[Allan Randall (930331.1730 EST)]

This is a response to various posts from Rick Marken and Bill Powers.

Rick Marken (930329.2000) (930330.1130) (930330.1500)

I think you have understood most of the experiment correctly, but a

few points need to be clarified:

Let me see if I have this right. I assume that cutting the "perceptual

line" means that the system is now operating open loop.

Yes. The system is now simply getting percepts and producing outputs.

The output no longer has any effect on the input. The whole objective

is to replicate the closed-loop result in an open loop, using only

the percept.

Also, what is the length of a perceptual

input? Are you referring to the length of the vector of values that

make up Pi?

Basically, yes. The length is the number of bits actually used to specify

the percept in the computer simulation. This is not a single value of

the percept at one moment in time, but a vector of values over a time

period.

...So if D had 1000 samples and I could recover all

1000 values by plugging in a Pi with 500 samples, then the entropy of

D is 500? Is that right?

Yes, provided that 500 is the *minimum* required. That is why we

need to start with the shortest and work up.

Is this right? ... Is P coming in via the usual closed

loop perceptual input or "for free" as an open loop input?

Open loop, for free. Your intuition here was exactly right. The

procedure for H(D|P) is identical to that for H(D) except that we are

given P for free. Both entropy calculations are done strictly open

loop, since the algorithmic definition of entropy we are using is

defined in open loop terms: input, program, language and output.

I know you

expect to need only one P0 value -- that any a P0 value of length

0 can be added to P' and maintain the ability of P' to produce O=D.

But you do have to try adding at least ONE P0 string (other than

the zero length one) to show that this is true.

I think we have a notational confusion here. Pi, *not* P0, is my

notation for an arbitrary value in the set {P0,P1,P2...}. There is

only one P0, and it is of zero length by definition. It is the Pi

values that can be of arbitrary length. Since the procedure is to

look for the shortest successful string, there is no need to try

*any* non-zero-length Pi's *if* P0 is successful. Otherwise, we

try as many as needed to find a successful contender. The procedure

is simple: try all Pi's in order of shortest to longest, and STOP

as soon as one is successful. Also, I assume your P' is the same

as my Pk - it represents the first successful Pi in an open

loop search.

Do you disagree about the results that I have assumed we would get

from this experiment?Yes. But you'll see when you actually do the experiment.

As I said in a message to Bill, I think this experiment is almost

redundant, since P' will turn out to be zero length (P0) for H(D|P),

and this is the very first Pi we will try. This part of the

experiment is essentially identical to Martin's proposal for a

"mystery function." If I am right, the rest of the experiment

will be unneccesary. If you are right, I will have to continue with

the search. So I guess I'll just have to do the experiment.

Do you understand my assumption that we will not actually be looking

for D in the open loop exponential search? Since you have already agreed

that nearly 100% of the information about D is in the closed loop

output, I have assumed it will be sufficient to look for replication

of this closed-loop output in the open loop phase (without using the

closed-loop, of course). I am trying to show your claim that the

output has 100% of the information about D and the input has 0% of it

to be inconsistent, so I think this assumption is valid for now.

Rick ("There's no information about disturbances in controlled

perceptions") Marken

Ashby's whole point was that perfect control was not possible, and

that error control relies on the detection of minor imperfections

to prevent major imperfections. This is why it's called "error

control," after all. If perfect control were possible, then yes,

there would be no information about the disturbance in the

perception. This is the goal of the control system. But it will

always fall short of this, since it controls via the detection of

imperfections. As Ashby said, perfect control would perfectly block

the very channel through which the control system gets its useful

information. Thus, error control is inherently imperfect. Its very

power lies in its inherent imperfection.

...The length of the first candidate i vector that produces

o values that match d perfectly is H(D) (There is a problem here;

What if you don't get any perfect matches to d, Allan?.Nothing

but an infinite loop gain system could produce outputs that

match d perfectly anyway: how about doing this just until a

candidate i vector produces o values that match the disturbance

to the same degree as did the o values generated in the closed loop

case.

This is basically what I did - I specified that, since we agree

that the closed-loop output has near-100% of the information about

the disturbance, then we are simply going to look for a replication

of the closed-loop output, rather than the disturbance.

Assuming you can find H(D) using this decidedly peculiar technique

(why not just measure the variance of d?)

Because what I have described, as "peculiar" as it may seem,

corresponds directly to the technical definitions you will find

in the literature on algorithmic information theory. Variance does

not. Information is defined in terms of probability, not variance.

Allan is

assuming that the output resulting from the original i vector

along with the "null" (0 length) candidate i vector will produce an

output that perfectly matches the disturbance (or, at least, matches

the disturbance as well as the output did in the original run).Is this a correct description of the experiment Allan?

Yes, exactly.

Bill Powers (930330.2000 MST)

Allan is using information about o in his method,

as I vaguely understand it now, with your help. It shouldn't be

surprising if he can also reconstruct d.

No, I am not using information about o to reconstruct d (or in this

case the original o).

By the way, I think this notation H(p|d) is not just an ordinary

function, but represents some sort of probability calculation

with base-2 logs and all that. Allen?

H(P|D) = -log prob(P|D) (see my original posting)

Rick Marken (930330.2100) responding to Bill Powers:

Allan is using information about o in his method,

as I vaguely understand it now, with your help. It shouldn't be

surprising if he can also reconstruct d.I don't think Allan is going to use information about o (the

o generated in the real, closed loop run) at all. In fact, the

way I conceive of Allan's study, you don't need to save o at

all; all you need is d and p from the closed loop run.

Your description is essentially accurate, except that I suggested

looking for the original closed-loop o, rather than an equal

d-correlation with it, as you suggest. Either strategy would

be okay as far as I am concerned. Given that we assume the

original o has 100% of the information about d, it really doesn't

matter terribly much which strategy we use.

...are you with us, Allan?

I think so. As far as I can tell, your description is accurate.

Bill Powers (930331.1030 MST) responds:

As I understood it, Allen was going to assume that control was

perfect, so o = -d with a constant reference signal. So the

procedure depends on knowing o, which amounts to knowing d.But let's wait to see what Allen says.

No, the procedure that replicates the closed-loop o in the open

loop phase uses only the Pi value, not o. The closed-loop o

value (or d if we choose Rick's version) is used only for

comparison purposes to check the result. It is not used in any

capacity at all to produce the open-loop result. I believe Rick

has understood the proposal, and so I will now attempt to

implement it. I probably will not post anything else on this

subject until I have the results.

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Allan Randall, randall@dciem.dciem.dnd.ca

NTT Systems, Inc.

Toronto, ON