[From Bill Powers (930621.1600 MDT)]
[Martin Taylor 930621 11:00]
(To Rick:) We keep trying to tell you that we NEED the circle,
and you keep trying to tie us up with the chain.
I'm curious about this. I haven't seen anything yet in the
information theory stuff you've sent to me that requires a closed
loop for anything. Got any examples?
I think it was in a very early posting on this information
theory bit that I pointed out that the fact of control demanded
that the uncertainty about the world remained stable over time.
That, in iteself, demonstrated the need for the completed
feedback circuit.
Your argument was philosophical and verbal, not mathematical. I'm
still waiting to see information calculations done with a closed-
loop system.
It isn't the uncertainty that has to remain stable, is it? If
control is taking place under conditions of 10% uncertainty about
the state of the controlled variable, it won't disappear if the
uncertainty decreases to 1%, will it? It should get better even
if the amount of uncertainty changes.
···
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RE: Power gain
Bill was talking about the power gain between the error signal
and the resulting effector operations that directly affect the
CEV.
No, I was talking about the power gain between any variable in
the loop and that same variable after a complete trip around the
loop. There can be power gains and losses at various points in
the loop; what matters is that there be a net power gain around
the WHOLE CLOSED LOOP. If you're measuring at a point where
signals carry low power, then the returned signal (with the
circuit broken) must have a much larger amplitude than the
original signal. Where you're measuring power at the same place
you're measuring amplitude, power ratios are just the square of
amplitude ratios.
My point was that if we are to designate a particular part of the
loop as the control system, the power gain in that part of the
loop must be much larger than the power gain in the remaining
part. The details of gains and losses within these parts are
irrelevant.
It seems to me that just as a power gain is an essential
element of the outflow side of a control system, so a
corresponding power loss is an essential element of the inflow
(perceptual) side.
Not so. Control systems with no power loss at the input work
quite well. However, if you're going to measure the state of a
variable without significantly perturbing it (not really
important in a control system because of the presence of much
larger disturbances from elsewhere), you'll want to draw as
little power from it as possible.
Poisson-distributed.
Gaussian, actually.
No, Poisson. The Poisson distribution is appropriate for
measuring signals composed of discrete impulses averaged over
many parallel independent channels (electron flow, for example).
In the Poisson distribution, the RMS noise level (sigma) is
proportional to the square root of mean signal amplitude.
I'm sure you sometimes feel a little frustration at people
telling you that you said things that are the opposite of what
you tried to tell them. So am I, but at least on this point
you have come around to the "correct" view.
It gets even greater when people think I have come around to the
correct view when I'm expressing things I've known since I was
18. Everyone in electronics has to learn about signals and noise
and how measurements are affected by noise, integration time, and
so on. All this can be expressed quite accurately without
information theory, and much more simply. Maybe it's not as
philosophically correct, but it gets to the same answers.
The resolution IS infinite under certain conditions, which
include infinite observation time and *a priori* certainty that
the thing observed does not change over the observation
interval.
The resolution is infinite if the measuring device is an analog
instrument, period. Resolution is simply the minimum difference
between two readings that can be indicated. Whether it indicates
a real difference of the measured variable is beside the point;
that is a question of precision or accuracy, not to mention
epistemology.
If you take a snapshot of pure noise using an analog meter, you
will get a reading on the instrument. If you take another
snapshot, you will get another reading. The two readings can
differ by any amount whatsoever all the way down to zero. The
number of possible different readings is infinite. If you can't
see any difference by eye, get out an electron microscope. There
is no point on the scale at which the needle could not come to
rest.
Resolution is finite for a digital measuring instrument. It is
the least change in the readout that can occur: one Least
Significant Digit. The digital meter simply cannot indicate any
reading between two consecutive numbers on its readout. Looking
at the world through a digital measuring device, one sees a
discrete universe in which nothing can change by an amount
smaller than one least significant digit, and in which
measurements are always an integer multiple of that least digit.
There are no values at all between the indicated readings.
In the nervous system, all signals pass through a pure analog
stage, where the signal amplitude is represented as a chemical
concentration in a cell body. The least amount by which the
chemical concentration can change is by one molecule inside the
cell. If you measure in terms of synaptic potentials, the least
difference is even smaller than that, because it also depends,
then, on the positions of molecules within the cell. For all
practical purposes, therefore, the resolution of a neural signal
measured in impulses per second is infinite. There is no signal
magnitude that cannot be represented by a neural frequency; two
signal magnitudes can differ by as little as the amount
represented by a single molecule inside the cell body, or less,
which is far smaller than the amounts by which natural neural
signals fluctuate -- for all practical purposes, infinitely
smaller.
Changing the observational interval and the integration time and
all those other factors relates to the precision and
repeatability of a measurement, not to its resolution. I think
you're just using the word resolution with a nonstandard meaning.
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Greg Williams (930621) --
I have told them that assigning such "importances" is not a
part of PCT modeling ...
I've forgotten the original context, and have lost most of my
interest in finding out, but just for the record:
"Importance" in many contexts can be translated to "error
sensitivity." That is, the ratio of effort to error. If keeping a
variable at a specific level is "important" to someone (and
importance is always importance to someone), then only a small
error is sufficient to result in a large corrective effort. A
variable of less importance to the person will show a lower ratio
of effort to error.
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Rick Marken (930621.1000 PDT) --
I was trying to guess why you think there is information in
perception.
If information is defined as log(R/r), where R is the range of
fluctuation of a signal and r is the least discernible change in
the signal, then there is information in the perception. It can
be calculated in bits per second. This measure does not represent
knowledge; it is not "about" something. It is just information, a
quantity calculated from measurements of R and r that can be
represented by a single scalar number. Do not confuse the word
information with the word information. Those are two quite
different words with entirely unrelated meanings. We just happen
to spell and pronounce them the same way. Perhaps we should
capitalize one of them to remind us of the difference, like
coulomb and Coulomb, the first being a unit of charge and the
second a person. Information is a measure of the range and
resolution of any signal, regardless of its causes or detailed
behavior through time. On the other hand, information deals with
relationships to other signals and is concerned with the meaning
of a signal's behavior through time: what the behavior of one
signal tells us about the behavior of another one. The most we
can get from a measure of Information is a relationship between
the _range_ of one signal and the _range_ of another, in terms of
the least measurable unit. From knowledge only of the range and
resolution of a signal, we cannot of course reconstruct its
detailed behavior through time.
v
Martin and Allen are saying there is Information in the
v
perceptual signal. You are saying there is no information in the
perceptual signal. You are both right. You cannot get information
from Information.
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Bruce Nevin (Mon 930621 14:44:33 EDT) --
Bill:
Your comments on the grape and the elephant seem to assume
that the grape and the elephant are distinguished because they
REALLY ARE different, whereas phonemes are distinguished
because they only SEEM different.
Bruce:
Nope. There are no phonemes there to be distinguished.
Phonemes are apparently present only because a perception of
contrast is controlled.
Try again; we'll get there yet. If you control and perceive
contrast, then what you perceive is contrast. You say, "Wow,
there's some contrast." You may decide that it's too much, and do
something to make it less, or that it's too small, and do
something to make it greater. But when you're done, you still
have only what you started with: contrast.
There's a very important principle of modeling lurking here, so
let's not give up on this.
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Best to all,
Bill P.