[Martin Taylor 2009.06.04.09.12]

[From Bill Powers (2009.06.04.0624 MDT)]

Martin Taylor 2009.06.04.00.46–

`I don't really see any value in continuing`

that discussion, which hinged around a claim that if (as is true in any

control system) o = f(d), and one can work with d = g(x), then o =

(f(g(x)).That’s not quite the argument.

o = f(qi)

qi = g(o) + d

o = f(g(o) + d)

Suppose

o = k*qi and

qi = -o^2 + d

where o^2 means o*o

then

`1 + SQRT(1 + 4k^2d)`

o = ------------------

`2k`

So the output will be observes as that rather complicated function of

d, and the appearance is that a stimulus produces a response according

to that formula. In fact, the forward (organism) function is a simple

proportionality but the feedback effect greatly modifies the

relationship between d and o.If k (the forward gain) is large, and d does not go negative, we have

the approximation`1 + 2k*SQRT(d)`

o ~ --------------, or

`2k`

o ~ SQRT(d)

so the output appears to be related to the disturbance as the inverse

of the relationship in the feedback path, and the forward relationship

between qi and o is not visible at all.Whether we use the approximation or not, the apparent relationship

between d and o is different from the forward equation describing the

behaving system. That is the basis of Rick’s objections to the proposed

analysis of the Schouten experiment. To work around this problem it is

necessary to identify qi, the controlled variable, which was not done

in the Schouten experiment. My objection is also based on lack of

identification of qi – in fact, on the impossibility of identifying it

given only the experimental results. Your proposed analysis of the

information content of the perceptual signal is based on using the

apparent relationship between d and o, which is not the same as the

actual relationship between qi and o. Without additional data, it’s not

possible to deduce the value of qi or the form of the forward function,

though of course by assuming a form of the forward function one can

deduce the rest. However, every different assumption will yield a

different value of the measured information, so the model is

underspecified.

I believe I understood this argument, and agree with all of it except

for the last sentence: “However, every different assumption will yield

a different value of the measured information, so the model is

underspecified.”

This sentence is false when substituting one monotonic function for

another. It is not totally false, since there exist functions y = f(x)

that have regions of x for which dy/dx = 0, or for which more than one

value of x can give the same value of y. These are functions that have

saturation or threshold plateaus, or that

are non-monotonic. Such functions will give different results for

H(y|x) than will the monotonic functions that are more likely “f” in qi

= f(o, d) in any experiment on perception. Control become problematic

in case where d(qi)/do = 0 or switches sign as a function of the

magnitude of qi, so we usually assume d(qi)/do retains a constant sign.

Since qi = f(d,o), we make the same assumption in respect of

d(qi)/d(d).

Your specific example of a proportional control system in which d

cannot go negative is not likely to represent many real-world control

situations. The principle you use it to illustrate, of nonlinear

relationships in various pathways of the control system, is likely to

represent almost all real-world situations, but so long as the loop

gain doesn’t change sign as a function of the magnitude of the

disturbance, it won’t affect the informational argument, which starts

with the relationship H(o | d) that is computed from the experimental

data, and in the Schouten experiment d(H(o | d))/dt.

Unless there are separate influences from a common source on o and d,

the maximum information rate d(H(o | d))/dt is determined by the lowest

channel capacity of any influential link connecting them.

One possible, though highly improbable, interpretation of the Schouten

experiment is that the * form* of the function perception =

f(presentation) changes consistently as a function of the time after a

light is turned on. It is more parsimonious, following Occam’s razor

and the usual principles of PCT, to assume that the form of the

function does not change when its argument (the turning on of a light

in the “presentation”) changes.

If this is true, then d(H(o | d))/dt as actually measured from

Schouten’s data is a fair measure of the lowest channel capacity in the

forward path from presentation to output. That path contains complete

control loops, but it also contains one uncontrolled link from

presentation to perception. One may argue about the functions in the

control loops, and about what perception precisely is being controlled,

but the resolution of such arguments has no influence on the analysis

of the information transmission unless it can be shown that the form of

the control loop changes as a function of time since the onset of the

light. It could, however, be argued that the limiting channel capacity

is in the control loop rather than in the link from presentation to

perception. That’s a different argument than the one you present above,

and I would suggest that it, too, is implausible. I have argued against

it in the earlier discussions.

Since you read Science, I assume you have studied the paper by Kiani

and Shedlen, May 8 2009, Vol 324 p 759. It most certainly is not done

or interpreted within the PCT framework, but I found it interesting,

nevertheless. I’d be interested in your comments on it, especially the

last part.

Martin