[Martin Taylor 2009.04.09.00.33]

[From Bill Powers (2009.04.06.1547 MDT)]

Martin Taylor 2009.04.06.14.11] –

Sorry, but you’ve left

out a

vital function, the environmental feedback function in that path that

loops from Output O to S2. S2 is not equal to O, but to some function

g(O).Could that be why I said: “

S2 is a function of O”? I

admit I called the function “f” rather than

“g”, but I didn’t think that distinction was as

important as you seem to make it. “A rose by any other name would

smell as sweet.”You said that, but you didn’t include it in your equations.

I don’t understand why you say that. I went back to my original message

to check, and there it was.

S2 depends on

O, so everywhere you see S2 you need to substitute f(O) (or g(O) if you

like). This emphasizes the point that you don’t know the form of that

function, either, so you’re stymied.

I think you are missing the point more than a little. The point is that

the perception p is being controlled. The variable p is a function of

two inputs, which we call S1 and S2, so if control were perfect, the

trace of S2 vs S1 would follow a simple curve of constant p in the

S1-S2 space, a different curve for each different reference value for

p. Since S2 is a single-valued function of O, the same is true for the

trace of S1 versus D in the S1-D space under the condition of constant

p. We may not know the form of any of these functions, but we can

determine quite a bit about them, especially when we set things up so

we can analyse differences. Everything you say, above or below, applies

equally when you are modelling a pursuit tracking experiment, because

the formal setup is the same.

When I first learned golf, one important trick everyone had to know was

how to get over a stymie without damaging the green. So many greens

were damaged by people who didn’t do it very well that the rule was

changed, and now golf has no stymies. Maybe we could change the rules

here, so that they apply equally to all experiments we might consider,

rather than damaging the “green” of the discussion group by being

applied selectively only to experiments I mention as being interesting.

The observed

relationship of O

to S1, therefore, will be determined by the reference level R and the

inverse of the environmental feedback function.Yes. Do you think that might possibly be why I said: “

The form of”

the function relating S2 to S1 is controlled by the form of p(S1, S2).

For example, if p == (S1 * S2), the answer would be quite different

than

it would if p == (S1-S2). Under some circumstances, we may be able to

assume the form of p, or test it by modelling the control

loop.I do admit that I did not mention the possibility of varying reference

levels.But when you say “is controlled by the form” you’re alluding to

a relationship without spelling it out. The result is that you’re

talking

about a mathematical derivation without actually producing it. I want

to

see the derivation itself, not statements about what it might

demonstrate. I have no idea how you arrive at your conclusions, and

showing the derivation might show that.

I did explain what I meant by “the form of p(S1,S2)”. Since that could

be anything at all, it’s quite impossible to specify. “The Test” can

approximate what that form might be in any particular case (I’ll have

things to say about “The Test” at some later time, when this is all

sorted out). “The Test” is one element of modelling the control loop,

anyway. So I think that in the part you quoted I covered your

objection. I specifically mentioned two possibilities p == (S1-S2) (the

pursuit tracking and the answer-matching cases) and p == (S1*S2)

(controlling an area). I could add the possibilities Rick tested in his

size study, and a myriad of others, but how would that affect anything

about the argument?

It will not indicate a

property

of the forward path through the organism.Which is not under examination. What is under examination is S1 =

f(D).That is part of the forward path through the organism. You can’t

“examine” that path because you can observe only one end of

it.

Aaargh!!! Isn’t the whole point of the demonstration to show how you

can? I really don’t care for critiques of a proof that X = Y that

consist simply of the assertion “you can’t say X = Y”. That’s rather

like Rick’s dismissal of all methods of studying the human other than

“The Test” for the controlled variable. You are saying that you can’t

tell whether something is hot or cold because you can only see a

reading on a thermometer and that’s not a feeling of warmth or cold.

Well, if you want to make that kind of argument, there’s little more to

be said, I think.

Because this is a

control loop

for which we assume an invariant reference value set by the

experimenter,

if control is perfect, thenp(S1.0, S2.0) = p(S1.1, S2.1), which is equivalent to p(S1.0, S2.0) -

p(S1.1, S2.1) = 0.What we are interested in is not really the transfer function between

S1

and O, but the one between the disturbance D introduced by the

experimenter and S1. We want to know whether we can deduce anything

about

this transfer function by observing the relationship between variations

in D and variations in O. In conventional psychology, D is the

independent variable (IV) and O the dependant variable

(DV).This is where the problem lies for me. What you deduce about the path

from D to S1 depends entirely on the kinds of functions you assume to

lie

in that pathway. You haven’t drawn any box between the disturbing

variable and S1, but that’s what we’re talking about

here.Yes, that’s what the experiment is presumed to be wanting to determine.

It’s why I said: “if we can determine the form of p by other.”

experiments (such as by allowing the subject to influence both S1 and

S2,

for example, and modelling the control loop), then we could determine

the

function D -> S1I think we are using the word “determine” very differently

here. If the function S1 -> O is postulated, we can then measure the

function D -> O and by dividing the second function by the first,

arrive at an expression for the function D -> S1. But we can’t claim

to have determined the actual functionD -> S1 because that function depends entirely on what function

we assumed for S1 -> O. We can make it whatever we want it to be,

just

by changing what we assume about S1 -> O.

Is that really what you mean by “modelling the control loop”? Making an

assumption? Don’t “other experiments” and “modelling” have a little

more behind them than that?

By the way, just when did the object of the study become necessarily

“to have determined the actual function D -> S1”? I guess one could

try to do that, but I don’t remember ever suggesting it as the object

of any study I have mentioned. You and Rick did claim to have done

that, I believe, in respect of Steven’s power law, but it’s not

something I would normally want to do.

Also, notice what is

observed

after D: nothing, until we get to O. So the measurements you’re making

could include any function between D and O, and might well include more

than one function. You can imagine any number of arrangements inside

the

system, including a simple S-O-R connection – and that is clearly what

traditional psychologists often assumed. There is nothing to say that

it

is a relationship of the disturbing variable to the output that is

being

controlled. You can’t be sure of that without doing some tests

specifically to see what is being controlled, and at what reference

level. You know what you told the subject to do, but that is not

necessarily what the subject is doing. Only the proper tests can show

what the subject is actually doing.Yes. Is that contradictory, confirmatory, or irrelevant to what I

wrote?

I think it is a restatement of: "_Knowing that O is a function of

D, what can we say about S1 as a function of D? That depends on what we

can determine about the control loop. The form of the function relating

S2 to S1 is controlled by the form of p(S1, S2). For example, if p ==

(S1

- S2), the answer would be quite different than it would if p ==

(S1-S2).

Under some circumstances, we may be able to assume the form of p, or

test

it by modelling the control loop. If we can determine the form of the

function that relates O to S1, and the function that relates O to D,

then

we can determine the function D -> S1, which is what we wanted to

know._"You’re assuming that we can observe S1 and S2. We can’t.

I made no such assumption. I made the contrary very clear, or so I

thought.Then how can we “determine” anything about S1 and S2? We don’t

know the form of the function relating S1 and S2, …

until we perform “The Test” to estimate it … (and “estimate” or

“approximate” is the best the Test can ever do). Since the function

relating S1 and S2 is derived from the function that creates the

controlled perception, we either have to assume it (as is reasonable

when we are doing pursuit tracking or answer matching, when p = (S1-S2)

or (S2-S1)), or Test for it. Notice that when we are doing a pursuit

tracking experiment, there is no way to distinguish between perfect

tracking of a changing reference level, imperfect tracking of a fixed

reference level, or anything between. The problems are exactly the

same, because the setup is exactly the same.

and we don’t know the

values of S1 and S2, …

until we experiment and model the experimental results …

so … well, I think we’re suffering a severe

difference in the way we use language. You speak in ways that make me

think you’re talking about verifying the model by comparing its

behavior

with real behavior, …

Yes, that’s what I’ve been talking about…

but then you claim to know things as if you could

measure the real counterparts of the modeled variables.

When?

I am beginning to think that when I think you’re talking about the real

system, the one we’re trying to model, you’re really talking only about

the model. Here it is in a nutshell: you say "If we canand my immediate response is to think

determine the form of the function that relates O to S1, and the

function

that relates O to D …"

“But you can’t determine the function that relates O to S1 because

you don’t know the function that relates D to S1 and the only

measurement

you have is the relationship of D to O.”Both of those imagined functions exist inside the organism and you

can’t

see either of them. The best you can do is assume a form for one of the

missing functions and see what that implies about the other one. That

means you’re working entirely with the imagined model, where you can’t

“determine” anything. All you can do is see what the logical

consequences of making assumptions are.

There are three answers to that.

(1) in the most general case, you are quite correct, and would be

equally correct when talking about a pursuit tracking study. That’s why

we see whether the logical consequences of the assumptions are

consistent with the observations – the data, in experiments. That’s

true of all modelling based on observations of what happens outside the

physical boundary of the organism. It’s one reason why I believe

Bayesian analysis is the most appropriate way to approach the relation

between theory and experiment.

(2) Very often (as is the case in the pursuit tracking study, and in

many detection and discrimination experiments) the experiment is set up

so as to make it highly likely that the function relating the sensory

input to S1 is very similar to the function relating sensory input to

S2. Furhtermore, when both S1 and S2 are category perceptions

themselves, the function generating the category is hardly relevant to

whether the categories are the same or different. In the case of a

discrimination experiment, one of them is derived from a current

sensory input, the other from a past instruction by an experimenter.

That does not affect the control of p(S1, S2), where p(x,y)==(x-y),

using a reference value of zero for the difference between the

categories X and Y.

(3) When (2) is not the case, one can often make reliable differential

estimates, without knowing the forms of the functions. From them, if

such is desired, one can determine certain aspects of the functions

themselves. That’s what you do when modelling tracking experiments. In

the case of detection and discrimination experiments the experiment is

deliberately designed so that only differential measures are available

(was the presentation class 1 or class 2?). When you and Rick did your

re-analysis of Stevens’s power functions, you had to make some

assumptions, and found that if S1= f(sensation) had a logarithmic form

regardless of the sensation, the results of the experiments would be

power functions. But the same would have been true for an infinite

number of other input functions, within experimental error. That

doesn’t in any way invalidate the results of your analysis, since it is

consistent with the data. In the case of a detection or discrimination

experiment, the differential inputs D1 and D2 allow for an estimate of

the precision of the channel D->S1. If the perception of S2 were

also uncertainly derived from the initial instructions (the subject

wasn’t quite sure how to react), the results would put a lower bound on

that precision (as indeed they do in any case, since one never knows

whether there are noise sources one has not incorporated in the

analysis, such as hitting the unintended button).

All we can observe are

D and O,

and the form of the environmental feedback function without the input

perceptual function that generates S2.## Apart from the dubious question of whether we can observe the

environmental feedback function – we may, but we may have to model itthat was my starting point.

Yes, but you didn’t seem to stick to those initial stipulations.

I thought I did. I don’t think your comments demonstrate the contrary,

though your language suggests they do.

Martin