[From Bill Powers (2012.01.05.1035 MST)]
Martin Taylor 2013.01.03.23.02 –
BP earlier: This formal
definition doesn’t help me much, I’m afraid. I can see what the
calculation is, but I still don’t understand what the phenomenon is to
which the word “uncertainty” is attached.
MT: You often use the example of “energy” to illustrate an
abstraction. The calculation of “energy” depends on measuring
various other quantities, themselves abstractions. But everyone has a
vague intuitive idea about it, something that probably was not true 300
years ago. The same is true of “frequency”, or
“average”, or “power”.
BP: Yes. I’ve used such terms often, as most of us have. But now I find
myself looking at them and asking what I mean by them. This really
started when I entered college physics and the same questions came up for
the second time (the first time they came up was in high school physics,
but I believed everything I was taught then). When I asked the same
questions in college, the professor got impatient with me – he wasn’t
teaching a course in philosophy.
BP earlier: You’re showing how
to compute how much of it there is, but you’re not saying what
“it” is.
MT: “It” is what the computation produces. That’s why I said
“No more, no less”. Have you ever seen an “average”?
It’s what a computation produces from certain data, no more, no
less.
BP: It’s the “no more” part that is slowing me down. Aren’t all
these computed entities supposed to be proposals about the nature of
reality? I asked my professor in college if we really needed such terms
– wouldn’t it be possible to say everything we knew about the physical
world without using the term “energy?” As I remember it, that
is the one I was asking about then. For example if a certain number of
kilograms of force were used to move a paddle around and around for some
distance through one kilogram of water, that would raise the temperature
of the water by some amount, and we would find that the amount per
kilogram-meter remained constant. Why imagine that something called
energy has been transferred from the stirring paddle to the water?
There’s no way to measure the energy directly – we always have to look
at some specific example and then imagine the energy, don’t we?The
smart-assed kid, needless to say, didn’t get a straight answer about
that, other than “that’s the way it’s done in
physics.”
MT: But as with
“uncertainty” or “energy”, people have a vague
intuitive idea of “a man of average height”. You CAN tie in
these concepts with some rather ill-defined everyday concept, and
sometimes it helps if you do that. But it can be misleading as well, just
as using an intuitive concept of “perception” can be misleading
when considering the implications of PCT.
BP: Yes, and I don’t think I ever disputed the usefulness of such
short-cuts. It was only when I had taken psychology courses for a while,
and learned that this sort of thing was done in psychology a lot more
than in physics, that I learned about the word “reification,”
which was used almost as a name for a disorder.
BP: What does the “everyday
meaning” of uncertainty have to do with the
calculation?
MT: No more than the “everyday meaning” of
“perception” has to do with the meaning of
“perception” in “Perceptual Control
Theory”.
BP: I don’t think I agree with that. In PCT, the parts of the model
having to do with perception are intended to amount to an explanation of
what we can see directly: the observed fact of a world in and around us.
If that world is outside our skins or hidden deep inside, how is it that
we can know about it, when no knowledge can get into a brain, as far as
we know today, except in the form of neural signals? What does energy
mean that corresponds directly to something in the world I experience? I
can answer that question for the term “perception.”
BP earlier: Are you saying that
the everyday meaning is obtained when a person has a perceptual input
function that performs one of the above calculations?
MT: No perceptual input system I ever heard of could make such a
computation. I suppose it would be possible to imagine one, but usually
it is a mathematical analyst performing numerical calculations, or more
commonly doing algebraic manipulations, that would do the calculations.
We do perceive our own uncertainty about things, but I seriously doubt
that the formal computation is involved in creating such a
perception.
BP: Wait a minute. How can the analyst do the calculations without being
able to perceive them? Symbols are perceptions, aren’t they? The rules of
mathematics are perceptions, too, and when we do math we are using some
sort of neural network when we use the rules to manipulate the symbols,
or so I think. We first learn how to manipulate the physical symbols by
writing them on paper or a blackboard, moving them around in
configurations we call equations, canceling them on one side of the equal
sign by striking through them and then writing them as appropriate on the
other side in the right place. Then we learn to do all but the most
involved manipulations in imagination – we can just see how a factor
multiplying the whole numerator disappears if the same factor multiplies
the whole denominator. It’s like playing with blocks, but with different
rules.
BP earlier: Does your formula
apply when the system involved, such as a cruise control, does not
contain the circuitry necessary for experiencing
uncertainty?
MT: “Uncertainty” is indeed something one experiences, a lot.
But I don’t think that has to do with any explicit calculation. I don’t
know why you bring up “experiencing”
uncertainty.
BP: Because when you try to say what the word means, it’s that experience
of being uncertain (a perception) that you have to rely on. You look at
the solution of the equation, which says “x = 7.233,” and you
think “OK, something close to that but not exactly and not every
time.” And you wonder just how far x might get from 7.233 if you
took new data and calculated it again. Isn’t that how statistics got
started?
MT: No control system that
“experiences” (i.e. has a perceptual function that produces a
value of) any perception also “experiences” another perception
such as that of “uncertainty”.
BP: But you just said it does! When you write U(X), that perception (or
an associated meaning) is what you’re experiencing as its meaning.
I’m wondering now if you’re going through a stage of understanding of PCT
that is like one I went through. There was a time when I did NOT say
“It’s all perception.” I made exceptions for the familiar
things that made up my world – my thoughts, for example, or just my room
in the house where I lived. Some things were just there, and I had
perceptions about them. In here are my thoughts and interpretations and
goals and all that, and Out There are the things the thoughts,
interpretations and so on are about.
MT: A control system for which
the perception was simply “uncertainty” would not be
experiencing any other perception. The cruise control
“experiences” various flight parameters, none of which is
“uncertainty” so far as I know. Nor does it experience a
spectrum of output variation, despite the fact that a Fourier analysis of
any of its output variables has such a spectrum, which can be computed
from a time series of its output values.
BP: So are you saying that certain things exist, and you know they exist,
which are not perceptions? That’s what I mean by making exceptions. I
just gave you an example where this could happen: I wrote “x =
7.233” and then started talking about being uncertain of getting
that same number from new data. Right at that moment, I was treating the
relationship of equality as actually existing like a piece of data
independent of me (and whether I was aware of it or not, I also treated
the way I treated it as simply existing, not as something I perceived
myself doing). I was aware of being uncertain of the repeatability of
that relationship, and that is what I was describing as the content of my
thinking processes – but again, not as the content of a thinking
process.
MT: What is meant by
“probability”? I could get into an interesting discussion on
that, which I hinted at in my message, but to do so would, I think, be a
side track. The computations of uncertainty work the same no matter how
you choose to think of “probability”, provided you wind up with
mutually exclusive possibilities whose total probability sums to 1.0. It
works the same for frequentists and for subjectivists.
============Topic Change===========
What follows is a side-track into the “what is probability”
discussion, which I do not want to pursue while I’m working on the
uncertainty, information, and control series of messages. I put my
position here, and expect to leave it at that.
BP: Can’t have that – this is a background thought that you’re talking
about, and it explains why you are going to say what you’re going to say,
doesn’t it?
BP earlier: It would seem that
there must be a random variable involved, and repeated trials must be
involved, in order for us to speak of probabilities.
MT: Here is where I deeply disagree with you, and with a whole school of
thinkers. Your statement reflects the “frequentist” or
“objectivist” school of thought about probability. I think that
is odd, given your strong assertion that all we know of the world is what
we perceive, not what is “objectively” out there in the real
world. The word “random” also raises a reddish flag, but I’m
not going to go into that other than to say that in most cases where that
word is used, it is used as a substitute for “I don’t know the
influences that determine what it will be”.
BP: You’re quite right about what I said, but what I mean is what you
said: probability is simply a word we use when we don’t know all the
deterministic influences at work. But what is wrong with seeing this as
if frequencies of occurrance were the primary data?
MT: Does the statement “I
think the probability is less than .001 that the next bird I see in my
garden will have blue wings and a red breast” mean anything to you?
If it does, you are not a frequentist, because no random variable is
involved and there can be no repeated trials.
BP: Yes, it has meaning to me, but only in the sense that I would say we
can’t estimate the “actual” probability without some data about
how often any particular bird visits my garden at this time of the year.
I have no trouble translating that statement into perceptual terms, so I
know I’m really asking how often I could expect to experience – perceive
– something under the current perceived conditions.
What I mean by “random variable” is simply any variable that
varies for reasons I can’t observe, and in ways in which I see no
regularities.
MT: Nor can there be a
repeated trial for a more mundane statement that “I think the
probability is close to 0.5 that the very next time I toss a coin it will
come up heads.”
BP: So how did you find out that tossing coins in fact favors neither
heads nor tails? Either you tested this premise or you imagined a
principle that predicts equal numbers of each on repeated trials. In
fact, a properly-designed coin-tossing machine could be able to alter the
50-50 balance.
Or “I think the probability
is about 0.2 that I will take a tour to South Africe next Spring”
(that happens to be a true statement, but if you had asked in October,
the probability would have been around 0.8). I have never been to South
Africa, so where are the repeated trials?
BP: Now you’re not talking about calculated probabilities, but only about
imagined outcomes. Of course you can give yourself such numbers, but they
don’t have the same kind of meaning they would have in a formal
experiment. There is no basis other than your own desires and guesses for
picking a number.
MT: You might well say that I
have tossed a coin many times and have observed that heads have come up
on about half the trials, but in truth there were NO repeated trials,
because each time I tossed a coin, the surrounding circumstances were
different.
BP: Different enough to make a difference? If you simply blurt out a
guess about the probability, you can’t answer that question, but neither
can you say that the circumstances were sufficient different. And you
can’t say what deviations from “fairness” could be expected
under any circumstances. When you say you don’t know what is causing the
variations, you can’t also conclude “… and therefore heads are as
probable as tails.” The only way to find out what the distribution
of the perceived outcomes will be is to experiment – first to see if the
observed frequencies are stable, and then if they are equal.
MT: The individual tosses
were different in many, many, ways, all of which you might judge to have
had no influence on the fall of the coin. But it is only your judgment
that allows you to call the different coin tosses “repeated
trials”. There has never been a repeated trial of anything at all,
though people have judged that the differences among the trials don’t
matter enough to make a difference in the trial result. Failures of such
judgments have caused not a few scientific errors.
BP: True, but there is also a judgement involved in deciding whether any
variations in the causes of changes are large enough to matter. You’re
crossing the line between qualitative judgment (no trial can ever be
“repeated” exactly) and quantitative judgments (… but trials
can be repeated so as to minimize unexpected variations).
Many people make this mistake about PCT. It is sometimes carelessly said
that control systems “correct errors.” That leads people to
question the whole theory, because if errors are in fact corrected, the
first guess would be that there can be no output from the control system.
To understand control properly, you have to convert the qualitative idea
of correcting error to the quantitative form of making errors small
enough.
MT: I say that the reason I can
make the probability statement (0.5 heads) about the coin is that I have
a mental model of what may or may not influence the coin the next time I
toss it, and nothing in that model favours head or tail. If I give the
coin to a conjurer, I might change my assessment of the probability of a
head on the next toss if I know that the result makes a difference to the
coin tosser.
Aren’t you forgetting that when we estimate probabilities from data, we
are allowing influences in the (hypothetical) external world to act
according to whatever properties exist in that world? What you believe to
be the probabilities can be refined by doing experimentsm, and your
beliefs will then predict perceived outcomes better. A riori
probabilities are untrustworthy, or at least less trustworthy than those
derived from competently done experimentation.
…
MT: Your questions are useful,
because I had no idea that such issues might cause you problems. I will
have to address them more than I had intended. Perhaps doing a parallel
intuitive introduction to Fourier analysis might make things clearer. I
hesitate to do that, though, because the current draft of an introduction
only to Uncertainty and Information is too long, even without addressing
the use of the concepts in control analysis.
BP: I haven’t had much luck with making thing clearer through changing
the example by any large amount. What seems “parallel” to you
may not be, and in my experience is usually not, obvious to anyone
else.
Best,
Bill P.