[Allan Randall (930618.1400 EDT)]

Bill Powers (930612.0930 MDT)

When we say "disturbance" in equations, we mean the STATE OF THE

DISTURBING VARIABLE:

...We do NOT mean the amount by which the knot (X) is displaced from

the target position.

In spite of all the other misunderstandings, I don't think this is a

problem. I do not mean the amount by which the knot is moved. I mean

the force(s) acting on the knot from the external environment, whether or

not the knot actually moves.

Bill Powers (930613.2200 MDT)

>Bill, I have a real problem with this whole log(D/r) thing...

I got it from Martin Taylor... I'm just following orders.

Well, all I can say is that this is not the standard definition, which

is -log(probability), as I've described many times in the past. I suspect

you are using log(D/r) differently than Martin, but I will let him speak

for himself. The way you are using it, it is not equivalent to information.

>If two signals have wildly different scalings of amplitude, but

>are otherwise identical, then I can write a very short program

>to convert one to the other...Suppose you have the relationship X(t) = Y(t)/10. ... No matter

how you write the program, when you compute X from Y you will get

a waveform with less relative resolution than there was in Y.

If you multiply X by 10, you will not get Y back; ...

Not necessarily. It depends on the representation used for the numbers. Eg:

00.51928374 / 10.00000000 --> 00.05192837 * 10.00000000 -->

00.51928370 LOSS

But:

.51928374E+00 / .10000000E+02 --> .51928374E-01 * .10000000E+02 -->

.51928374E+00 NO LOSS

So if I scale a sequence of these numbers, I do not have to also scale

the resolution at which they were measured nor the resolution at which

they are represented in the computer in order to retain information.

I don't want to make a federal case out of this - its not a really

important point. I was only trying to illustrate that your statement

about log(D/r), while true about THAT measure, is not IN GENERAL true

about information. D/r is just the number of possibilities inherent in

the numbering system or the measuring apparatus. If you considered this to

be the inverse of the probability, then I would have no problem. But this is

not how you've been using it. You are tying the whole thing to the way

numbers are represented - to the precision at which measurements are made.

But many other schemes are also possible - this is why the ideas of computer

programs/languages and probabilities/distributions are much better ways to

think about it.

Another way of viewing it: D/r *can* be used to compute information on a

sequence of numbers *after* they have been compressed to their shortest

possible representation in whatever language we're using. Only then does

it make sense to use the precision of the numbers themselves to compute

the information. Then your comments would be more or less accurate.

But this is a BIG difference. This compressibility has not been

addressed in the measure log(D/r) as you are using it.

>...tell me how to compute log(D/r) for the

>following sequence S:> .0 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .2 .7 .7

>.7 .7 .7 .7 .7Before I could do that, you would have to tell me the size of r.

There isn't any inherent "resolution" in an arbitrary sequence of

numbers like the one above.

Sorry - I had thought it implied that the resolution was .1.

If r had been .001, I would have written .000 .500 .500, etc.

...What they mean depends on the

physical situation they are taken to represent.

Sure - and that depends on our *model* of the physical situation.

When you say: "...depends on the physical situation they are

taken to represent," it sounds like the same thing I mean when I say

"...depends on the chosen language."

If this is a

series of measurements of some physical variable, then we have to

talk about the measuring device's resolution.

If that is the extent of your model of the physical situation, then fine.

As I said, if your model or language is restricted to the number system or

measuring apparatus, then log(D/r) will work. However, the "physical

situation" the numbers are "taken to represent" is generally more involved

than just the measuring resolution. In a human, it is likely to be something

like a hierarchy of control systems.

My problem is that I can't find any link between the

manipulations you talk about and any PHENOMENON.

You misunderstand. The arbitrary hypothetical examples are my attempt to

explain a mathematical system called information theory. This is not

meant to be connected to any particular physical phenomenon. It has

nothing to do with control theory or thermal systems or any other

physical system. It is a mathematical notion. Now I have ALSO talked about

applying this theory to control systems, but in that case, I was not making

up arbitrary hypothetical examples - I was using the control hierarchy as

the language.

... the only answer I've received so far is

"Well, that depends on what you assume."

This is not a fuzzy, ill-defined answer. *Given* a language or model to work

with, information *is* well-defined. But, you said yourself that what the

numbers mean depends on the physical situation we take them to represent.

And this *does* depend on what is assumed.

... How do you

decide what is the most plausible way to set it up for a control

system model of a specific example of behavior? So far I'm

drawing a complete blank on that. And, apparently, so are you.

No - I've said many many times that the control hierarchy is the language.

The general notion of information *is* completely open-ended as

you describe, but as applied to control theory, it is much better defined.

## ยทยทยท

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Allan Randall, randall@dciem.dciem.dnd.ca

NTT Systems, Inc.

Toronto, ON