X-Envelope-to: CSG-L@vmd.cso.uiuc.edu
X-VMS-To: @CSG
MIME-version: 1.0
Content-transfer-encoding: 7BIT
Newsgroups: bit.listserv.csg-l
Date:         Sat, 19 Dec 1992 03:06:21 -0700
From:         "William T. Powers" <POWERS_W%FLC@VAXF.COLORADO.EDU>
Subject:      Information theory vs control theory

[From Bill Powers (921219.0130)]

Martin Taylor (921218.1830) --

Your remarks about the predictability of the world and PCT are
mostly cogent. This is particularly true if you include the
predictability of the output effectors as well.

I can quibble, however, about a number of points.

>In a choatic world, delta matters.  If delta is very small, the
>probability distribution of states at t+delta is tightly
>constrained by the state at t. If delta is very large, the
>probability distribution of states at t+delta is unaffected by
>the state at t ...

It matters a heck of a lot more to a plan-then-execute model than
it does to a control model. Remember that control of a variable
depends only on the ability of the system to affect that
variable, directly, in present time. It isn't necessary to
produce an output and wait to see its future effects. If progress
doesn't follow the intended path, correction occurs right away,
after a short delta. So in properly-designed control systems,
delta is always small. Even if the stated goal is far in the
future, the path to the goal can be defined as a reference path,
and control can assure that progress stays on that path. This, in
fact, is the only practical way to control for long-delayed ends.

In circumstances where perceptions are uncertain and there are
long delays in the control loop, control is simply going to be
poor. Predictive control with a large delta is doomed to be
lousy.

>The central theme of PCT is that a perception in an ECS should
>be maintained as close as possible to a reference value.  In
>other words, the information provided by the perception, given
>knowledge of the reference, should be as low as possible.

I think you'd better take one that back to the drawing board. The
reference in no way predicts the perception by its mere
existence. The best control requires the widest bandwidth in the
system, including its input function, up to the point where noise
begins to become significant. I don't see how this is consistent
with saying that the information provided by the perceptual
signal should be as low as possible.

It may be that given an excellent control system, the state of
the reference signal does predict the perceptual signal well, so
in that case an observer will find knowledge of the actual state
of the perceptual signal redundant. The information provided by
the perceptual signal TO THAT OBSERVER is low. But that would
have absolutely nothing to do with the design of the control
system. That's a calculation by an external observer, not a
property of the system. The only reason for which the perceptual
signal provides low information to the observer is that it
provides a great deal of information to the comparator.
I still think that your analysis is simply descriptive, and has
nothing to do with the design of control systems. It may apply to
a successful design, but it can't provide a successful design.

>In other words, the information argument does not specify what
>Bill's eleven levels are, but it does make it clear why there
>should BE level of the hierarchy that have quite different
>characteristics in their perceptual input functions.

>It is that kind of thing that I refer to as "understanding"
>PCT, not the making of predictions for simple linear phenomena.

Dennis Delprato, here is another addition to your list of myths
about PCT: that we can predict only simple linear phenomena.
Martin, have you looked at the Little Man? It is chock full of
nonlinearities. We have actually tested it using a much more
realistic muscle model -- in fact one that proved to be far more
nonlinear (6th power) than the actual muscle (2nd power) owing to
a misunderstanding of a published model (which tried to include
the limits of limb travel in the same equation!). And the
feedback path from output torques to visualized position of
fingertip, and the method of depth perception for the target, are
highly nonlinear.

We can't solve the nonlinear equations analytically (nobody can),
but that is not a constraint on the simulations. In my psych
review article so long ago, I showed tracking data with a cubic
relationship between handle position and cursor position that
actually reversed slope in the middle. The model handled it just
the way the real subject did, by skipping across the region of
positive feedback. Rick Marken's experiment with size control
modeled one case in which the controlled variable changed as the
square of handle position. No problem. We have tried all sorts of
nonlinear functions. But there's no point in teaching control
theory using nonlinear equations that nobody can solve.

>But the results of the (almost) a priori argument agree with
>the (despised) results of experiments in reading that I
>discussed in our 1983 Psychology of Reading.

What do you mean, they "agree?" Do you mean that you predicted
the reading performance of every single subject with an error of
five percent or so? Or even 20 percent? Come on, what are you
calling "agreement?"

>I said that if you don't understand Shannon, you won't
>understand PCT.  I didn't say you won't be able to use PCT to
>make predictions.

Why am I reminded of that poem about Hiawatha?

>>And in a control
>>model, the signals in the various paths normally carry far
>>less information than the theoretical limits allow.
>Dubious.  I would like to be able to figure out how to test
>that assertion.

It's easy. Most perceptions occur on a scale between 0 and
maximum magnitude, and vary at a rate between 0 and some maximum
cutoff frequency. To accomodate the maximum magnitude and
frequency, the perceptual channel must have a certain information
capacity. As perceptual signals can be controlled at any level
within the whole range and can be varied at any rate up to the
maximum, it follows that unless the perception is being
controlled at maximum magnitude and the reference signal is
changing at the maximum rate that still permits control, the
actual information flow must be much less than the channel
capacity. Most perceptions are not controlled at their extremes;
hence most perceptions must use less than the whole channel
capacity.

>> But
>>information theory doesn't tell you there will BE an error
>>signal.

>As I pointed out to Tom, it does.

Show me where Shannon's theory says there must be a comparator, a
reference signal, and a perceptual signal.

>Natural laws are no use without
>boundary conditions to describe particular situations.  But if
>you understand the abstract principles, you can make better
>[bridges/kettles/radios/control systems].

This brings us right back to Tom's challenge. We have, for
example, a simple tracking model that predicts behavior with an
accuracy of better than 5%, measured as RMS error between model
and real handle behavior divided by peak-to-peak real handle
excursion (a signal-to-noise ratio of 20:1, by a standard measure
in electronics). What can we do to this model, using information
theory, that will make it predict any better?

I think that information theory is by its very nature a post-hoc
description, not a model. You can't start with information theory
and come up with a system design. Or so sez I.

Backing up to pick a sentence I passed over:

>The reason is that things in the world can change continuously
>and (despite Bill's claim) most perceptions are based on very
>poor information--the SNR is very low.

I think you're letting theory triumph over observation. I have
just shown that we can predict behavior with an SNR of 20:1. We
do this in a routine way. The SNR of the perceptual channel
certainly can't be _lower_ than that. Do you consider 20:1 very
low?
I come back to my basic statement about the levels I've defined
in the hierarchy: they refer to the world you observe. The world
you observe does not have a predominance of noise in it. I know
that you have said, "Oh, you're talking about _conscious_
perception, which is a different matter." But I see no reason to
suppose that conscious perception consists of anything but neural
perceptual signals, the same signals that are there when we are
using the same control systems unconsciously, as in standing
erect. My theory of perception agrees with a largely noise-free
experienced world; yours appears to predict a world in which
perception barely stands out over the background noise. If your
model were correct, precise control would be impossible. Yet we
manage to control variables of all kinds with great precision,
even variables like the form of an algebraic  expression (as in
proving trigonometric identities). We don't control ALL variables
precisely, but the reason is usually not that there is some
inherent imprecision in the process itself, but that we're
attempting to control something we have conceived poorly, or that
nature dictates is not amenable to control (like another person).

One last observation:

>The way around this is that category boundaries are not
>thresholds, but fold catastrophes.

That's a pretty fancy term for a Schmidt trigger. Anyway, saying
that categories are fold catastrophes says nothing that my
description of hysteresis didn't say. Categorizing categorizing
doesn't tell us how it works. It doesn't work the way it does
because it's a fold catastrophe. It's a fold catastrophe because
of the way it works, which remains undisclosed.
--------------------------------------------------------------
Best,


Bill P.
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU