Feedback Functions

[rsmarken]

[From Rick Marken (2007.09.06.1030)]

It just struck me that one of the most mathematically precise
demonstrations of the effect of a person's purpose on the results of
an experiment is the difference between the Fechner's and Steven's
laws of psychophysics. According to Fechner, the relationship between
stimulus intensity (I) and sensation magnitude (R) is R = k log I.
According to Steven's the relationship is R = I^k. I refer to
sensation as R because the measure of sensation is based on the
subject's response. In Steven's case R is an actual numerical response
to a stimulus presentation; in Fechner's case R is derived from the
proportion of times the subject correctly reports the more intense
stimulus as being more intense.

What's interesting is that these two different laws -- which can be
viewer as disturbance-output functions from a control theory
perspective -- are obtained when subjects have different purposes
(different variables they control). In the Fechner studies what is
controlled is the relationship between a binary response ("more" vs
"less" intense) and the relative intensity of pairs of stimuli
presented on each trial. In the Stevens studies what is controlled is
the relationship between a numerical response and the intensity of
single stimuli presented on each trial.

Since, according to control theory, the functional relationship
between disturbance and output is the inverse of the feedback function
connecting output to controlled variable, the difference in the
Fechner and Stevens laws is a reflection of the difference in the
feedback functions in these two cases. And the feedback function is
different in the two cases because the controlled variables and the
outputs that affect the states of these variables are different in the
two cases.

I would love to be able to describe, quantitatively, the controlled
variables and the feedback functions in the Fechner and Stevens
experiments so I could show how control theory explains the difference
in the relationships found between input (I, which is actually a
disturbance variable) and output (R) found in the two cases. But I'm
finding this a difficult task. So that's why I'm posting this. I'd
love to get some help on this. I guess the first thing is to get
confirmation that my basic premise is true -- that Fechner's log law
and Steven;s power law are just the inverse of the different feedback
functions that exist in Fechner's discrimination approach to measuring
R versus Stevens' numerical judgment (he called it magnitude
estimation) approach.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[Bill_Powers1]

It just struck me that one of
the most mathematically precise

demonstrations of the effect of a person’s purpose on the results of

an experiment is the difference between the Fechner’s and Steven’s

laws of psychophysics. According to Fechner, the relationship
between

stimulus intensity (I) and sensation magnitude (R) is R = k log I.

According to Steven’s the relationship is R = I^k. I refer to

sensation as R because the measure of sensation is based on the

subject’s response. In Steven’s case R is an actual numerical
response

to a stimulus presentation; in Fechner’s case R is derived from the

proportion of times the subject correctly reports the more intense

stimulus as being more intense.
[From Bill Powers (2007.09.06.1525 MDT)]

From Rick Marken (2007.09.06.1030) –

Long ago, I wrote a paper on this (not what you’re doing, just the
relationship between the two hypotheses) and sent it to Science.

Fechner: Rf = k log(I)

Stevens: Rs = I^k

Take the log of both sides of Stevens’ law:

     log(Rs) = k

log(I)

So Fechner is saying the response is k log(I), and Stevens is saying the
log of the (numerical) response is k log(I). From this we get

log(Rs) = Rf.

I pointed out in the submitted article that this relationship between Rs
and Rf could be completely accounted for if subjects perceived the sizes
of numbers in the same way they perceive the intensities of stimuli. I
didn’t carry it any farther than that.

I got a very snippy reply relayed by Science from Stevens saying that
“Logarithms have nothing to do with the power law” and the
paper was rejected. So the moral is, ignorance is its own justification,
if the ignorant person is famous enough.

Here is my present analysis, which seems to be irrefutable given the
premises:

Let Pi be the perception of stimulus intensity, and Pn be the perception
of the size of a number.

Suppose that Fechner is right in saying that the perceptual signal is
proportional to the log of the stimulus intensity.

Pi = k1 log(I).

Suppose, too, that the magnitudes of numbers are perceived in the same
way:

Pn = k2 log(N).

In Stevens’ experiments, the task is to generate a response N such that
Pn = Pi. The perceived size of the number is adjusted to match the
perceived size of the stimulus.

When Pn = Pi, we have k2 log(N) = k1 log(I), or

Log(N) = (k1/k2)Log(I).

The right side is equivalent to Log(I^(k1/k2)).

Using both sides for powers of 10, we have

10^Log(I^(k1/k2)) = I^(k1/k2) on the right, and

10^log(N) = N on the left, leading to

N = I^(k1/k2) (N is Stevens’ Rs)

Thus the exponent in Stevens’ power law is the ratio of the perception of
the stimulus to the perception of the number when the number is said to
be equal to the stimulus.

This can obviously be extended to all the other applications of the power
law, in which one stimulus is evaluated as N times the magnitude of
another stimulus. So Stevens was simply repeating Fechner’s law in a
disguised form. His law is the same law. The power law is Fechner’s
law.

Why don’t you write an article titled “Stevens’ Power Law is
Fechner’s logarithmic law of perception.”?

Perhaps you can see how this would relate to your larger project – I
don’t quite see the connection, but there is probably one.

Best,

Bill P.

I presume Stevens is dead by now (yes, I see he died in 1973), so perhaps
it will be true that “the advance of science is maintained by
waiting for the right funerals.”

[rsmarken]

[From Rick Marken (2007.09.06.1740)]

Bill Powers (2007.09.06.1525 MDT)--

Fechner: Rf = k log(I)

Stevens: Rs = I^k

Take the log of both sides of Stevens' law:

          log(Rs) = k log(I)

So Fechner is saying the response is k log(I), and Stevens is saying the
log of the (numerical) response is k log(I). From this we get

log(Rs) = Rf.

I pointed out in the submitted article that this relationship between Rs
and Rf could be completely accounted for if subjects perceived the sizes of
numbers in the same way they perceive the intensities of stimuli. I didn't
carry it any farther than that.

Perhaps you can see how this would relate to your larger project -- I don't
quite see the connection, but there is probably one.

Yes, I see the connection quite clearly!! Thanks.

Basically, we just assume that, in both cases, the controlled variable
is Pn - Pi, that it's reference is 0, (so the subjects goal or purpose
is to keep Pn = Pi, as you say) and that Pi is a log function of I.
The feedback function from the output (the number N) to controlled
variable (Pn-Pi, say) is Pn = k log (N) as you say. But in Fechner's
case the log relation between N and Pn doesn't effect the result
because the subject is only asked to use the numbers to classify the
Pi; 0 = no perceived difference between Pi1 and Pi2 and 1 = a just
noticeable difference between Pi1 and Pi2. Presumably, the is a
constant difference between Pi's -- that is, a critical difference, c,
between log(I[n]) and log (I[n+1] -- that leads to a 1 rather than a
0. So equal log differences of a particular size between intensities
result in a 1 (different) response rather than a 0. Accumulating the
unit responses over these log differences gives the log function R = k
log (I) where R is the value that result of adding up just noticeable
differences (jnds), each of which I am setting to 1, which is how
Fechner got his log law.

Your derivation above gives the results for Stevens power law, which
results from the fact that subjects controlled for Pn-Pi by saying a
number, N, resulting in a Pn which is a log function of N. So the
observed relationship between N and I is the inverse of the
logarithmic feedback connection between N and Pn -- the power
function: N = I^(k1/k2).

Does that seem right?

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[Bill_Powers1]

Yes, I see the connection quite
clearly!! Thanks.
[From Bill Powers (2007.09.06.1850 MDT)]

Rick Marken (2007.09.06.1740) –

And so you do! This is wonderful. Your analysis has nothing wrong with it
that I can see, and I’m very relieved to see that you understood what I
said and agree with it, as shown by your paraphrases. Your review of how
Fechner reached his conclusion using Just Noticeable Differences
completes the picture. I think this is eminently publishable.

Do we get Richard Kennaway’s blessing?

Best.

Bill P.

[Martin_Taylor2]

[Martin Taylor 2007.09.06.19.466]

[From Bill Powers (2007.09.06.1525 MDT)]
Long ago, I wrote a paper on this (not what you're doing, just the relationship between the two hypotheses) and sent it to Science.

Fechner: Rf = k log(I)

Stevens: Rs = I^k

Take the log of both sides of Stevens' law:

         log(Rs) = k log(I)

So Fechner is saying the response is k log(I), and Stevens is saying the log of the (numerical) response is k log(I). From this we get

log(Rs) = Rf.

I pointed out in the submitted article that this relationship between Rs and Rf could be completely accounted for if subjects perceived the sizes of numbers in the same way they perceive the intensities of stimuli. I didn't carry it any farther than that.

I got a very snippy reply relayed by Science from Stevens saying that "Logarithms have nothing to do with the power law" and the paper was rejected. So the moral is, ignorance is its own justification, if the ignorant person is famous enough.

Interesting. My first submission to Science, on the PEST psychophysical method we developed, was also rejected. One reviewer said it was too trivial, and any scientist would know it with with a couple of minutes thought, the other said it was too complex, and nobody would understand it or use it. Both said no, so it was rejected.

My very first presentation at a scientific meeting was on the kind of topic under discussion. It was a summer student project, while I was working on my thesis in the winter. I had Stevens in the audience, and he didn't like it one bit. I remember his comment on my answers to his questions: "It's like trying to pin a jellyfish to the wall". But he was nice afterwards, and invited me to tea with him and his wife.

My work had to do with the matching of different perceptual scales, including the magnitude of numbers. As part of my thesis work I had theorised that the magnitude of extents was non-linear, first increasing more than linearly near an anchor point (such as zero), and then less than linearly (possibly logarithmically). I used a whole range of perceptual dimensions as stimuli and as responses (the task was to set an amount of one extent that the subject felt corresponded to a presented amount of another). The dimensions I remember included visual length, orientation over a 90 degree arc, position of a nonsense word in a partly learned word list, hue (adjusted by changing the voltage on an electroluminescent panel), grey-scale value, time between light flashes, and one or two more.

I used most of the perceptual dimensions both as stimuli and as responses, because in the "stimulus" arrangement there is an extra anchor point, and the idea was to compare dimension A with dimension B when both were stimuli (responses being dimensions X, Y, Z) and when both were responses (stimuli being X, Y, Z).

The point was that the perceptual scales would be different for things viewed as stimuli and things being set as responses, becasue in the response case the subject had control over something that was fixed when viewed as a stimulus. Being fixed allowed the point to act as an anchor. Having control over it (I thought) changed the perception of it. And so it proved.

I never published this work, because the summer turned out to be too short. For the thesis, I used only the two-dimensional placement of a dot on an index card (and the most difficult part of the work was to persuade them to buy me clean 3 x 5 index cards without little blemishes which could act as anchor points -- I showed that having one extra dot on the card made a huge difference to where people placed their responses).

I was working on the principle that the perceived extent was a function not of the physical magnitude of the extent, but of its discriminability (d' for those who know what that means). I didn't and don't buy Fechner's proposal that discriminability can be determined by summing just noticeable differences (jnds). (Actually, I once published a study showing that it simply could not be correct). Theory and practice suggest a slower than linear growth of perceived magnitude as a function of physical magnitude after the near-zero acceleration. I never did buy into Stevens' power law, either.

So my work of a half-century ago probably isn't relevant to this discussion, since it amounts to saying "A plague on both your houses!" And it was never completed in any case.

You might like to look at the Wikipedia article "Stevens Power Law".

Martin

[Bill_Powers1]

[From Bill Powers
(2007.09.06.1525 MDT)]

Long ago, I wrote a paper on this (not what you’re doing, just the
relationship between the two hypotheses) and sent it to Science.

Fechner: Rf = k log(I)

Stevens: Rs = I^k

Take the log of both sides of Stevens’ law:

     log(Rs) = k

log(I)

So Fechner is saying the response is k log(I), and Stevens is saying the
log of the (numerical) response is k log(I). From this we get

log(Rs) = Rf.

I pointed out in the submitted article that this relationship between Rs
and Rf could be completely accounted for if subjects perceived the sizes
of numbers in the same way they perceive the intensities of stimuli. I
didn’t carry it any farther than that.

I got a very snippy reply relayed by Science from Stevens saying that
“Logarithms have nothing to do with the power law” and the
paper was rejected. So the moral is, ignorance is its own justification,
if the ignorant person is famous enough.

Interesting. My first submission to Science, on the PEST psychophysical
method we developed, was also rejected. One reviewer said it was too
trivial, and any scientist would know it with with a couple of minutes
thought, the other said it was too complex, and nobody would understand
it or use it. Both said no, so it was rejected.

My very first presentation at a scientific meeting was on the kind of
topic under discussion. It was a summer student project, while I was
working on my thesis in the winter. I had Stevens in the audience, and he
didn’t like it one bit. I remember his comment on my answers to his
questions: “It’s like trying to pin a jellyfish to the wall”.
But he was nice afterwards, and invited me to tea with him and his
wife.

My work had to do with the matching of different perceptual scales,
including the magnitude of numbers. As part of my thesis work I had
theorised that the magnitude of extents was non-linear, first increasing
more than linearly near an anchor point (such as zero), and then less
than linearly (possibly logarithmically). I used a whole range of
perceptual dimensions as stimuli and as responses (the task was to set an
amount of one extent that the subject felt corresponded to a presented
amount of another). The dimensions I remember included visual length,
orientation over a 90 degree arc, position of a nonsense word in a partly
learned word list, hue (adjusted by changing the voltage on an
electroluminescent panel), grey-scale value, time between light flashes,
and one or two more.

I used most of the perceptual dimensions both as stimuli and as
responses, because in the “stimulus” arrangement there is an
extra anchor point, and the idea was to compare dimension A with
dimension B when both were stimuli (responses being dimensions X, Y, Z)
and when both were responses (stimuli being X, Y, Z).

The point was that the perceptual scales would be different for things
viewed as stimuli and things being set as responses, becasue in the
response case the subject had control over something that was fixed when
viewed as a stimulus. Being fixed allowed the point to act as an anchor.
Having control over it (I thought) changed the perception of it. And so
it proved.

I never published this work, because the summer turned out to be too
short. For the thesis, I used only the two-dimensional placement of a dot
on an index card (and the most difficult part of the work was to persuade
them to buy me clean 3 x 5 index cards without little blemishes which
could act as anchor points – I showed that having one extra dot on the
card made a huge difference to where people placed their
responses).

I was working on the principle that the perceived extent was a function
not of the physical magnitude of the extent, but of its discriminability
(d’ for those who know what that means). I didn’t and don’t buy Fechner’s
proposal that discriminability can be determined by summing just
noticeable differences (jnds). (Actually, I once published a study
showing that it simply could not be correct). Theory and practice suggest
a slower than linear growth of perceived magnitude as a function of
physical magnitude after the near-zero acceleration. I never did buy into
Stevens’ power law, either.

So my work of a half-century ago probably isn’t relevant to this
discussion, since it amounts to saying “A plague on both your
houses!” And it was never completed in any case.

You might like to look at the Wikipedia article “Stevens Power
Law”.

Martin

–

No virus found in this incoming message.

Checked by AVG Free Edition. Version: 7.5.476 / Virus Database:
269.13.5/988 - Release Date: 9/4/2007 9:14 AM
[From Bill Powers (2007.09.07.0245 MDT)]

Martin Taylor 2007.09.06.19.466 –

I think our experiences with referees and famous people are part of a
good education. Hero worship and submission to authority are never a good
idea in science, and probably not anywhere else, either.

My discussion of the relation between Fechner’s law and Stevens’ law is
mainly an examination of the independence of these laws, showing that if
numbers are perceived according to Fechner’s law, then Stevens’ way of
measuring perception is entirely equivalent to the simple logarithmic
hypothesis. But I agree with your view of perception, which is that it is
probably nonlinear relative to any external counterparts, with pure
mathematical forms being only approximations of limited usefulness. And
if there is any common transformation between ALL senses and external
variables, we will never discover it this way.

I think this makes even clearer the fact that it is the environment that
is hypothetical, not our perceptions of it. One must never forget to ask
how the “true” state of a stimulus is measured. That
measurement is also a perception. When we compare subjective perceptions
with objective measurements, we are still comparing one perception with
another, and never are comparing a perception with reality. I wish
physicists were more aware of this.

The Wiki article you cite is a good reminder that these “laws”
are really just curve-fitting. It would be interesting to see the raw
data behind those tables of Stevens’ in the wiki article. How many
different curves would fit the same data just as well? One possible curve
would be a simple square root: if the just noticeable difference or JND
is assumed to depend on signal-to-noise ratio, and if noise is considered
to rise as the square root of the mean signal amplitude, then the noise
is proportional to the square root of the signal (Poisson distribution)
and the JND increases with the signal – not linearly, but the square
root might fit as well as the log over some range since it bends the
right way.

If you took the mean of all those exponents in Stevens’ table, I’ll bet
it would come out pretty close to 1. (OK, I did it and it’s 1.06 if I did
it right – may have skipped or duplicated one or two, but that would’nt
make too much difference in 33 entries). I wonder what the standard
deviation of the measured exponents was.

Best.

Bill P.

[rsmarken]

[From Rick Marken (2007.09.07.1040)]

Bill Powers (2007.09.06.1850 MDT)--

> Rick Marken (2007.09.06.1740) --

> Yes, I see the connection quite clearly!! Thanks.

And so you do! This is wonderful. Your analysis has nothing wrong with it
that I can see, and I'm very relieved to see that you understood what I said
and agree with it, as shown by your paraphrases. Your review of how
Fechner reached his conclusion using Just Noticeable Differences
completes the picture. I think this is eminently publishable.

Actually, I want this to be just one components of the paper I'm
planning to write (with the help of a student who will be going
through the literature with me and identifying CV, disturbance and
output in a sample of psycholofy experiments) on the role of purpose
in experimental research. A paper just on reconciling the log and
power "laws" would be nice but I think you (we?) would have to wade
through a lot of scholarship on this, much of which is mathematical in
the most set theoretic way that is incomprehensible to me (perhaps
Richard Kennaway could, indeed, help with this). An example of the
kind of paper we would have to deal with is

http://aris.ss.uci.edu/~lnarens/1996/Narens_JMP_1996.pdf

This is a Journal of Math Psych article by Louis Narens, who I met
when I was doing some of my thesis work back in the 1970s at UC
Irvine. What I can understand of it suggests that he misses the
possibility of a log function for numbers but it's so set theoretic
that I just can't make it past the first page. Other articles in this
area are of the same character. But we really have to do the
scholarship on this because I can't believe that no one else has
realized that a log function relating N to Pn would account for the
power law assuming a log relation between I and Pi.

I will use the idea of a log feedback function from N to Pn as one
example of how understanding the subjects's purpose in an experiment
can explain the nature of the observed relationship between S
(external disturbance variable) and R (the output that influences the
controlled input that is affected by both S and R). My point will not
be a psychophysical one -- I don't want to get involved with that
group again -- but a methodological one, showing where the concept of
purpose fits in to existing experimental research and how to
incorporate it properly into future research.

Do we get Richard Kennaway's blessing?

I think it would be great of you and Richard would write the paper on
how a log relationship between N and Pn would explain the power law
relationship between N and I if Fechner's log law of the relationship
between Pi and I is correct. Then I would have something to reference.
I might be able to help you find the papers you should reference but I
think only Richard could understand them!

Best regards

Rick

···

Best.

Bill P.

--
Richard S. Marken PhD
rsmarken@gmail.com

[Martin_Taylor2]

[Martin Taylor 2007.09.07.13.57]

[From Rick Marken (2007.09.07.1040)]

But we really have to do the
scholarship on this because I can't believe that no one else has
realized that a log function relating N to Pn would account for the
power law assuming a log relation between I and Pi.

Nor can I, not because I can put my finger on such a paper, but because it gave me such a strong feeling of Deja Vu, from the deep past. Maybe Green and Luce said it. Maybe I read it when I was doing my summer student work. Maybe I just figured it out for myself. Maybe my professor (W.R.Garner) told me. Wherever it came from, I'm sure it's something I've known for a long time, just brought back into memory by Bill's posting. I thnk you would have to search the literature from the 50s and early 60s, if not before.

Martin

[Richard_Kennaway]

[From Richard Kennaway (2007.09.07.2036 BST)]

[From Rick Marken (2007.09.07.1040)]

An example of the
kind of paper we would have to deal with is

http://aris.ss.uci.edu/~lnarens/1996/Narens_JMP_1996.pdf

My eyes glaze over at this as well, but I'll have a go and see what it says.

···

--
Richard Kennaway, jrk@cmp.uea.ac.uk, Richard Kennaway
School of Computing Sciences,
University of East Anglia, Norwich NR4 7TJ, U.K.

[Bill_Powers1]

Nor can I, not because I can put
my finger on such a paper, but because it gave me such a strong feeling
of Deja Vu, from the deep past. Maybe Green and Luce said it. Maybe I
read it when I was doing my summer student work. Maybe I just figured it
out for myself. Maybe my professor (W.R.Garner) told me. Wherever it came
from, I’m sure it’s something I’ve known for a long time, just brought
back into memory by Bill’s posting. I think you would have to search the
literature from the 50s and early 60s, if not before.
[From Bill Powers (2007.09.07.1255 MDT)]

Martin Taylor 2007.09.07.13.57 –

My hunch would be that if “log perception of N” had been
published, nobody would have taken the power law seriously after that.
Since people still talk about it, I doubt that any refutation was ever
published. It was in the 60s that I submitted that paper to Science, and
neither the editors nor Stevens mentioned that anyone else had already
said it all. Stevens himself said that logs had nothing to do with the
power law, which convinced me that he wasn’t all that sure what a
logarithm is.

Anyway, despite the pretentious mathematicizing in that article Rick
cited, I think there’s a place for a nice simple little note, hardly
longer than my post, just laying out the logic of the argument. We don’t
have to compete with the mathematical show-offs.

Best,

Bill P.

[Bill_Powers1]

My eyes glaze over at this as
well, but I’ll have a go and see what it says.
[From Bill Powers (2007.09.07.1450 MDT)]

Richard Kennaway (2007.09.07.2036 BST) –

Our hero. You don’t really have to, but I would be curious whether they
are saying anything at all in that article.

Best,

Bill P.

[rsmarken]

[From Rick Marken (2007.09.07.1440)]

Bill Powers (2007.09.07.1255 MDT)--

My hunch would be that if "log perception of N" had been published, nobody
would have taken the power law seriously after that. Since people still talk
about it, I doubt that any refutation was ever published.

You know, I bet you're right! In order to get the Pn = log N
explanatoin one would have to accept the idea that it is the
relationship between Pi and Pn that is controlled in magnitude
estimation. And it seems very unlikely that anyone would get that
since psychologists are so firmly comitted to the idea that N is a
response to Pi.

Anyway, despite the pretentious mathematicizing in that article Rick cited,
I think there's a place for a nice simple little note, hardly longer than my
post, just laying out the logic of the argument. We don't have to compete
with the mathematical show-offs.

I agree. Perhaps the four of us (you, me Martin and Richard) should
write it, submit it to a relevant journal and when it's published I'll
refer to it in my more general paper! Why don't you write the note,
I'll figure out where to submit and handle the submission details,
Martin will get a list of the relevant references and RIchard can make
it look good mathematically.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[Martin_Taylor2]

[Martin Taylor 2007.09.07.17.54]

[From Rick Marken (2007.09.07.1440)]

Bill Powers (2007.09.07.1255 MDT)--

  My hunch would be that if "log perception of N" had been published, nobody
would have taken the power law seriously after that. Since people still talk
about it, I doubt that any refutation was ever published.

You know, I bet you're right! In order to get the Pn = log N
explanatoin one would have to accept the idea that it is the
relationship between Pi and Pn that is controlled in magnitude
estimation. And it seems very unlikely that anyone would get that
since psychologists are so firmly comitted to the idea that N is a
response to Pi.

I don't see why. The reason I did the multidimensional study as a summer student was the notion that all the perceptual scales were not only nonlinear but also dependent on what the subject was doing with the perception. That applied to ALL the dimensions, not only the numbers or the positions of nonsense syllables in a partially learned list.

Obviously I didn't have any ideas about controlled perception, but the idea that the perceived magnitude of a number was roughly a logarithmic function of its numerical magnitude isn't a long step, any more than it's a long step to suggest that discrimination jnds are often logarithmic with the magnitude of the quantity.

Anyway, I'm afraid I can't do better right now than say I had this feeling of Deja Vu when reading Bill's analysis. It seemed like something I had known, but not thought about for decades.

As for not taking the power law seriously after that, you have to remember that lots of people didn't take it very seriously anyway. Other people did, an do. Not everyone reads everything.

  Anyway, despite the pretentious mathematicizing in that article Rick cited,
I think there's a place for a nice simple little note, hardly longer than my
post, just laying out the logic of the argument. We don't have to compete
with the mathematical show-offs.

I agree. Perhaps the four of us (you, me Martin and Richard) should
write it, submit it to a relevant journal and when it's published I'll
refer to it in my more general paper! Why don't you write the note,
I'll figure out where to submit and handle the submission details,
Martin will get a list of the relevant references and RIchard can make
it look good mathematically.

References are, scientifically, my worst point. I tend to remember the facts, not who found them or where they were published.

Martin

[rsmarken]

[From Rick Marken (2007.09.07.1545)]

Martin Taylor (2007.09.07.17.54) --

>Rick Marken (2007.09.07.1440)]
>
>You know, I bet you're right! In order to get the Pn = log N
>explanatoin one would have to accept the idea that it is the
>relationship between Pi and Pn that is controlled in magnitude
>estimation. And it seems very unlikely that anyone would get that
>since psychologists are so firmly committed to the idea that N is a
>response to Pi.

I don't see why.

Because otherwise it makes no sense at all. It's just a bunch of equations.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[Bill_Powers1]

[Bill] My hunch would be
that if “log perception of N” had been published, nobody

would have taken the power law seriously after that. Since people
still talk

about it, I doubt that any refutation was ever
published.

[Rick] You know, I bet you’re right! In order to get the Pn = log N

explanatoin one would have to accept the idea that it is the

relationship between Pi and Pn that is controlled in magnitude

estimation. And it seems very unlikely that anyone would get that

since psychologists are so firmly comitted to the idea that N is
a

response to Pi.

[Martin] I don’t see why. The reason I did the multidimensional study as
a summer student was the notion that all the perceptual scales were not
only nonlinear but also dependent on what the subject was doing with the
perception. That applied to ALL the dimensions, not only the numbers or
the positions of nonsense syllables in a partially learned
list.
[From Bill Powers (2007.09.08.0659 MDT)]

Martin Taylor 2007.09.07.17.54 –

But as you say in the next sentence, you weren’t thinking that behavior
was produced to control perception. Most psychologists, I think, would
have assumed that when the subject uttered the name of a number –
“2” – that this was a response, an output rather than an
input. And the idea that the response was selected and produced to
control a perception of the meaning of that utterance, to make it match
another perception, does not seem ever to have been part of psychology
prior to PCT. You don’t seem to be saying it was:

Obviously I didn’t
have any ideas about controlled perception, but the idea that the
perceived magnitude of a number was roughly a logarithmic function of its
numerical magnitude isn’t a long step, any more than it’s a long step to
suggest that discrimination jnds are often logarithmic with the magnitude
of the quantity.

It’s not a long step, but it has to be taken before you can claim to have
taken it. It was a tiny step in the 1960s, but at least I tried to take
it. It may not be a big important discovery, but leave me
something to brag about, at least.
The wiki article says:
The principal methods used by Stevens to measure the
perceived intensity of a stimulus were magnitude estimation and
magnitude production. In magnitude estimation with a standard, the
experimenter presents a stimulus called a standard and assigns it
a number called the modulus. For subsequent stimuli, subjects
report numerically their perceived intensity relative to the standard so
as to preserve the ratio between the sensations and the numerical
estimates (e.g., a sound perceived twice as loud as the standard should
be given a number twice the modulus).

I take this to mean that a subject was given a standard stimulus
and told to consider it as having, for example, a magnitude of 1. Then a
different amount of the stimulus would be given, and the subject would
give a numerical estimate of its magnitude relative to the first
magnitude, say 2. This was taken to mean that the perceived magnitude of
the second stimulus was twice the magnitude of the first. And indeed it
was – but the perception of “twice as large” is not what
arithmetic would suggest. “Twice as large” is perceived as a
perceptual ratio of k*log(2). Steps of “twice as large” in
intensity are perceived as (roughly, over some range) equal in size, as
the frequency of octaves on a piano is perceived. A piano has eight
equally-spaced octaves, plotted neatly on a logarithmic scale of
frequency as a row of keys.

Anyway, I’m afraid
I can’t do better right now than say I had this feeling of Deja Vu when
reading Bill’s analysis. It seemed like something I had known, but not
thought about for decades.

And you kept it to yourself, despite the big to-do over Stevens’
discovery of the first new law of psychophysics in 100 years? Come on.
What you’re saying is that this analysis is so simple and obvious that
you’re embarrased for not having seen it. I admit that it’s simple and
obvious,

As for not taking
the power law seriously after that, you have to remember that lots of
people didn’t take it very seriously anyway. Other people did, an do. Not
everyone reads everything.

Logical, but probably not true. I doubt that any professional
psychologist hadn’t heard of Stevens’ Famous Power Law. It received
prominent play in his 1950 Handbook of Psychology, and my article was
written in response to an article about the power law by Stevens in
Science (I think it was a technical comment). I doubt that very many
psychologists had any grounds for thinking it untrue – only the ones
with some serious committment to Weber-Fechner. When psychologists say
“nobody believes that” or “everybody believes that”,
I have found that what they usually mean is that everybody or nobody in
their own tiny school of thought does. Like “modern control
theory,” which sounds like a huge new movement, but isn’t. A bunch
of little frogs in a small pond always try to sound larger than they
are.

References are,
scientifically, my worst point. I tend to remember the facts, not who
found them or where they were published.

OK, then, we’ll go ahead and do it with three of us, if
Richard comes aboard. Three little frogs can probably make k*log(3/4) as
much noise as four little frogs.

Best,

Bill P.

[rsmarken]

[From Rick Marken (2007.09.08.0935)]

Bill Powers (2007.09.08.0659 MDT)--

> Martin Taylor --
>
>Obviously I didn't have any ideas about controlled perception, but the idea
> that the perceived magnitude of a number was roughly a logarithmic function
> of its numerical magnitude isn't a long step,

It's not a long step, but it has to be taken before you can claim to have
taken it. It was a tiny step in the 1960s, but at least I tried to take it.
It may not be a big important discovery, but leave me something to brag
about, at least.

You have lots to brag about. But this little gem is one of the best!

OK, then, we'll go ahead and do it with three of us, if Richard comes
aboard. Three little frogs can probably make k*log(3/4) as much noise as
four little frogs.

I agree. Why don't you write up the paper. I'll do the scholarship and
the submission paperwork. Richard can check the math. I would also
suggest that it be framed in terms of the behavioral illusion. It's a
perfect illustration thereof. The title could be "The Power Law: A
Behavioral Illusion". That way we don't have to frame it as an
argument with these mathematical psycho physicists and we won't be
"poaching" on their territory. Why not write up a first draft and I
can make some suggestions/ changes and eventually we can get to a
finished product off line. I'll also be in touch with you off line
with my suggestions for where it should be submitted. Probably a
theoretical note in _Psych Review_.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[Martin_Taylor2]

Re: Feedback Functions
[Martin Taylor 2007.09.08.12.51]

[From Bill Powers (2007.09.08.0659
MDT)]

OK, I’ll concede that you are the first person with an interest
in perception to notice that if plog(x) = qlog(y) then x =
y^(q/p).

Anyway, I’m afraid I can’t do better
right now than say I had this feeling of Deja Vu when reading Bill’s
analysis. It seemed like something I had known, but not thought about
for decades.

And you kept it to yourself, despite the
big to-do over Stevens’ discovery of the first new law of
psychophysics in 100 years? Come on. What you’re saying is that this
analysis is so simple and obvious that you’re embarrased for not
having seen it. I admit that it’s simple and obvious,

I didn’t claim to have invented it, just that I remember knowing
about it, and that it was possible this memory might have come from my
own observation. Apparently I must not have had that memory. I
apologise for saying that I had, simply because I believed that I
had.

In any case, the existence of the logarithmic relationship in no
way disproves the power law. It simply says that you can’t prove the
power law from the data at hand, because not only is it true that if
plog(x) = qlog(y) then x = y^(q/p), but it is also true that if
f(plog(x)) = f(qlog(y)) then x = y^(q/p), at least if f is an
invertible function (such as a power law). So, even if everyone had
known that the logarithmic relation holds, they would have had no
reason to deny that the power law holds. And on what basis could you
be more jsutified in asserting that the perceived magnitude of numbers
is a logarithmic function of their value than Stevens was in asserting
that the perceived magnitude of numbers is a linear function of their
magnitude?

All that the logarithmic relationship demonstrates is that the
apparent perceptual function depends on how the subject perceived the
response dimension. And THAT was the subject of my summer work and
thesis work.

As to whether rational arguments are sufficient to alter the
zeitgeist, you have only to look at the penetration of PCT into
mainstream psychology to see why a mention of the logarithmic
relationship would be insufficient to dispel the rather seductive
notion of the power law. The necessity of PCT is provable on physical
grounds, but what kind of effect does that have on the
mainstream?

Power laws are all over the place in comparative physiology, and
had been for a century or so. Can you reasonably argue that the main
body of psychophysiscists should have been expected to dismiss it
because of a logarithmic relationship which does not disprove it, but
merely offers a plausible alternative?

···

---

In my thesis work, I wasn’t at all concerned with the power law,
because I had already determined from the reversible cross-modality
matching summer experiments that if it was ever true it was very
context-dependent, which made it uninteresting. What I was concerned
with was the apparent curvature-reversal of the function relating
physical to perceived distance near an anchor (which neither a power
law nor a logarithmic law could accommodate). That reversal apparently
accounted for a consistent pattern of what were thought to be memory
errors in re-placing a dot on an index card, but which I argued to be
perceptual differences between the stimulus situation and the response
situation. (Effects of anchoring and distance
perception on the reproduction of forms. Percep. Mot. Skills, 1961,
12, 203-239)

To my mind, in 2007 all we can say is that we assume the subject
has the purpose of peceiving the experimenter to be pleased with
his/her performance in the experiment. Some subjects may try to cheat,
not having that purpose, or may misinterpret the instructions; but
many experimenters are aware of that possibility and try to design the
experiments so that it is hard for the subjects to get consistent
results if they do try to cheat. So we maintain the assumption that
the subjects try to perceive the experimenter to be pleased – to
cooperate.

Assuming the subject understands the instructions in the way the
experimenter intended them to be understood in a perceptual magnitude
matching experiment, if the subject reports a value on the response
dimension, then that value truly is perceived as matching the
magnitude of the stimulus (disturbance, if you like) that is presented
(allowing for random error). The subject can control the response, but
cannot affect the presentation that was to be matched.

To this point, I see no difference between a PCT approach and the
conventional approach. All that is asserted is that somewhere inside
the subject there is the possibility of comparing the magnitudes of
two perceptions of different kinds. Again, that’s conventional
psychophysics.

The problem arises when there is some assertion about an ABSOLUTE
scale or function, in particular the notion that the perception of the
magnitude of a number is given by a linear function of the number
itself. That’s Stevens’ assumption, and using it, the data lead to the
power law. Another plausible assumption is that the perceived
magnitude of a number is proportional to the logarithm of the number.
That assumption leads to a logarithmic law for other perceptual
dimensions matched against numbers, using the same data. In each case,
it’s an unsupported assumption about how numbers are perceived.

In order for the logarithmic assertion to have some impact, one
has to show some different kind of data that would discriminate
between the two hypotheses. It makes intuitive sense that the
perceived difference between 1,000,005 and 1,000,010 is less than
between 5 and 10, but intuition is only a guide in science. Perhaps
differentials might work, but given the effects of local anchors the
analysis would be quite subtle, and probably not persuasive. For
example, to me the difference between 99 and 101 is substantially
larger than the difference between 66 and 68. You would have to do
something to handle that kind of effect.

For myself, I’m reasonably happy with the notion that the
perceived magnitude of an extent in most perceptual dimensions is a
function for which the slope increases from zero and then decreases
gradually as a function of distance from an anchor. If the anchor is
itself non-zero in that dimension, this function modulates whatever
other functions apply inthe neighbourhood. Such a function does seem
to explain some perceptual effects, but there’s no way I’d assert it
to be a universal truth along the lines of the assertions that the
perceived magnitude of a number is a linear or alternatively a
logarithmic function of its numerical magnitude.

Martin

[Bill_Powers1]

In any case, the existence of
the logarithmic relationship in no way disproves the power law. It simply
says that you can’t prove the power law from the data at hand, because
not only is it true that if plog(x) = qlog(y) then x = y^(q/p), but it
is also true that if f(plog(x)) = f(qlog(y)) then x = y^(q/p), at least
if f is an invertible function (such as a power
law).
[From Bill Powers (2007.09.08.1240 MDT)]

Martin Taylor 2007.09.08.12.51 –

You misunderstood the import of my derivation. If we say that Weber’s
logarithmic law applies to the perception of the sizes of numbers, then
Weber’s law predicts Stevens’ power law. It doesn’t “disprove”
it.

So, even if
everyone had known that the logarithmic relation holds, they would have
had no reason to deny that the power law holds. And on what basis could
you be more jsutified in asserting that the perceived magnitude of
numbers is a logarithmic function of their value than Stevens was in
asserting that the perceived magnitude of numbers is a linear function of
their magnitude?

I have no reason to deny it, either. Stevens’ law IS Weber’s law under
the assumption that number magnitudes are also perceived
logarithmically.

To my mind, in 2007
all we can say is that we assume the subject has the purpose of peceiving
the experimenter to be pleased with his/her performance in the
experiment.

I don’t think that’s all we can say. We can say that if subjects are
asked to produce numbers indicating the relative magnitudes of two
perceptions, and if all size perceptions including perceptions of number
sizes follow Weber’s law, then they will produce numbers that also follow
Stevens’ law. Whether either law is correct or even good enough to
predict other data is irrelevant.

In order for
the logarithmic assertion to have some impact, one has to show some
different kind of data that would discriminate between the two
hypotheses.

There are not two hypotheses: the two hypotheses are the same, given
Weber’s law applying to numbers.

As to the effect of your “anchors,” I think that brings in new
perceptions, perhaps at a different level. Interesting, though.

Best,

Bill P.

[rsmarken]

[From Rick Marken (2007.09.08.1520)]

Martin Taylor (2007.09.08.12.51)

OK, I'll concede that you [WTP] are the first person with an
interest in perception to notice that if p*log(x) = q*log(y)
then x = y^(q/p).

But that's not what Bill noticed. What Bill noticed is that, in
magnitude estimation, people control a perception of Pn = Pi and that
if (as per Weber/Fechner) Pn = k log N and Pi = m log I then N will be
a power function of I.

To my mind, in 2007 all we can say is that we assume the
subject has the purpose of peceiving the experimenter to be
pleased with his/her performance in the experiment.

The subject is asked to have the purpose of keeping N proportional to
I (N = a*I). We control theorists know that the subject's purpose must
have been to keep Pn proportional to Pi. But psychophysicists ignored
the subjects' purpose and assumed that N is a behavioral output caused
by I: N = f(I). That's why they never saw what Bill pointed out: that
the power relationship between N and I predicts the power law because
both Pi and Pn would be log functions of the environmental variables,
I and N, respectively.

Assuming the subject understands the instructions in the way the
experimenter intended them to be understood in a perceptual
magnitude matching experiment, if the subject reports a value
on the response dimension, then that value truly is perceived as
matching the magnitude of the stimulus (disturbance, if you like)
that is presented (allowing for random error). The subject can
control the response, but cannot affect the presentation that
was to be matched.

To this point, I see no difference between a PCT approach and the
conventional approach.

The difference is the assumption that it is the value on the response
dimension -- N in this case -- that is perceived as matching the
magnitude of the stimulus. What we are saying is that it is the
_perceived_ value of N -- Pn -- that is perceived as matching the
perceived value of the stimulus. The difference between the PCT and
conventional approach is that, in PCT, it is assumed that the subject
is controlling for Pn = Pi; in the conventional approach it is assumed
that the subject responds with N = Pi.

All that is asserted is that somewhere inside the
subject there is the possibility of comparing the magnitudes of two
perceptions of different kinds. Again, that's conventional
psychophysics.

I don't believe that conventional psychophysics treats the subjects'
"responses" as a perception, let alone one that is part of a
controlled perception.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[Martin_Taylor2]

[Martin Taylor 2007.09.08.22.13]

[From Bill Powers (2007.09.08.1240 MDT)]

Martin Taylor 2007.09.08.12.51 --

In any case, the existence of the logarithmic relationship in no way disproves the power law. It simply says that you can't prove the power law from the data at hand, because not only is it true that if p*log(x) = q*log(y) then x = y^(q/p), but it is also true that if f(p*log(x)) = f(q*log(y)) then x = y^(q/p), at least if f is an invertible function (such as a power law).

You misunderstood the import of my derivation. If we say that Weber's logarithmic law applies to the perception of the sizes of numbers, then Weber's law predicts Stevens' power law. It doesn't "disprove" it.

...

In order for the logarithmic assertion to have some impact, one has to show some different kind of data that would discriminate between the two hypotheses.

There are not two hypotheses: the two hypotheses are the same, given Weber's law applying to numbers.

Sorry. I've been misreading, or perhaps reading too much into your intentions. I know you said it, that all you were doing was to show that Weber-Fechner implies Stevens if number magnitude is perceived logarithmically. I read into that a dismissal of the power law, wrongly.

The Deja Vu still is there, though. Unfortunately, Ina and I have discarded most of the books we had in graduate school, so I can't check whether it was in them. I do know that the logarithm to power law relation was part of my undergraduate engineering education, if not before. Maybe the Deja Vu feeling comes from there rather than from a direct application to the two laws.

Martin