Weber-Fechner (was Feedback reciprocity ...)

[Martin Taylor 2010.07.09.00.48]

The following was rejected by the mailer because the attachment was too big. I have uploaded it to my web site, so where you see a reference to an attached document, you can download it from <http://www.mmtaylor.net/Academic/GeomOfIllusion1961.pdf&gt;

Martin

···

[Martin Taylor 2010.07.08.17.08]

[From Bill Powers (2010.07.08.0905 MDT)]

Rick Marken (2010.07.07.1920) --

>If you now look at how much change in stimulus is needed to
> generate a constant step-change in magnitude estimate, you will generate a
> table that looks like a difference threshold table.

Sure, but now that assumes that a step change in magnitude estimate
reflects a step change in the magnitude of the perception. And all
this is based on an input-output model of the behavior in these
experiments.

It doesn't assume that the two step-changes are of the same size. One is the log of the other (but wait a couple of paragraphs).

I would propose that there is a perception Ps of the stimulus variable, which is one component of qi. The perception varies (to try out the W-F hypothesis) as the logarithm of stimulus intensity. ...

Going back in this thread, you have both been assuming that perceived magnitude can be equated to a summation of just-noticeable differences. There's no logical or psychological reason why this should be so, and there's evidence that it is false. I mentioned one bit of evidence that it is false some years ago: that in the Mueller-Lyer figures, the distance between the end-points of the arrowheads is perceived as longer for the >< figure than for the <> figure, whereas small intervals all along those extents are perceived as longer in the <> figure than in the >< figure. The implication is that the longer extent is not perceived as the sum of the smaller extents of which it is composed. If this is so, then the perceived extent cannot be a function of the sum of JNDs along the extent. The Weber-Fechner idea was a perfectly reasonable hypothesis, but that's all it has ever been.

One of my favourite books of all time was given to my grandfather as a school prize. It is undated, but from internal evidence of what the author knew and did not know, I'm guessing it was written around 1882. The anonymous author must have been a scientist of the highest calibre, on the level of Faraday or Maxwell, and must have been versed in a very wide range of sciences that he could illustrate by way of experiments that a child could do, and that provided a great basis for discussion of the implications -- rather like Bill's demos. I used it for the very first reference in my first solo paper. Here's the citation from the paper:

---------------

"Anonymous, "Popular Scientific Recreations, a storehouse of
instruction and amusement: in which the Marvels of Natural
Philosophy, Chemistry, Geology, Astronomy, etc., are explained
and illustrated, mainly by means of pleasing experiments and
attractive pastimes." Translated and enlarged from Gaston Tis-
sandier, Les Recreations Scientifiques, New and Enlarged Edition
[Ward, Lock, Bowden and co., London, 1882 (?)],p. 113.The
statement does not appear in the original"

----------------
Incidentally, I searched in many libraries for the Tissandier original without success, but one day I was casually waiting in the stacks at the University of Michigan for a newly minted Ph.D. to get signed out before I drove him to Baltimore, and found myself facing the Tissandier book on the opposite shelf. It is about 1/3 the size of my grandfather's school prize, and contained none of the interesting things that I looked for in it! Even more incidentally, Tissandier may have been the inventor of military aerial surveillance, having used balloons to scout out the German positions during the siege of Paris in 1870.

The "statement" mentioned in the citation was that the discriminability of an extent was not necessarily the summation of just-noticeable differences along that extent, nor necessarily any function of that sum. I used that observation, together with the hypothesis that discriminability was monotonically related to the perceived magnitude of an extent to predict various properties of a novel visual illusion, including the counter-intuitive prediction that people who observed more precisely would show a greater illusion than those who were less precise. I attach the paper in the form of a PDF with selectable OCR derived text. I don't imagine the OCR will have worked well on the equations, but the quoted citation was cut-and-pasted from the PDF, so it's not too bad.

The point of bringing this up was to show that there is at least a little evidence, in the form of the prediction of this illusion, that perceived extent may be a monotonic function of discriminability, but there is also evidence that perceived extent is NOT a function of the summation of JNDs along the extent. In other words, you can't simply integrate and hope to get a useful answer.

Martin

(Gavin Ritz

[Martin Taylor 2010.07.09.00.48]

Well do PCTers regard
Weber-Fechner Law as a law or not?

So the method you used
in your test was it a casting net or a testing specimens.

Regards

Gavin

[From Bill Powers (2010.07.09./1610 MDT)]

Martin Taylor 2010.07.09.00.48 –

Going back in this thread, you
have both been assuming that perceived magnitude can be equated to a
summation of just-noticeable differences. There’s no logical or
psychological reason why this should be so, and there’s evidence that it
is false.

BP:
(1) If there is a logarithmic relationship between perceived magnitudes
and stimulus intensities and
(2) if a just-noticeable-difference always corresponds to the same linear
difference in perceived magnitude, then
(3) it follows that the first derivative of perceived magnitude divided
by the magnitude is a constant. That’s the nature of a log function.
And
(4) if (3) is true, then the integral of dI/I would reconstruct the
function connecting perceived magnitudes to stimulus magnitudes.
I don’t accept either (1) or (2) as a fact – yet – and I have found
some data showing that the relationship between numerical magnitude
estimates is neither a log relationship nor a power-law relationship for
some sensory modalities. For that kind of data, dI/I is not a constant.
All the data I have seen so far are averaged over multiple subjects, so
there is no indication of the degree to which subjects differ, and plenty
of room for many other forms of nonlinearity to fit the data equally
well.
What I’m really trying to do here is to get rid of this whole subject so
we can go on to other things. I don’t want to abandon it arbitrarily just
because it doesn’t interest me, so I’m trying to give it an honest look,
which means looking at the internal logic and at the experimental data
that I can find. Testing it by incorporating it into a model of behavior
is another way to see if there’s any reason to think that log or
power-law nonlinearity is of importance, but that would take more time
than I have to give it right now.
Long ago I voiced my suspicion that if there is any aspect of perceptual
transformations common to all perceptions, there is no way we could
discover what it is by comparing one perception to another. This is what
I was trying to say when I wrote that letter to Science so long
ago. And since all our knowlege about reality is derived by comparing one
perception to other perceptions, I don’t think there is any way at all of
doing real psychophysics. To do that we would have to be able to measure
reality itself without observing through any perceptual input functions,
and I don’t think we can do that. The best we can do is compare one model
with another and see first if they are mutually compatible, and if not,
which one fits experience better.

I mentioned one bit of
evidence that it is false some years ago: that in the Mueller-Lyer
figures, the distance between the end-points of the arrowheads is
perceived as longer for the >< figure than for the <> figure,
whereas small intervals all along those extents are perceived as longer
in the <> figure than in the >< figure. The implication is
that the longer extent is not perceived as the sum of the smaller extents
of which it is composed.

I believe that the integral of the derivative of a function is the
original function plus a constant. I don’t know what this implies about
your findings; possibly that the small extents without enclosing end
marks are perceived differently from the way large extents enclosed with
end marks are perceived. Unfortunately it’s very hard to determine how
they are perceived without making an unprovable assumption about the
magnitude of an experience and how it relates to the means by which it is
described. There is always an assumed model, and the best we can do is
find the model that behaves the most like the real system. Not knowing
how the distance between end-points of a line with or without arrowheads
is perceived, I can’t help with that.

Best,

Bill P.

···

If this is so, then the
perceived extent cannot be a function of the sum of JNDs along the
extent. The Weber-Fechner idea was a perfectly reasonable hypothesis, but
that’s all it has ever been.

One of my favourite books of all
time was given to my grandfather as a school prize. It is undated,
but from internal evidence of what the author knew and did not know, I’m
guessing it was written around 1882. The anonymous author must have been
a scientist of the highest calibre, on the level of Faraday or Maxwell,
and must have been versed in a very wide range of sciences that he could
illustrate by way of experiments that a child could do, and that provided
a great basis for discussion of the implications – rather like Bill’s
demos. I used it for the very first reference in my first solo paper.
Here’s the citation from the paper:


"Anonymous, "Popular Scientific Recreations, a storehouse
of

instruction and amusement: in which the Marvels of Natural

Philosophy, Chemistry, Geology, Astronomy, etc., are explained

and illustrated, mainly by means of pleasing experiments and

attractive pastimes." Translated and enlarged from Gaston Tis-

sandier, Les Recreations Scientifiques, New and Enlarged Edition

[Ward, Lock, Bowden and co., London, 1882 (?)],p. 113.The

statement does not appear in the original"


Incidentally, I searched in many libraries for the Tissandier original
without success, but one day I was casually waiting in the stacks at the
University of Michigan for a newly minted Ph.D. to get signed out before
I drove him to Baltimore, and found myself facing the Tissandier book on
the opposite shelf. It is about 1/3 the size of my grandfather’s school
prize, and contained none of the interesting things that I looked for in
it! Even more incidentally, Tissandier may have been the inventor of
military aerial surveillance, having used balloons to scout out the
German positions during the siege of Paris in 1870.

The “statement” mentioned in the citation was that the
discriminability of an extent was not necessarily the summation of
just-noticeable differences along that extent, nor necessarily any
function of that sum. I used that observation, together with the
hypothesis that discriminability was monotonically related to the
perceived magnitude of an extent to predict various properties of a novel
visual illusion, including the counter-intuitive prediction that people
who observed more precisely would show a greater illusion than those who
were less precise. I attach the paper in the form of a PDF with
selectable OCR derived text. I don’t imagine the OCR will have worked
well on the equations, but the quoted citation was cut-and-pasted from
the PDF, so it’s not too bad.

The point of bringing this up was to show that there is at least a little
evidence, in the form of the prediction of this illusion, that perceived
extent may be a monotonic function of discriminability, but there is also
evidence that perceived extent is NOT a function of the summation of JNDs
along the extent. In other words, you can’t simply integrate and hope to
get a useful answer.

Martin