[From Bill Powers (930307.0930)]
Bob Clark (930306.1930) --
Very nice, Bob. The "error" in a control system exists in but is
not perceived by the control system itself. Ergo, the opinion
that an error exists must be a perception in some other system.
There's a related problem with language here. People have often
interpreted the target position in pursuit tracking as the
reference signal, because the person doing the tracking seems to
be reacting to the target-to-cursor distance as an "error."
Behind this there is an unconscious assumption, which is that a
distance of zero between target and cursor is the only possible
reference condition. To show that this assumption is incorrect, I
usually just ask the tracker to maintain the cursor at some fixed
distance from the target for a while. The apparent reference
condition suddenly disappears. Now there is only the target-
cursor distance to be seen, with no reference distance visible.
No observer can now see what the reference condition is, except
by inferring it from the way the person changes the cursor
position when the target position changes. The reference
condition has no physical existence in the environment.
Still, there is a problem yet to be solved here. When a control
system fails to operate properly, for example because something
has changed in the external part of the loop or because there is
an overwhelming disturbance, the person as a whole certainly
knows that something has gone wrong -- that there is an error
larger than errors should be when control is successful. Somehow
something in the whole system knows that a subsystem has failed
to bring its perceptual signal to the reference level requested.
I don't have any neat solution to that problem. The idea of
model-based control might lead to a solution, but so far I
haven't been able to work out the details that would make such a
system work.
ยทยทยท
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Martin Taylor (930307.0050) --
Bill issues a challenge to compare PCT versus information
theory. I don't understand this. If by "information theorist"
Bill means me and Allan (since I have had much to say on this
topic I suppose he does), then there is hardly likely to be a
situation in which PCT and information theory could make
opposite predictions.
All I'm asking is that you show me that this is true in a
specific experiment.
Information theory as I understand it leads directly to PCT as
Bill understands it, or so I believe and have tried to explain.
OK, I have heard your words. I am asking that you take the
specific experiment I described, analyze it in terms of
information theory, and tell me what information theory says
about the expected result. If information theory leads directly
to PCT, then obviously information theory will lead to the right
prediction. Just show me how it does, in this specific case.
The difference I see between PCT with information theory and
PCT without is that with IT it should be possible to make
models that need not have as much arbitrary fitting of
parameters in order to account for real data. If you know that
the acquisition rate of information from say a cursor position
for one condition is 60 bps, and for another condition is 20
bps (perhaps because of viewing distance or brightness contrast
or something) then you should be able to derive the predictions
for the second condition from those for the first.
Fine. That would be very helpful. The problem I have set is much
easier, though, because you only have to analyze two conditions.
What you must do to meet the challenge, I presume, is to lay out
the way you would determine the number of bits per second in the
various data sets, how you would compute the degree of regulation
of the essential variable in terms of information theory, how you
would treat the person doing the Regulating in information terms,
and finally how you would calculate the predicted result, as
defined in the challenge. If PCT follows from information theory,
and PCT can make a general prediction before the fact, then
information theory should also be able to make a prediction
before the fact if it is fundamental to PCT. When the actual data
become available for the experimental run, you should be able to
plug them into your quantitative formulas and show that they
match the predictions. I expect to be able to do that with the
PCT model.
I'm a bit puzzled at the claim that information theory would
allow fitting the model to the data with fewer arbitrary
parameters. In the model we use most often, we obtain an
excellent fit for an individual by changing just one parameter,
the integration factor of the output function. It is hard to
imagine fitting the data with fewer parameters.
If you're offering an alternative test experiment, then lay it
out for me and I will try to meet the counter-challenge. In the
meantime, how about meeting my challenge?
In respect of the specific experiment Bill proposes, the
information theorist has to worry about the information
transmission between R and T as well as T and E. If the task
is to keep E small, information about the future of D might be
useful, but E is where the action is. Given a rigid robot so
that there is no loss of information between R and T and
between T and E, information about D just might permit keeping
E within reasonable tolerance, but the time lags would have to
be very small compared to the bandwidth of the disturbance for
condition 2 to work at all well even under such unrealistic
conditions.
What the information theorist has to worry about is the
information theorist's problem. It strikes me that you're talking
about determining rather a large number of parameters from the
data, probably more than one. How many parameters are you allowed
to adjust in an information-theoretic model order to reduce the
number of parameters the PCT model must adjust?
We are not given a rigid robot, but a person doing a real
behavior in a real experiment. It is up to you to assume what you
must in order to apply information theory to your own
satisfaction. I have laid out the data that will be available and
how they will be obtained. PCT can arrive at a prediction on the
basis of those data. If information theory is fundamental to PCT,
it should be able to do the same thing with the same data and
come out with the right answer. The conditions are not
unrealistic: I have spelled out exactly how they will be
accomplished, and it will be perfectly possible to do an
experimental run under those conditions.
Look, I'm offering to do as much of the actual detailed work for
you as you wish, including writing the program, running the
experiment, and providing you with the raw data. You can do any
part or all of this for yourself to determine that the experiment
was properly done and reported. You can even send me the formulae
for the analysis of the raw data, and I will write the required
program and apply them for you.
You say that information theory is capable of offering
explanations more fundamental than what PCT offers. Good. But I
want to see it actually done, not just talked about. Just to up
the ante a bit: until my challenge is met, I will feel free to
claim that information theory is incapable of predicting ANY
behavior that PCT can predict.
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Challengingly,
Bill P.