain of sand flowing into virtual glass,
is the whole world of information, radically
primitive precursors of which show up in the
way that four signs inform us of two objects.
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| s^1 |
| · |
| · |
| · |
| · |
| p · · · · s^2 |
| · · |
| · · |
| · · |
| · · |
| q · · · · s^3 |
| · |
| · |
| · |
| · |
| s^4 |
| |
o-----------------------------------------------------------o
In order to develop the information-theoretic aspects of this example,
let us imagine an experimental scenario of the following description.
We do an experiment that has two possible outcomes, p and q. We do
not observe the outcomes directly but only the signs that indicate
how they turn out, where the association between the outcomes p, q
and the signs s^j, for j = 1 to 4, is as shown by the dotted lines
in the Figure. Let us now ask a pair of questions that will lead
us to the information-theoretic features of our present example:
| 1. What is the informational value of a sign in this setting?
|
| 2. What is the informational value of the whole set of signs?
Now, we have no particular reason as of yet to associate definite
probabilities with the various outcomes and sundry indications in
this situation, so I will just do what is usually done by default
in such cases, namely, to assume the "all things being equal" or
the "maximum entropy" option, in which p and q are split 50-50.
Putting all of the pieces together, we have the following setup:
| We have an initial uncertainty of 1 bit,
| concerning whether p or q will turn out.
|
| s^1 says that p is certainly the case,
| thereby reducing our uncertainty to 0.
|
| s^2 says that p or q is the case,
| leaving our uncertainty at 1 bit.
|
| s^3 says that p or q is the case,
| leaving our uncertainty at 1 bit.
|
| s^4 says that q is certainly the case,
| thereby reducing our uncertainty to 0.
The "average uncertainty reduction on receiving each sign" (AURORES)
is (2 bits)/4 = 1 half bit, and this is the capacity of the channel
that is afforded by the medium of the 4 signs {s^1, s^2, s^3, s^4}.
Jon Awbrey
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