[From Bill Powers (940602.2040 MDT)]
I'm all written out for today, but here's a little more...
Martin Taylor (940602 17:30)--
RE: program level control
But there still seems to me to be a flavour of choosing among
outputs, that is not there when different perceptual conditions
(as compared to their reference levels) lead to different
outputs in the lower levels.
Remember that control action just about guarantees that perceptions
will match reference signals; outputs change with disturbances, but
perceptions do not, to a first and good approximation.
"Choosing outputs" is much more easily understandable in a
continuous-variable model, especially one with a number of systems
being simultaneously active at several levels. In general, _all_
systems of one level are involved in the control processes of _each_
system of the next higher level. This means that there is no simple
choice between using lower system A and lower system B. No one
higher-level system determines the reference signals of any one
lower-level system. When you watch such hierarchies in (simulated)
action, it becomes clear that the effects of disturbances on
reference signals of lower systems vary with the direction of the
disturbance and its point of application. It looks as though some
system is choosing to emphasize one set of reference signals over
another, depending on the kind of disturbance. But in fact there is
no difference in the way any system operates; the shifts are the
natural tradoffs between higher control systems acting through a
shared set of lower systems. The operating hierarchy represents the
ongoing solution of a set of equations in n variables at m levels,
the solution being the best attainable minimum of overall error in
the whole system.
There must be situations in which context changes require something
like switching control systems in and out of action -- but these
would entail switching not just one control system, but a whole set
at several levels. I'm not prepared to deal with that. Many of the
apparent choice-of-output effects seem to come naturally out of a
hierarchical model in which there is no actual choice of outputs.
Maybe we should try to understand those situations first, so we can
recognize genuine exceptions.
ยทยทยท
--------------------------
RE: the Marken Effect.
Discovering that the model had to reproduce the real subject's
transport lag in order to get this effect did, as you suggest,
reveal the conditions under which this effect is seen. But it also
_explained_ the effect.
The explanation is this. The auxiliary control system, with a modest
loop gain, simply tried to keep the controlled variable constant,
operating in mild conflict with the main control system, either Rick
or the model of Rick. What made control a little better with the
auxiliary system in operation was the fact that it did NOT have a
transport lag in it. Thus, on the average, a disturbance that caused
a change in the controlled variable was counteracted, to some
extent, by the auxiliary control system _during Rick's transport
lag_. This reduced the effective disturbance that Rick or the model
of Rick experienced, resulting in slightly but reliably better
control.
A test of the Marken Effect by simulation failed at first because
the model used for Rick's behavior did not include a transport lag.
The model of Rick could therefore act just as fast as the auxiliary
control system could, so there was a simple conflict and no
improvement in control. When the model of Rick was changed to match
Rick's actual transport lag, the improvement reappeared.
This explanation goes considerably beyond merely noting the
conditions under which the Marken Effect appears.
------------------------------------------------------------
Thomas Baines --
RE: what kind of uncertainty?
When a control system experiences a perceptual signal that tends to
deviate from the reference level, an output is generated that acts
against those deviations. As the cause of those deviations is not in
general perceivable, the control system has no way of knowing
whether these fluctuations are systematically related to specific
disturbance waveforms (one or more), or are being caused by
unsystematic effects -- noise somewhere in the system or its
environment. An observer sitting inside the control system and
having available only the perceptual signal to observe could not
measure the "uncertainty" in the perceptual signal relative to the
environment. That is, the fluctuations might be due to random noise,
or might be a faithful and noise-free representation of systematic
effects from the environment, or anything in between.
An omniscient external observer, on the other hand, could
simultaneously know the state of the environment and the state of
the perceptual signal. Now it would be possible to compute the
probablity of p given d, or d given p, and other conditional
probabilities involved in calculating uncertainty. If this
calculation were done continuously, it would continuously provide a
number indicating the uncertainty. This number, however, would be
available only to the omnisicient observer, not to the control
system.
The uncertainty calculated by the omniscient observer we could call
the "objective uncertainty;" the uncertainty relative to objective
measurements inside and outside the control system. Perhaps that is
what you mean by "data/perception" uncertainty.
That suggests, by verbal symmetry if not logic, that there is a
"subjective uncertainty." As no control system can calculate the
uncertainty in its own perceptual signal, subjective uncertainty has
to involve more than one level of perception and control. One
control system, evaluating the perceptual signals from a number of
lower systems, infers the presence of a disturbing variable and from
this inference constructs a perception of the disturbing variable.
At the same time, systems at the same level receive copies of the
perceptual signal, so both the perceptual signal and an inferred
disturbance are represented as higher-level perceptions. Now still
higher systems can do the equivalent of computing conditional
probabilities and can construct a perception of the uncertainty in
the relationship between the lower perceptual signal and the
inferred disturbance. The higher systems could say, "That perceptual
signal is being disturbed by that disturbing variable", and by
extension, "That perceived output is counteracting that
disturbance." Some degree of uncertainty could then be attached to
these statements.
Perhaps this is what you meant by "source/perception" uncertainty.
Martin came up with two different interpretations. So you have a
total of four interpretations to choose among: two things taken two
at a time. I think that means you are lacking exactly two bits of
information, but of that statement I am less than certain.
----------------------------------------------------------------
Martin Taylor (940602.2020) --
RE: Paul Revere
One issue that has been in the discussion is whether Paul
Revere perceives directly the uncertainty. I think he does.
In the situation as you initially described it, there was no
uncertainty to be perceived; the only uncertainty would be in what
Paul imagines, and then only if he tries to anticipate what the
outcome will be.
To judge this, let's change the situation slightly.
As you have changed the situation, you have indeed introduced a lot
of uncertainty that was not there before. But there is still no
_necessary_ occasion for Paul Revere to perceive any uncertainty.
Since you are allowing a change in assumptions, let me add a few
more. Let's assume that Paul is an accomplished statistician, and
furthermore has experience with many forays to determine enemy
movements, so he knows the probabilities of a correct observation
based on observations under similar conditions. He can say that
given his observations of apparent British movements, the
probability of actual movements is some number. Doing this
calculation over all his experiences, he can state the summed
uncertainty of the actual troop movements given the apparent troop
movements, and can thus compute the uncertainty. He can do this for
each possibility, so he can compare the uncertainty in the
perception that the route will be by sea and the perception that it
will be by land, given the apparent troop movements. The logical
thing to do will then be to choose the least uncertain perception
and act on it.
So where is there any reason for Paul Revere to feel uncertain? The
formal method of computing uncertainties is well-defined, and the
decision criterion is simple. All he has to do is turn the crank and
act as the result indicates. No choices are left to make.
There is one way in which Paul could experience an uncertainty, and
that is if he tries, before the data are all in and the computations
are made, to guess how the uncertainties are going to be
distributed. This could lead him into the sorts of feelings we call
uncertainty, but of course this would not affect his actions in the
slightest, because they will be taken only when the data are
complete and the calculations are finished, and will be completely
dictated by the result. The feelings of uncertainty are simply side-
effects of trying to guess at what can't be known.
It is also possible, of course, that Paul Revere is not an
accomplished statistician, has not done the necessary prior
observations, and does not know how to compute uncertainties. In
that case he has no knowledge of the conditional probabilities, and
even if he did he would not know how to convert them into measures
of uncertainty. So what does he do? He acts on the basis of
appearances, unaware that he might be deceived and issue the wrong
warning. Even if he is aware of that, there is nothing he can do
about it but go with appearances. If he realises that, he will also
see that he is doing the best that can be done, making uncertainty a
self-indulgence.
Even if Paul Revere could compute and perceive the relevant
uncertainties, how could his subsequent action of alerting the
countryside with a specific message reduce that uncertainty? His
data are no better than they were before he reached a decision. The
specific uncertainty that existed still exists. The only way to
reduce the uncertainty would be to improve the observations. Of
course eventually he will find out which way the British went, but
that information will come too late to be corrected if he guessed
wrong. The only variable he can control is the relationship between
the perceived movements of the British and the perceived deployment
of the Minutemen.
When faced with a truly uncertain situation, in which perceptions
are known to be inaccurate, one really has only two choices. If
there is time, the obvious thing to do is to reduce the uncertainty
to a negligible amount by making better observations. That will lead
to accurate control.
If there is no time, then all one can do is go by appearances,
whether benefitting from past experience or not, whether using
statistical calculations or not. And one would be well advised not
to pin a lot of hopes on the outcome. Poor control is poor control,
and is not made any better by explaining that it is the best that
could be accomplished.
----------------------------------------------------------------
Best,
Bill P.