There's no paradox to explain away. That's the very starting point from which
the whole discussion began. It's not a paradox but a prior prediction, almost
a description of "control." Just as you can't explain control by starting
with the perceptual signal and using it to decide the output, because that's
a backwards analysis, so your statement of what is "contrary to reason"
relies on a backwards analysis. You should be stating that "as the precision
of opposition to the disturbance increases, so the information about the
disturbance remaining in the perceptual signal decreases" and then you
would see it as a perfectly straightforward, self-evident proposition, in
place of a paradox contrary to reason.
···
=======================================
--(Comments of today are indented with "--" marking)
[Martin Taylor 921218 15:30]
Remember my point about the reduction of information rate as we go up the
hierarchy. It is this that gives play to predictive memory models.
----------------------
[Martin Taylor 921218 18:30]
I guess I'd better try to describe, as I did a year or so ago, wherein
information theory helps in the understanding of PCT. I didn't succeed
in getting across then, and I'm not sure I'll do any better now. ...
Now consider the interchanges of a week or two ago about planning and
prediction, continued in Bill's post of today to Allan Randall. In those,
the situation is greatly different. The information required from the
lower level for the upper level to maintain control through a hiatus
in sensory acquisition depends greatly on the accuracy of control maintained
at the lower level. Where does that come from, and where does it go?
We come back to the fundamental basis of PCT. Why is it necessary, and is
it sufficient? Let's take two limiting possibilities for how a world might
be. Firstly, consider a predictable world. PCT is not necessary, because
the desired effects can be achieved by executing a prespecified series of
actions. No information need be acquired from the world. From the world's
viewpoint, the organism is to some extent unpredictable, so the organism
supplies information to the world. How much? That depends on the probabilities
of the various plans as "perceived" by the world.
At the other extreme, consider a random world, in which the state at t+delta
is unpredictable from the state at t. PCT is not possible. There is
no set of actions in the world that will change the information at the
sensors.
Now consider a realistic (i.e. chaotic) world. What does that mean? At
time t one looks at the state of the world, and the probabilities of the
various possible states at t+delta are thereby made different from what
they would have been had you not looked at time t. If one makes an action
A at time t, the probability distributions of states at time t+delta are
different from what they would have been if action A had not occurred, and
moreover, that difference is reflected in the probabilities of states of
the sensor systems observing the state of the world. Action A can inform
the sensors. PCT is possible.
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 (remember, we are dealing with observations and
subjective probabilities, not frequency distributions--none of this works
with frequentist probabilities; not much of anything works with frequentist
probabilities!). Information is lost as time goes by, at a rate that can
be described, depending on the kinds of observations and the aspect of the
world that is observed.
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. But in the chaotic world, simple observation of the CEV
provides a steady stream of information. The Actions must provide the same
information to the world, so that the perception no longer provides any
more information. Naturally that is impossible in detail, and the error
does not stay uniformly zero. It conveys some of the information inherent
in the chaotic nature of the world, though less than it would if the Actions
did not occur. The Action bandwidth determines the rate at which information
can be supplied by the world, the nature of the physical aspect of the world
being affected, and the delta t between Action and sensing the affected CEV
determines the information that will be given to the sensors (the unpredicted
disturbances, in other words), and the bandwidth of the sensory systems
determines how much information can be provided through the perceptual
signal. Any one of these parts of the loop can limit the success of
control, as measured by the information contained in the error signal.
So far, the matter is straightforward and non-controversial, I think.
Think of a set of orbits diverging in a phase space. The information
given by an initial information is represented by a small region of
phase space as compared to the whole space. After a little while, the
set of orbits represented by the initial uncertainty has diverged, so the
uncertainty has increased. Control is to maintain the small size, which
means to supply information to the world.
Things become more interesting when we go up a level in the hierarchy.
Now we have to consider the source of information as being the error
signals of the lower ECSs, given that the higher level has no direct
sensory access to the world, and that all lower ECSs are actually controlling(both restrictions will be lifted later, especially the latter). Even
though the higher ECSs may well take as sensory input the perceptual
signals of the lower ECSs, nevertheless the information content
(unpredictability) of those perceptual signals is that of the error, since
the higher ECSs have information about their Actions (the references suppliedto the lower ECSs) just as the lower ones have information about theis
Actions in the world. The higher ECSs see a more stable world than do
the lower ones, if the world allows control. (Unexpected events provide
moments of high information content, but they can't happen often, or we
are back in the uncontrollable world.)
What does this mean? Firstly, the higher ECSs do not need one or both of
high speed or high precision. The lower ECSs can take care of things at
high information rates, leaving to the higher ECSs precisely those things
that are not predicted by them--complexities of the world, and specifically
things of a KIND that they do not incorporate in their predictions. 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. ...
Look, for example, at the attention and alerting discussions, which
come absolutely straight from the Shannon theory. 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. had I known
about PCT then, I could have made a much stronger case than I did, but only
because of Shannon.
---------------------
[Martin Taylor 921218 19:45]
Bill to Tom Bourbon (921218.1438 CST) --
In the past, both of us wondered how, specifically, Shannon's
ideas, or any of the major concepts from information theory,
would improve any of the quantitative predictions we make with
our simple PCT models.
This is the right question about information theory -- not "does
it apply?" but "what does it add?" The basic problem I see is
that information theory would apply equally well to an S-R model
or a plan-then-execute cognitive model -- there's nothing unique
about control theory as a place to apply it. Information theory
says nothing about closed loops or their properties OTHER THAN
what it has to say about information-carrying capacity of the
various signal paths.
You are right, but that "OTHER THAN" is a pretty big place to hide very
important stuff. I had not previously realized that you wanted me to use
Shannon to differentiate between S-R and Plan-then-execute. I think I did
incidentally make that discrimination in my posting in response to the
same posting by Tom. At least I think I showed how applying Shannon
demonstrated that neither S-R nor Plan-then-execute could be viable. But
we knew that already, so I didn't play it up.
------------------------
[Bill Powers (921219.0130)]
Martin Taylor (921218.1830)
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.
--To which I replied:
[Martin Taylor 921221 12:00]
The word "should" seems to be ambiguous. It refers in my posting to the
results of having a good, properly functioning ECS. In your comment, you
take it to refer to how a functioning ECS is to be designed, and that the
perceptual bandwidth should be low. If the perceptual bandwidth is low,
then the ECS will have difficulty matching the perceptual signal to the
reference signal, and thus the error signal will have high information
content. Now it is true that if the perceptual signal has lower bandwidth
than the reference signal and the same resolution, then the error signal
will in part be predictable, thus having lower information content than
would appear on the surface. But I had the presumption that we are always
dealing with an organism with high bandwidth perceptual pathways, so I forgotto insert that caveat.
Well, given last year's experience, I didn't expect my information-theory
posting to be understood, and I wasn't disappointed in my expectation. Is
it worth trying some more? I'll give it a little shot, and then give up if
it still doesn't work (rather like CSG papers trying to get into conventional
psychology journals, isn't it!).
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.
The statement is completely independent of what is acting on or looking at
the world. It has to do only with the rate at which the world supplies
information that can be looked at. Of course it matters "a heck of a
lot more to a plan-then-execute model than it does to a control model."
Didn't I demonstrate that adequately in my posting?
-------------------------
[Martin Taylor 930325 11:30]
There are several reasons why the discussion of information in PCT has
explored so many blind alleys, but apart from the differences among
participants in the definitions of words, the most significant one is
in a general non-realization that we have been taking multiple views
simultaneously.
The "inside view": The result of applying the perceptual input function
to the set of sensory inputs--i.e. the perceptual signal. Other, but
different inside views could be at the ouput of the reference signal
collector function, the comparator, or even the output function. Any
one of these inside views is (not "represents") a scalar signal that
has a single momentary value that is a number. It exists, in and of
itself, representing, from the inside view, nothing at all. An inside
viewer recording the signal can see only a waveform, which according
to Fourier and subsequent mathematicians, can be represented exactly
by a discrete set of numbers. These numbers represent the waveform,
and are the most extreme "representative" view that can be taken from
inside the ECS.
The "outside view": from outside, any aspect of the universe can be seen
and incorporated into some analysis. This includes the signals within
the ECS, the state of the CEV that is defined by the Perceptual Input
Function (PIF) of the ECS, and the actions of any disturbing variables.
An outside view can incorporate all the structure of the hierarchy. It
is undetermined what is and what is not in a generic "outside view,"
though restricted outside views can be postulated for the purposes of
specific discussions.
Now, using those quasi-definitions, we can perhaps see why there has been
an argument. It is transparently obvious that from an inside view the
perceptual signal, being scalar, cannot distinguish between two independent
effects upon it. There is nothing there but a unidimensional waveform
(ignoring for the moment the fact that the successive independent samples
do define a multidimensional space). Nothing in any sample can indicate
more than that it has "that" value. From this viewpoint, it is
straightforwardly correct that "there is no information about the
disturbance (or disturbing variable) in the perceptual signal."
...
The situation is different if we take a full-blooded outside view of the
action of a CEV. It is from this kind of view that we argue that the
disturbance provides information that passes through the perceptual
signal to the output signal. From the outside we can see the disturbing
variable do whatever it does to affect the CEV, and we can see the ECS
modifying its output to bring the perceptual signal back to its controlled
value. From outside we can see the reference signal of the ECS changing,
and the ouput changing to move the CEV so that the perceptual signal comes
to its new controlled value. From outside, the arguments about there
being no information from the disturbance in the perceptual signal lose
their force.
---------------------------
[Allan Randall 930325 12:40] to Rick Marken
>Are we also agreed that this disturbance, while defined in this
>external point of view, is nonetheless defined in terms of the
>CEV, which is defined according to the internal point of view?
Say what? Why not just say CEV(t) = d(t) + o(t). If that's what
the above sentence means then I agree with it.
The point is that the disturbance d(t), if separated out from o(t),
is not a meaningful quantity to the ECS. It is meaningful only to the
external observer. By drawing an arrow marked d(t) you are talking
about something the ECS has no direct access to. From the perspective
of the ECS, only the variation in the CEV matters. It cannot separate
out its own output from the disturbance. On the other hand, this
disturbance is defined in terms of the CEV, since only things in the
world that affect the CEV can be said to be disturbance.
---------------------
[Rick Marken (930325.1100)]
Allan Randall (939325)
Now, if we preprogram P into the hierarchy and consider
it to be part of the language and not the program, then we have a
minimal program length of H(D | P,h). With P, the disturbance can be
generated at the output with far fewer additional program bits than
without P. I.e.: H(D | h,P) < H(D | h). The percept contains information
about the disturbance.
NOW WE ARE GETTING SOMEWHERE.
All you have to do now is show that your statements above are true. Show
me the minimal program required to generate D and the minimal program
required to generate D|P (are there some rules about what constitutes
a program step?). I predict that you will need EXACTLY the same size
program to generate D (given nothing) and to generate D|P. That is,
I predict that H(D|h) = H(D|P,h). If H(D|h) - H(D|P,h) is a measure
of the information about D in P then I claim (and you can now prove
me wrong) that the information about D in P is precisely zero.
---------------------
--To which, Allan posted an analytic rebuttal in the form of a thought
--experiment, concluding that
"your claim [that there is no infoprmation about the disturbance in the
perceptual signal] can only be correct if the disturbance can be
reproduced "nearly 100% perfectly" at the output with a completely
cut-off perceptual line. This *is* theoretically possible, if the
environment is completely predictable. But this is obviously not
the case in the real world. As far as I can see, Rick, you are forced
by your claim to conclude that the system can control blind."
----------------------
--[Jeff Hunter (930329-A)] also weighed in with a quite different proof
--inspired by Avery Andrews, and treated the discovery of d(t) from
--p(t) as a problem in cryptography, finishing with:
The conclusion is that in this special case I can find
almost complete information about d(t) solely from p(t), the form
of the ECS, and the assumption that no-one is manipulating d(t)
and r(t) and the internal noise in an attempt to spoof me.
Thus the statement "there is NO information about d(t)
in p(t)" is disproved.
----------------------
--It having been proved on two quite different grounds that under some
--conditions d(t) could be accurately reconstructed from p(t) along with
--other non-varying data, Rick again stated (930329.1500)
If, however,
the statement means "it is not possible to reconstruct the disturbance
given the perceptual input in a tracking task" then your criticisms of
my experiment are not germain and the results of the experiment do
provide strong evidence that there is, indeed, no information about the
disturbance in the perceptual input.
--and (here I quote the entire posting):
[From Rick Marken (930330.1130)]
Allan Randall (930330.1330 EST)--
I was hoping that we could agree on the outcome without the need to
actually do the experiment.
A very common sentiment when the cause-effect crowd meets the PCT
loonies. Hang in there; you're in for some BIG surprises when you
start to run the experiments.
Showing that H(D|P) = 0 is trivial and involves no exponential
growth.
Well, I think you'll find showing this to be quite NON-trivial. Try
the experiment!
Rick's claim has been
disproved before the experiment got past the starting gate.
That would make things a lot easier for your horse, indeed. But
don't scratch me yet. Just fire up the ol' simulator and see
what she does. Ol' "Stewball" Marken might surprise you.
Best
Rick ("There's no information about disturbances in controlled
perceptions") Marken
-------------------------
********************************************************************
--***Here is the "Magical Mystery" Experiment series of postings:***
[Martin Taylor 930330 11:20]
(Rick Marken various)
(930315.1500 as an example)
"... THERE IS NO INFORMATION ABOUT DISTURBANCE IN THE PERCEPTUAL SIGNAL
CONTROL SYSTEM. This means that THE PERCEPTUAL INPUT TO A CONTROL SYSTEM
CANNOT BE WHAT CAUSES THE OUTPUR OF THE SYSTEMN (sic) TO MIRROR THE
DISTURBANCE."
Let's consider a thought experiment to test this. If I understand the
claim, oft repeated, Rick means that no function that takes as input
(a) the perceptual signal and (b) any other signal that is agreed to have no
information about the disturbance can reconstruct the disturbance, but
that nevertheless the disturbance is mirrored in the output.
I'll leave the logical problem with the "mirroring" unaddressed, and
assume that Rick accepts as correct what he says, that the output
mirrors the disturbance.
In my thought experiment, I will take the ECS, and add a simple function
that takes as its input the reference signal to the ECS (which I think we
can agree has no information about the disturbance) and the perceptual
signal, which Rick CAPITALIZES as having no information abou the disturbance.Let us see whether a function can be constructed that takes these two
inputs and produces a signal that matches the disturbance. If so, I
would consider it conclusive evidence that information about the disturbance
is to be found in the perceptual signal.
------------> Signal X (which should match the disturbance)
>
mystery function M(r, p)
^ ^
> >
> > V (reference signal R(t) into ECS)
> > >
> <-------|
> V
>---------->comparator------- error = P-R
> >
perceptual output
signal P(t) function O(error)
^ |
> V
> output signal
> (accepted as mirroring the disturbance)
-------------------------------------------------------------------
If Signal X matches the disturbance, the perceptual signal must be the
route from which the mystery function M(r, p) gets the information about
the disturbance. Right?
Now let the function M be indentical to O(R-P). Signal X will then be the
negative of the output signal, which is the disturbance. The only question
here is whether O(error) is a function or a magical mystery tourgoodie. I
prefer to think we are dealing with physical systems, and that O is a function.
Therefore, information about the disturbance is in the perceptual signal,
and moreover, it is there in extractable form.
QED.
(Actually, QED is too strong, since I imagine most of us will want to
challenge Rick's claim that the output mirrors the disturbance. But that
way lies the argument to information rate, which I will pursue whether or
not Rick accepts that way out of the Q that ED.)
------------------------
[Bill Powers (930331.1030 MST)]
The diagram you gave, below, won't work:
----> Signal X (which should match the disturbance)
>
mystery function M(r, p)
^ ^
> >
> > V (reference signal R(t) into ECS)
> > >
> <-------|
> V
>---------->comparator------- error = P-R
> >
perceptual output
signal P(t) function O(error)
^ |
> V
> output signal
> (accepted as mirroring the disturbance)
If the reference signal is zero, the signal X won't mirror the
disturbance. In general, it won't. What you need is
----> Signal X (which should match the disturbance)
>
mystery function M(O,P) <-------------
^ ^
> >
> ----------------<------------------ <-----
> V (ref sig R(t) into ECS) |
> > >
> V |
>---------->comparator------- error = P-R |
> > >
perceptual output |
signal P(t) function O(error) |
^ | |
> V |
> output signal O -->-
> (accepted as mirroring the disturbance)
The output signal does NOT follow the reference signal, so must
be sensed directly.
...
In all our examples so far, the reference signal has been fixed
at 0. According to your diagram, the mystery function would then
receive the perceptual signal and a null signal, with only the
perceptual signal variations then producing signal X. This would
say that when the reference signal is zero, the perceptual signal
is the same as the disturbance, which is not true.
---------------------------
[Martin Taylor 930331 14:15]
(Bill Powers 930331.1030)
The diagram you gave, below, won't work:
----> Signal X (which should match the disturbance)
>
mystery function M(r, p)
^ ^
> >
> > V (reference signal R(t) into ECS)
> > >
> <-------|
> V
>---------->comparator------- error = P-R
> >
perceptual output
signal P(t) function O(error)
^ |
> V
> output signal
> (accepted as mirroring the disturbance)
If the reference signal is zero, the signal X won't mirror the
disturbance. In general, it won't.
Have another look. There are two inputs to function M. But there is
only one input to function O. The input to function O is (P-R).
M is defined as equivalent to O except that it incorporates (R-P)
as a first stage. It is functionally identical to O(R-P). If you
want to be even more general about it, you can remove the requirement
for odd symmetry in O, and define M as -O(P-R).
Why does the value R = 0 have any special quality? In what way will
X not mirror O?
Notice that I never claimed O mirrors the disturbance. That's Rick's
claim (I don't remember you making it without the necessary simplifying
assumptions). The diagram and thought experiment is intended to show
the inconsistency of simultaneously maintaining the two claims:
(1) The output mirrors the disturbance,
(2) There is no information about the disturbance in the perceptual signal.
This would
say that when the reference signal is zero, the perceptual signal
is the same as the disturbance, which is not true.
Neither is it claimed. Mystery function M is not a unity operator.
It has every characteristic that the output function has, including
any leaky integrators and nonlinearities. It has access, apart from
the specific inputs described in the figure, only to that information
available to the output function.
Function M simply takes the reference signal, which no-one has claimed
to contain information about the disturbance, together with the perceptual
signal, which some people have claimed to have no information about the
disturbance, and reproduces a mirror image of the output signal, which
some people have claimed reproduces the disturbance exactly.
There are three possibilities: (1) The reference signal contains information
about the disturbance; (2) The perceptual signal contains information
about the disturbance; (3) The output signal does not mirror the
disturbance, and is uncorrelated with the disturbance. At least one of
these three must be true. Rick's claim is that all are false.
...if we can recover the disturbance exactly out of the perceptual signal
by a demonstrable mechanism, there seems little need to fluster about
the abstractions of information theory, which are often misunderstood.
...the current value of the disturbance
is not recoverable from the current value of the perceptual signal. But
then, I know of nobody who thought it was. That's red herring that
Rick keeps asking Allan and me to accept as a fish fresh for cooking.
---------------------------------
--Rick liked this experiment when he was quite sure it wouldn't work the
--way I expected it to. NOTE, PLEASE, THAT THE ONLY CLAIM WAS THAT IF THE
--OUTPUT IS RELATED TO THE DISTURBANCE, THEN THE PERCEPTUAL SIGNAL IS
--SHOWN TO CONVEY INFORMATION ABOUT THE DISTURBANCE. If the output fails
--to relate to the disturbance (bad or no control), no claim is made.
[From Rick Marken (930331.0800)]
Martin Taylor (930330 11:20) --
Let's consider a thought experiment to test this.
EXCELLENT THOUGHT EXPERIMENT! Now let's do it as a simulation,
shall we?
If I understand the
claim, oft repeated, Rick means that no function that takes as input
(a) the perceptual signal and (b) any other signal that is agreed to have no>information about the disturbance can reconstruct the disturbance, but
that nevertheless the disturbance is mirrored in the output.
I'll buy it.
I'll leave the logical problem with the "mirroring" unaddressed, and
assume that Rick accepts as correct what he says, that the output
mirrors the disturbance.
What's the "logical" problem with the "mirroring"? You can look at
the data from our tracking experiments and see that o = -d to within
a few pixals throughout an experimental run. When both o and d are
measured in screen units, the time traces of these two variables will
be symmetrical about a line corresponding to the fixed screen position
of the target. I call this characteristic of the graph "mirroring".
In my thought experiment, I will take the ECS, and add a simple function
that takes as its input the reference signal to the ECS (which I think we
can agree has no information about the disturbance) and the perceptual
signal, which Rick CAPITALIZES as having no information abou the disturbance.
Excellent! For simplicity, let's make the reference signal a constant
when we simulate your model. But a variable r will work too -- just
trying to keep it simple.
Let us see whether a function can be constructed that takes these two
inputs and produces a signal that matches the disturbance. If so, I
would consider it conclusive evidence that information about the disturbance>is to be found in the perceptual signal.
OK!
------------> Signal X (which should match the disturbance)
>
mystery function M(r, p)
^ ^
> >
> > V (reference signal R(t) into ECS)
> > >
> <-------|
> V
>---------->comparator------- error = P-R
> >
perceptual output
signal P(t) function O(error)
^ |
> V
> output signal
> (accepted as mirroring the disturbance)
-------------------------------------------------------------------
If Signal X matches the disturbance, the perceptual signal must be the
route from which the mystery function M(r, p) gets the information about
the disturbance. Right?
Right!! I completely agree with your proposal as diagrammed above.
I think a good first candidate for M(r,p) would be the function
O(r,p), right? Ah, I see you think so too:
Now let the function M be indentical to O(R-P). Signal X will then be the
negative of the output signal, which is the disturbance.
It is at this point that experience will triumph over the "obvious"
conclusions of your thought experiment. I think it's time to fire up
the simulator; really!
The only question
here is whether O(error) is a function or a magical mystery tourgoodie.
Your magical mystery tour will really begin when you run the simulation!
I
prefer to think we are dealing with physical systems, and that O is a
function.
Therefore, information about the disturbance is in the perceptual signal,
and moreover, it is there in extractable form.
QED.
And a right excellent proof i'tis. Now try the simulation.
Best
Rick "There is no information about the disturbance in controlled
perception" Marken
----------------------------
[From Bill Powers (930331.1430 MST)]
Martin Taylor (930331) --
Rick is right. Simulate your proposed setup with Simcon and see
what happens. It is not what you say happens. The signal X does
not reproduce the waveform of the disturbance.
-------------------------------
--So, both Bill and Rick accepted the conditions of the experiment,
--and claimed that it would not turn out as I expected.
[Martin Taylor 930331 18:15]
(Rick Marken 930331.0800)
Rick accepts my thought experiment, and suggests doing a simulation, which
he suggests will not turn out as I claim. Seems reasonable, though I
am not clear how they could come out different.
----------------------
[RIck Marken (930331.2100)]
no information about the disturbance
can be recovered from the perceptual signal all by itself (or, as
you will see from the simulation, with the help of the output function).
Besides, if we can recover the disturbance exactly out
of the perceptual signal by a demonstrable mechanism, there seems little
need to fluster about the abstractions of information theory, which are
often misunderstood.
This is a very helpful way to formulate the problem; it let's us get
past definitions and into working simulations of control systems.
...
Your method does not use knowledge of o to extract
the information about d from p -- so I consider it quite a fair method.
It does use information about the output function (that transforms error
into the variable that affects the perception). But I think you imagine
that it is the output function that extracts (recovers) the information
from p. In fact it doesn't -- but that's what the simulations will show.
-------------------------------
--At this point Bill P produced code that showed the correctness of my
--prediction of 921218 that the correlation between perception and
--disturbance would be very low if control was good, but he used it to
--claim
--
-->d vs sp: -0.03156 /* no info about dist. in percept. signal*/
--
--which is an incorrect comment based on the success of a prediction that
--required that info about the disturbance was passed through the perceptual
--signal.
------------------------------
[Allan Randall (930401.1700 EST)]
Before doing my own experiment to compare H(D) and H(D|P), I've done Martin's
"Mystery function" experiment, since it is the simpler of the two. It is
based on the Primer code of Bill Powers. It is similar to the first example
used in the Primer, except that it runs for 50 iterations, with a
disturbance of 10 introduced over iterations 5 to 14, and a disturbance
of -6 introduced over iterations 26-29. The reference is 20.
Following is the C code and the results. You can see that the function
Mystery() uses only the percept and reference. Note that the mystery
function produces the negative of the output (compare the qo and the qX
columns). On the assumption, which Rick Marken has agreed to, that the
output contains almost 100% of the information about the disturbance, then
the percept must also contain this information.
So, if Rick is going to continue to insist that there is no information
about the disturbance in the percept, he will have to reject his claim that
the output has almost all of this information. These two claims are
logically inconsistent, which can be seen quite clearly in this
simulation.
...Martin's experiment has
an advantage over mine in being simpler and more intuitive. Mine has the
advantage of being based directly on the mathematics of information theory.
But in both cases, the functions being computed (Mystery() and H(D|P)) are
exact replications of the original closed-loop output. This is why the
claim that disturbance cannot be extracted from the percept, but *can* be
extracted from the output seems *so* contradictory to Martin and me. It
hardly seems like these experiments should be necessary.
------------------------------------
--Now, having seen that the results DID, contrary to prediction, conform
--with "the perceptual signal carreis information about the disturbance"
--we find Rick and then Bill discovering that they shouldn't have accepted
--the conditions of the experiment.
[From Rick Marken (930401.1800)]
Allan Randall (930401.1700 EST) --
Before doing my own experiment to compare H(D) and H(D|P), I've done Martin's
"Mystery function" experiment, since it is the simpler of the two.
Hooray! Too bad Martin said today that it doesn't test the question
of whether there is information about the disturbance in perception.
But I'll stick with it.
On the assumption, which Rick Marken has agreed to, that the
output contains almost 100% of the information about the disturbance, then
the percept must also contain this information.
If I made that assumption then I was wrong. What I have learned from
running these simulations is that when information is "blocked" by
the closed loop, then it is "blocked"; NO INFORMATION GETS THROUGH
TO THE OUTPUT.
...
The opposition of o to d is not the result
of information being transferred from d to o. It is (as I HAVE said)
a SIDE EFFECT of the operation of the closed loop. The bottom line --
a control system LOCKS INFORMATION OUT THE SYSTEM.
So, if Rick is going to continue to insist that there is no information
about the disturbance in the percept, he will have to reject his claim that
the output has almost all of this information.
Absolutely right! I am going to continue to insist that there is no
information about the disturbance in controlled perception so I hearby
RECANT and REJECT any claim I ever made that the output has ANY (let alone
most of the) information about the disturbance. The output (like Martin's
mystery function) has NO information about the disturbance.
---------------------------------------
[From Rick Marken (930404.1200)]
Rather than try to answer Martin's and Allan's last posts, let me
just try to explain again EXACTLY what I mean when I say "there is
no information about the distrubance in controlled perceptual input".
I mean that given a string of values of perceptual input there is
NO WAY to reconstruct with ANY confidence the disturbance that was
present when the perception was occurring. I consider it unfair to
use, as part of the process of extracting d values from p values, any
information that the control system itself could not possibly have . I am
willing to assume that the system can know r, the value of its reference
signal; I'll even allow that it can know O(), the output function (I'm
being REAL liberal here), and, of course, it can know p (and any function
thereof). What it can't know is d (of course), anything about d (like when
it starts or stops).
If there is information about d in p THAT IS RELEVANT TO THE
OPERATION OF A CONTROL SYSTEM then the control system MUST be able to
extract that information knowing only what the control system itself can
know.
-------------------------------
--(Which is, of course, what had been demonstrated)
-----------------------------------
[Martin Taylor 930402 11:10]
(Rick Marken 930404.1030) (day after tomorrow; good predictive control)
It would not be possible to use my thought experiment to test the
experimental question as to whether there is information from the
disturbance in the perceptual signal, because we have no access to
a perceptual signal or the related reference.
Well, I guess that does it. If you won't accept your own proposal --
the one you already accepted as a test of this question -- then I'm
sure you won't accept any of ours.
Your argument was about models, as was and is ours. My comment you quoted,
as you well know, is about processes in a living control system. I
proposed and Allan executed the experiment in a simulation, and proved the
point that you cannot jointly say that the output contains information
about the disturbance but the perceptual signal does not.
Up until today (or rather two days from now), you have said that the
disturbance and the output correlate 0.99xxx in many many postings. Now,
to retain your position that there is no information about the disturbance
in the perceptual signal, you have suddenly shifted to the position that
the correlation between disturbance and output is 0.00. There is experimental
fact, determined by you and acceptable as being a scientific fact even
to Bill Powers, that the 0.99+ figure is often correct, and 0.95+ usually
correct.
You can't have it both ways. Either the disturbance and the output are
correlated, as experiment and model both show, or they are not, which you
now claim in an extreme resistance to the simulation demonstration. I
find this resistance mind-boggling.
....
3) The correlations between p vectors in two closed loop tracking
experiments can be nearly 0 while the output vectors are perfectly
correlated (.99+) with each other and the disturbance.
Yes, that is as one would expect, given the slightest difference in
initial conditions of the runs. However, the correlations of p1 and p2
(different runs) do differ from zero at points where the disturbance
makes a rapid excursion. p1 and p2 do track moderately well over a
short period after such an excursion.
4) Simulation of Martin's Mystery function shows that the x vector
equals d vector only when p=o+d, r=o and there is no error input
to the output or to teh Mystery function.
Not so. P = O + D was a given because that is what has always been
claimed as a basic property of the loop. The other conditions are
irrelevant. The simulation was to show that X=O, not that X=D. Provided
O and D are correlated, X has information about D, but it is not equal
to D unless the correlation of O and D is 1.000. On our side, no claim
was made about the relation between O and D or between X and D. We relied
on you for that. In a scientific argument, you are allowed to change your
claims about the properties of systems, given new evidence. But you can't
change them just to retain an inconsistent set of beliefs.
----------------------------------------
[Allan Randall (930402.1000 EST)]
Rick Marken (930401.1800)
> On the assumption, which Rick Marken has agreed to, that the
>output contains almost 100% of the information about the disturbance, then> >the percept must also contain this information.
If I made that assumption then I was wrong. What I have learned from
running these simulations is that when information is "blocked" by
the closed loop, then it is "blocked"; NO INFORMATION GETS THROUGH
TO THE OUTPUT.
Yes! This was exactly the entire point of Ashby's Law. Ashby defined
control as the blockage of disturbance information. This is the
whole idea that I have been pushing from square one.
--At this point, Rick produced sequences of perceptual values and started
--to claim that we were proposing to deduce the values of disturbing variables
--without access to the output and feedback functions. The discussion has
--sustained itself on this degenerate plane ever since.
--And that's why I am so hesitant to try to launch back into a serious
--discussion of the relationship between information theory and PCT on
--CSGnet. There's lots of interesting stuff to discuss without getting
--into what can only be a mutually frustrating miscomprehension.