Newsgroups: bit.listserv.csg-l
Date:         Thu, 8 Oct 1992 21:03:55 EDT
From:         mmt@BEN.DCIEM.DND.CA
Subject: Re: Why 99%, hierarchical perception

[Martin Taylor 921008 20:30] In response to Rick Marken's suggestion that we
continue a private discussion via CSG-L.  In view of his suggestion, I hope
he will not object if I quote from that mail.  If I'm wrong, I apologise in
advance.

I said in a posting just sent that I was going to put this off.  But it might
be put off for a long time, so rather than do that, I'll try to be brief.

Rick (921005 15:55)

>>Why, for example, am I
>>wrong in suggesting that your data, which can be used (statistically) to
>>decide between x+y and x*y, could be used to discriminate between x*y and
>>log(x)+log(y)?
>
>Why don't we do this on the net. I would have to have a little better
>idea of exactly what you are deciding about (with respect to the model).
>Why, for example, would you expect the variance of the error of prediction
>to suggest anything about nature of the controlled variable? What is
>the statistical analysis that you think is worthwhile for my data?

I'm not sure of the detail of the study (you probably described it somewhere
but I have forgotten), but let's assume that subjects were asked to "keep
the figure the same size" while you disturbed some aspect of it.  You built
a simulation model that incorporated appropriate gains and delays, and tried
it with a perceptual input function of (x+y) and again with (x*y).  You found
that the best you could do with (x+y) as an input function left you with
2% unexplained variance, but you could halve that error if you used (x*y).
I doesn't matter whether this is exactly what you did, because it can serve
to illustrate what I am talking about in any case.

What are the follow-on possibilities to this study?  What have you found out?
I think you might be tempted to believe that you have shown the possibility
that perceptual input functions (PIFs) can include multiplication operators,
x and y both (presumably) being controllable CEVs in themselves.  Now I propose
to you that this is false, and I propose as a counter-possibility that addition
operators and logarithm operators are possible, and that the "correct" PIF
is log(x)+log(y).  It is (conceivably) important to distinguish these
possibilities, because they may have strong implications for the brain
structures involved not only with this percept, but with many other percepts.

What to do?  If you have tested only with one "size", you can't tell these
possibilities apart very well, since the controlled value of either (x*y)
or (log(x)+log(y)) is a constant, and the errors are (by assumption) due
to non-infinite gains in some control loops associated with the task, which
affect the apparent overall gain of the "size" control loop.  So, since
the error is a simple difference between reference signal and perceptual
signal, all we can do is estimate it and say "that's how good the control is."

But now let's introduce a new reference size, in which x and y are both
doubled.  So the new level of (x*y) is four times the old reference level.
But the gain is, we assume, undisturbed, so we expect to see much the same
error in prediction.  We have only changed the reference level, an additive
variable in the equation.  The same is true if the subject is controlling
log(x)+log(y), but instead of multiplying the reference signal by 4, we have
added 0.6 units to whatever it was beforehand.  If we now refer the variance
of our estimates to the size set as a reference for the subject, we find that
if the logarithmic hypothesis is correct, the residual variance is constant
in log size units, whereas if there is a multiplier operator in the PIF, the
residual variance is constant in area units.

I think the results would be contaminated by other effects, but the main idea
should be valid.  It's a very simple kind of case, but it is typical of the
sort of thing that psychologists want to know: is shape recognition initiated
in the Right Hemisphere and symbolic in the Left, before being transferred
across to the other hemisphere for syntactic and pragmatic integration?  That's
the kind of question of interest for solid practical reasons that apply to
cases of stroke.  How could you do a non-statistical PCT study to address
such a question?

Martin
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