# Re: Information Theory

Date: Thu May 22 1997 - 18:28:00 BST

(25) What is information?

a. any message that makes you feel informed

b. any message that makes you feel informed when the sender meant you
to feel informed

c. ***anything that reduces your uncertainty

d. anything that reduces your uncertainty when the sender meant to

e. none of the above

> From: Hawkins, Sean <swh196@soton.ac.uk>
> I do not believe c (anything which reduces your uncertainty)

But did you know that that was what had been given as the formal
definition of information in the lectures?

From Lecture Notes on Chapter 8

"I have used the food dispenser example before to explain what
information and communication are about: If there are six numbers and I
can only choose one number each day, then my chance of getting lunch is
1/6, but if someone tells me that the number is odd today (and that is
true) then my probability of lunch has gone up to 1/3."

From Lecture Notes on Chapter 9

"Information = the reduction of uncertainty between alternatives
that MATTER to you.

"The 6-choice lunch machine is an example of how any data that could
reduce the uncertainty about which of the 6 windows contains the lunch
is informative."

The best time to challenge that definition of information would be in
lectures, tutorials or skywriting. When you are taking a quiz, it would
be better to provide evidence that you knew what had been CLAIMED
(wrongly, if you like) to be the definition of information in the
lectures!

> Information may reduce uncertainty or increase it.

But what is this "X" that you think can both increase and reduce
uncertainty?

This is why the lunch machine is a good "mental model" for thinking
about information: Regardless of what you know or believe, if each of
the six levers had an equal probability of being lunch, and which one
it was on any given day was completely unpredictable -- and you NEEDed
to eat lunch daily -- then if you kept picking even numbers, wrongly
thinking (or having been wrongly told) that that would increase your
chances of lunch, you would still only eat one in six times, and go
hungry 5/6 times.

The uncertainty is not what you may or may not be thinking in your head,
it is the probability of going lunchless in such a situation.

So what might "increasing" uncertainty here?

With 6 options, you can't make it much worse than 5/6 uncertainty, but
if two new levers were added, that would increase your uncertainty to
7/8.

Someone could tell you (truly) that lunch is never 7 or 8. That would
keep your uncertainty at 5/6, as before (if you believed it, acted on
it, and it was true). Or someone could tell you (falsely) that lunch is
never 7 or 8 (whereas in reality 1-8 are all equally probable). Then
your probability of lunch would be 1/8, though you would think it was
1/6:

But it doesn't matter what you think your uncertainty is; what matters
is what it really IS, given your behavioural strategy: Even if you were
wrongly "sure" (= certain) that lunch was always 5, if in reality all
the levers were equally probable, you would still eat only 1/8 times if
you always chose 5. So your uncertainty would be 7/8 even though you
thought it was zero.

What this shows is that in information theory, "certainty" is not
subjective certainty (how "sure" you feel about it), but OBJECTIVE
uncertainty, namely, what your real chances of lunch are.

> The inference that information reduces uncertainty infers that
> there was a degree of uncertainty before receiving the
> information. For example, if you are provided with one
> piece of information - "The sky is blue", may reduce
> uncertainty if you were uncertain as to the colour of the
> sky, but if someone else subsequently provides you with
> contrary or conflicting information "The sky is red and I
> can prove it" can only increase your uncertainty.

Uncertainty is not a subjective matter here: You can be told, or can
think or believe all kinds of things. What matters -- if we are
speaking about the technical meaning of "information" and "uncertainty"
as they are used in the branch of probability theory called
"information theory" -- is what the alternatives and their
probabilities really are, and whether the "information" has reduced
your uncertainty between the alternatives. "Sky is blue" is vague: Think

> My other point concerns relevance of information in the
> context of reducing uncertainty. Information is a social
> construct and although registration is through one of the
> sensory modalities, consciously or unconsciously we decide
> whether the information is relevant or not and whether we
> wish to retain it. There is not necessarily a reduction in
> uncertainty - there may just be a rejection of the
> information because of its irrelevance.

Unfortunately this is a subjective concept of information --
and that it is a "social construct" sounds like a theory.
The technical definition of information came from analysing
signals and noise, and the probability of receiving signals
despite the noise. This corresponds only a little with
the layman's dictionary-definition notion of information --
and even less with social-construction theories of information.

> In addition, a new piece of information will not reduce
> uncertainty in subjects of which we have little or no
> knowledge - as there was no uncertainty to begin with! In
> other words, we do not know what we do not know, and
> being provided with information does not reduce uncertainty
> in subjects of which we had no uncertainty to begin with.

That's like saying that telling someone that "lunch is number 5 today"
is uninformative to an anorexic: Fine, so let's turn instead to
something that DOES matter to an anorexic...

> Taking these points into consideration, I therefore
> challenge the answer and await with some uncertainty (sic)