[Martin Taylor 951102 13:30]
francisco arocha, 95/11/02-10.15
+ Bill Powers (951102.0920 MDT)
The subjectivist interpretation of probabilties states that
probabilties reflect degrees of uncertainty of a hypothesis or a
proposition, not of an external event (that would be an objectivist
interpretation). ...
... To call guesses, probabilities is
confusing guesses or guesstimates with probabilities, which amounts
to confusing a psychological category with its mathematical
formulation.
+I'm curious; just how do psychologists think of this state we call
+uncertainty? ...What does being certain feel like? I should think you'd have
+to know that before you could say you're UN certain. Is uncertainty like
+a conflict?
+...
+Could somebody give me a restart on this meaningless noise?
Don't know whether this will be a restart or more meaningless noise, but
I'll have a go.
First off, let's make our usual assumption that there is a real world out
there, but that we can know nothing of it except through our perceptual
functions working (ultimately) on sensory data--that word "ultimately" is
intended to include imagination, memory, etc.
The consequence of that is to make it clear that the "probability" of
something, or its "uncertainty" is a perception. So it is legitimate to
ask, as Bill does, "what does being certain feel like," just as one may
ask "what does red look like." Red looks like "that folder," "those leaves,"
"that sunset," "that light"... You can't specify it in words, but you can
get across the idea by example, by trying to find a bunch of things that
have only their "redness" in common.
With "probability" as a perception, it may be more difficult to find examples,
since the probability is of a perception that is not a function _only_ of
immediately present sensory data. Probability is an attribute of some
fact or event not now present. I can get across the idea by trying to
choose facts or events about which I assume you have much the same notion
of probability as I do. If you don't, I'll fail, just as I will fail to
show a colour-blind person "what red looks like." I'll know that the
colour blind person didn't get it when he points out as red something I
would call green. So, if what I get across to you is not my idea of
uncertainty or probability, then at some point you will make an assessment
that is at variance with what I would say.
This happened in an earlier interchange on "uncertainty." Bill P. said
that the value of the consequence of a decision affected his perception
of the uncertainty of the decision. I've forgotten the example, but I'll
reconstruct an analogue. He would be more uncertain as to whether the
flower in a vase was an aster or a carnation (assuming unfamiliarity with
flower names) if its being an aster meant he had to do something very
important and its being a carnation meant he had to do the opposite, than
if the difference were just a matter of coming to know which was which.
In my use of the term, the uncertainty would be the same in both cases,
and the importance of the decision would be a separate issue.
So there's a problem of labelling. Bill P., for all I know, may not even
_have_ a perception that corresponds to the feeling I label with the word
"uncertainty". There's no "uncertainty" in the world, unless (possibly)
we are dealing with quantum effects, and even then, it's an interesting
issue as to whether the uncertainty is in the world or in our perception
of the world.
Second point. I assume we all have some kind of feeling that goes along
with the difference between "The sun will rise in the East and not the
West tomorrow," and "The weather will be fine for our picnic on Tuesday
next." We are less uncertain about the first statement than the second.
We believe strongly that the sun will rise in the East, but we may believe
equally that the weather will be fine and that the weather will be rainy
next Tuesday. I think it more likely that there will be 4 inches of snow
in Toronto on Jan 16 than on July 16 1996, but I'm almost certain that there
will be more than 4 inches total between Jan 1 and Dec 31 1996.
When the time comes, the sun will or will not rise in the East, and the
weather will or will not be fine enough to have the picnic. The uncertainty
will have become very low in both cases. But it is an attribute of a
single event, not of a set of many events. When this event has many
reasonable possibilities of how it will (or did) turn out, one is more
uncertain than if there is only one reasonable possibility. And that
word "reasonable" is also a label for a perception. There's no "reason"
out there in the world.
Starting (I think) with Buffon in the (?)17th century, people began to try
to put numbers on this feeling of uncertainty. Gamblers had been using it
for a long time, but without mathematics. For mathematics, you need some
kind of set of axioms, and those gave rise to (or were derived from) a
notion of "probability," which is quite different from "uncertainty" but
is still a label for a perception. One may be "uncertain" about what the
weather will be for the picnic, but that uncertainty depends on there being
several reasonable outcomes. One can associate a "probability" with each of
the outcomes. "Probabilities" are associated with numbers that conform
to certain axioms, no matter what the underlying perception may have been
before the concept of probability became mathematized. It's much the same
as the precision in the term "control" as used in PCT as compared to the
fuzziness of its everyday use.
Probabilities have certain requirements.
(1) If there is a set of possible outcomes for an event, their
probabilities must sum to 1.0 exactly. This immediately removes the
label "probability" from the perceptions that are an attribute of the
imagined event, and applies it to some symbolic perceptions derived from
those feelings.
(2) If one outcome of an event is perceived as being more likely than
another, then the "probability" of that outcome is the greater. "Likely"
is another label that has been mathematized, but here I use it in reference
to the perception that one associates with the imagined outcome. If you
perceive it as more likely next Tuesday will be sunny than that it will rain,
the "probability" of sun is greater than the "probability" of rain.
(3) If two events A and B have no causal connection (in the physicist's
sense), then the probability that A will have outcome i and that B will
have outcome j is eaxctly the product of the individual probabilties of
those outcomes.
(4) Since the probabilities are real numbers, the probabilities of
the outcomes of two disparate events can be compared. The probability
that it will rain next Tuesday is greater than, equal to, or less than
the probability that there is life under the nitrogen atmosphere of Titan
(and whether it is greater than or less than depends on who you are).
With these and maybe one or two other axioms, the whole mathematics of
probability is developed. And there remains a parallelism between the
results of the maths and the subjective feelings one has about the events
symbolized in the maths, at least most of the time.
Probability is entirely subjective and symbolic and deals with single
events ONLY. But how the underlying "likeliness" perception is acquired
is another matter (I don't use "likelihood" because that is also mathematized).
One often derives it from a model of the world together with current and
remembered perceptions of the world. Is the probability that the sun will
rise in the East tomorrow high because it always has, or because you have a
model of the spinning world that leads you to believe that the spin won't
change before tomorrow morning? A bit of each, I suspect. Is it improbable
that a camel could pass through the eye of a needle because, of lots of camels
that you have seen trying it, only a few succeeded? I doubt it. You have
modelled camels, needles, and the requirement for passage, and found in
your imagination few camels that could succeed. (Though as I remember, "The
eye of the needle" was the name of some city gate or other, with no
reference to a sewing needle; some camels might have got through).
"Frequentist" probability depends on the assumption that an event can be
repeated, sometimes with one outcome and sometimes with another. But no
event can ever be repeated. All we can do is to assert that two events
differ only in ways that do not matter to us. A coin has been tossed
a large number of times, and the more times it is tossed, the nearer the
proportion of heads comes to 0.5. It doesn't matter that the mail arrived
after the 651st toss, that Mercury completed three orbits around the sun
during the experiment, that Sam went to see the Pyramids during the 3,000 to
5716th toss... But we, personally, choose to see certain events as "the
same" for purposes of measuring the proportions, and when we imagine a
new event to be "the same" again, we can take that historic proportion and
use it to estimate the _probability_ of the outcomes of the new event.
"Frequentist" probability (or as it is sometimes, absurdly, called,
"objective" probability) substitutes historic proportion for the probability
of the new event. And if you have no model of what is going on, historical
proportion may indeed be the best way to get a probability number. But
historical proportion should never be confused with "probability."
Enough "meaningless noise" on "probability." How about "uncertainty?"
Above, I asserted that "uncertainty" is a perception about an event. Whereas
"likeliness," or in symbolic terms "probability," is associated with a
possible outcome, "uncertainty" refers to the relationship among possible
outcomes. The more nearly the probabilities are the same, and the more
of them there are, the greater the uncertainty about the event. But to
use "probability" in this way is already to mathematize "uncertainty."
We come around in a circle. "Uncertainty" as a feeling of, shall we say,
indecision, is a very vague, non-numeric kind of thing (except, in PCT,
every perception has a numeric value, but we ignore that for the moment).
"Uncertainty" as a technical, mathematical, term, captures that feeling
in much the same way that "probability" captures the feeling of "likeliness."
In the technical sense, the "uncertainty" of an event has no relation to
the importance of the various outcomes. It depends only and exclusively
on their probabilities. Shannon (The Mathematical Theory of Communication,
Urbana, U of Illinois Press, 1949) specified three axioms, which, like those
of probability, seem to capture much of the flavour of our feeling of
"uncertainty," if we leave out the importance of a decision. Shannon's
axioms are (paraphrased):
(1) Very small changes in the probabilies of the various outcomes lead to
very small changes in the uncertainty of the event.
(2) If all the outcomes have equal probabilities, then the more outcomes
there are, the greater then uncertainty about the event. (Remember that
the probabilities must sum to 1.0).
(3) If two or more outcomes are combined into one sub-event and their
probabilities as outcomes of the sub-event are then considered, that
combination and re-division does not affect the resulting uncertainty
of the original event. For example, if a light may possibly be shown
as "white or coloured" and "if coloured, it will be red or blue," the
uncertainty about the light colour will be the same as if it had been
thought of as "white or red or blue" initially.
These axioms are sufficient to define a mathematization of "uncertainty"
based on the prior mathematization of "probability." If the N different
outcomes of an event are labelled (1,2,...j,...N), and the probability
of outcome j is pj, then the uncertainty about the event is
H = -K(sum-from-1-to-N (pj log pj)).
In that formula, K is an arbitrary constant. The logarithmic form is
forced by the three "common-sense" axioms. Shannon also developed the same
form when the event can be considered as a measurement, with a continuous
range of possibilities, rather than as a choice among N discrete possible
outcomes. In the continuous form, the sum is replaced by an integral,
and probability is replaced by "probability density."
What is "probability density?" Simply put, if you define for yourself
an outcome as being "the measurement will be between X and X+deltaX",
then the probability density at X will be the probability of that outcome
divided by deltaX, as deltaX is reduced ever closer toward zero.
To return to Bill's original plaint about what "uncertainty" feels like:
Nothing mathematical really expresses what something "feels like." What
does "2" feel like, or "plus?" Both are abstractions from a whole mess of
ordinary perceptions of ordinary events and relationships. So it is with
uncertainty. Sometimes we "just know" what's going to happen (though we
may well be wrong when the time comes), and sometimes we "haven't a clue
how it will turn out." These feelings are abstracted into a mathematized
form as "probability" and "uncertainty." And on those symbolic
representations we can operate, with useful (or pointless) results.
As in any other technical area, terms may be used with precision in ways
that capture only a part of the range of everyday usage. Sometimes we
trip up when we use the everyday sense and are understood to be using the
technical sense, or vice-versa. But as with "Perception" in PCT, there
really isn't a satisfactory alternative to using the everyday terms with
which the technical concepts most closely relate.
···
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I hope there isn't too much gibberish there. I had no intention of making
it so long when I started, or of taking so much time on this issue. But I
think it does matter, so I won't apologize.
Martin