[From Bill Powers (950609.1545 MST)]
Bruce Abbott (960108.0900 EST) --
>Why "number of response-seconds"...?
Well, it has to do with the fact that you said that response-
seconds/second would be unitless, because the seconds in the
numerator and denominator cancel. This may be technically true but
is misleading: it is a rate (at which seconds of potential
responding accumulate) whose denominator is expressed in seconds.
I get the idea, but this is not in the mathematics. If you want to
propose this idea as part of the model, then you have to express it
explicitly in terms of variables and functions relating them.
"Technically" true means mathematically true, doesn't it? I presume we
want the mathematics to be correct.
In my engine analogy, imagine that the pump is spurting the
gasoline into the carburetor float bowl rather than directly into
the engine. As the engine runs, the gasoline gets pumped into the
bowl faster than it is being consumed. If we measure the gasoline
in terms of number of seconds of engine-running, then gasoline is
being added at a rate of x seconds of (potential) running per
second. After t seconds, you have x*t seconds of running-time in
the bowl. It is true that rate at which x increases is in
seconds/second, which is unitless, but retaining seconds/second
helps keep what is going on conceptually clear. By labeling the
top quantity "response-seconds," Killeen is just trying to help the
reader to understand what the model is saying: seconds of
responding accumulating at some rate.
What you're doing here is alluding to something that is not expressed in
the mathematics. If it's not in the mathematics, it's not in the model.
When you DO put it into the model, the result may not be what you
expect; in isolation, this little process makes sense, but what happens
when you write the equations for it and combine them with the equations
you already have?
Perhaps what you want to say that there is a quantity that builds up
with every incentive, and that it can produce responses for some time
after the incentive (God, these words!). We can say that the amount of
the quantity is Q, and that it increases by m with each incentive. Q
also decays at some rate, which we can call c. So with R and B in terms
of rates,
dQ = (m*R - c)*dt, and
B = Kb*Q
where Kb is an arbitrary constant expressing responses/sec per unit of
Q.
Now if m is large enough, a single jolt of reinforcer will produce
enough Q so that the behavior rate does not fall to zero for some time
afterward, and multiple responses occur.
This may not be what you want to propose, but at least it expresses a
similar idea. Note that in my interpretation of your idea, the duration
of behaving after a single incentive depends is m*dt/c, and the number
of responses depends on Kb*Q.
The only units needed are seconds, but it is still important to
keep seconds of (potential) responding logically distinct from
seconds of real time.
Obviously, introducing this hypothesis is going to change the model and
the predicted behavior. That's OK if you want to do it; the important
test is whether the resulting model matches observations. If you want to
propose a different meaning for what you say, go ahead -- but put it
into the model rather than alluding to it as a background idea. If
"Potential responding" has any signficance, it belongs in the model.
There is no "potential responding" in Killeen's model as it stands.
In Killeen's hunger model, h declines as responding supplies food
and thus decreases the level of deprivation. Since v and s are
fixed quantities, it is certainly possible that v*h/s will not
equal n.
If so, there is a mistake in the mathematics, such as a definition that
contradicts another definition (subtly). The equations as presented
propose that v*h/s = n. Since we also have already that a = n*s, it
follows that v*h = a. From other equations we have
(6) h = Y*d, where Y (gamma in Killeen's paper) is just a constant of
proportionality.
so now _a_ is being proposed to depend on the level of deprivation.
Bruce, this is just too much of a mess. It's exactly the sort of tangle
one gets into when trying to make sense of a mathematical development
that is flawed from the start. Juggling quantities around and giving
different combinations of them different names is just wishful thinking;
nothing new is being introduced.
When you divide the minimum time for n responses by the minimum time
for 1 response (pair of responses), you end up with n*s/s, or simply
n. And of course n*R = B.
This would be true if a were computed from n*s, but I take it that
it is derived empirically.
No, it is true because the mathematics says it is true. Given the
definitions and relationships Killeen proposed, this is the mathematical
outcome. If Killeen does not come up with this answer empirically, then
he is not interpreting the data according to his own equations. The
moment you write down your mathematical expressions, ALL the
implications are decided. The subsequent manipulations simply reveal
what the implications are. If those implications don't fit your
empirical measurements, then you have done something wrong; either your
model is wrong or you measured incorrectly.
I recall your having said in a post a few weeks ago that if
Killeen's model contained a control system, it was pretty well
hidden.
At that time I hadn't seen the part dealing with deprivation -- only the
stuff relating to "specific activation," which is still just a
tautology.
But I hope you will find some way to remark that his "mechanics of
behavior" model can be disregarded.
If Killeen's model is taken seriously in EAB then I can't just
disregard it. My strategy is first to understand it, then to expose
its problems and Killeen's mistakes in analyzing his own model,
then to develop an alternative PCT model which can be directly
compared to Killeen's. My food-level control model represents the
first stage in the development of that alternative model.
Well, good luck. You may expose Killeen's mistakes, but it will be a
miracle if he doesn't find some way to claim that they aren't mistakes.
By the way, have you worked out whether the reinforcement loop in
Killeen's model has positive or negative feedback?
Not yet.
···
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Hans Blom, 960108 --
The Test For Model-Based Control null hypothesis: a model-based
control system's performance will not appreciably degrade when no
feedback is available. (Sloppy formulation, but you get the gist, I
presume. Plenty of qualifications apply, for instance that no model
is perfect and that the period of no feedback should not be too
long, due to the accumulation of prediction error over time).
Excellent. The qualifications are OK -- the question of _how good_ the
model-based control is can bew treated separately from the question of
whether it occurs at all.
One example of a practically possible test: ask the subject to
perform some cursor tracking test of a predictable periodic signal
(e.g. square, triangle or sine wave) in which the feedback is
sometimes unavailable, but where the periods where the feedback is
missing are observable, e.g. because the trace on the display
disappears. Otherwise there is no noise. If the subject keeps
tracking more or less satisfactorily, control is model-based.
Initial practice for some time is required to build up the model in
the first place.
This isn't quite a clean test from the PCT point of view, because it
omits the usual random disturbance. In a predictable tracking task, the
subject can learn to produce a relationship between target position and
_hand_ position, when the cursor is directly coupled to hand position.
This leaves the possibility that the subject has simply turned to
controlling a different perceptual variable, rather than employing a
model of the original variable. It's possible to use a feedback function
such as an integrator that prevents the hand movements from being
directly related to the cursor movements, to reduce this source of
alternative hypotheses.
However, the best test would use smoothed random disturbances. Even with
regular and predictable movements of the target, the hand movements
would no longer have a predictable relationship to the now-invisible
cursor position. You have demonstrated that it is possible for a model-
based control system to deal with disturbances when the perceptual
pathways are intact, so the presence of a similar disturbance should not
be an unfair test of model-based control.
As usual in tests like these, the problem is how to rule out
explanations other than the main one. I think there will probably be
some ways of doing the experiment that show a person continuing to
produce actions calculated to control something, with other hypotheses
reasonably well ruled out. After all, we do get around in the dark for
brief periods, and we do have world-models! There are many situations
where it would be hard to explain behavior without introducing model-
based behavior through control of an imagined world when the real one
can't be sensed.
What I anticipate is that there will be some cases in which model-based
control will turn out to be the most reasonable explanation of what we
observed, while in others it won't work at all. I think there are many
control tasks in which a person deprived of present-time perceptual
information will immediately fail, so it would not be a good idea to
insist that _all_ behavior involves model-based control. However, if we
can learn more about when this kind of control does and does not happen,
we will know more about the structure of the control hierarchy than we
knew before.
if feedback available then
perception := f (action) {f is the world's transfer function}
else
perception := g (action) {g is the model's transfer function}
Neatly put.
Am I clear?
Transparent! Shall we do some experiments?
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Shannon Williams (960108.2am) --
I think we'd better take things one step at a time. Would you define
what you mean by a "loop?"
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Best to all,
Bill P.