What's a statistic

[From Rick Marken (940830.1420)]

Martin Taylor (940830 13:40) --

What's a statistic? I learned the word as meaning the reduction of a set
of values to one or a few representative values, such as, indeed, means,
variances, etc.

Actually, a _statistic_ is a measure of some characteristic of a sample (such
as the mean or variance of the sample values) that is used as an _estimate_
of the corresponding characteristic of the population from which the sample
was taken. The corresponding characteristic of the population is called a
_parameter_. Sample statistics are used to estimate ppoulation parameters.
The sample mean is used as an estimate of he population mean. Remember?

So, when you say that the perceptual input function computes a statistic I
hear you saying that perceptions are _estimates_ of what is really out
there. Of course, this is exactly the information theory view of perception;
the senses transmit information to the brain about what is actually out
there. The brain is a statistician using perceptual statistics to estimate
real world parameters. This is quite a different picture of perception
than that given by PCT. In PCT, perception is a functional analog of an
aspect of the external world that is defined by the perceptual function. The
perceptual function could compute the mean or variance of its inputs; so the
perceptual signal's value could represent variations in the mean or variance
of the inputs to the perceptual function. But this perceptual signal is not
an _estimate_ of anything (the mean or variance it represents if not computed
as a statistic); it is simply a perceptual variable.

One interesting question is just how far the computation of these statistics
reduces the number of degrees of freedom available in the input, because
it is that reduction that determines the magnitude of the alerting problem.

What is an "alerting problem"? How do you "alert" a control system?

Best

Rick

[Martin Taylor 940901 13:45]

Rick Marken (940830.1420)

Martin Taylor (940830 13:40) --

What's a statistic? I learned the word as meaning the reduction of a set
of values to one or a few representative values, such as, indeed, means,
variances, etc.

Actually, a _statistic_ is a measure of some characteristic of a sample (such
as the mean or variance of the sample values) that is used as an _estimate_
of the corresponding characteristic of the population from which the sample
was taken.

You mix up two things here. What IS a statistic, and what is a statistic
used for (or why would somebody compute one). The statistic is the mean,
for example, and somebody chooses to use it as an estimate of some
population from which the items used in the computation are asserted to
be representative choices. There is a whole mess of problems associate
with this "use" of the statistic, but there is no problem about what the
statistic is. It is one number, representing one degree of freedom out
of N degrees of freedom inherent in the N items measured. When you have
computed N independent statistics that's all you can get.

You tempt me into addressing the further questions of "what is an estimator?"
and "what is a population?" Both are interesting and often misunderstood,
but perhaps not very relevant to PCT (...maybe, though....let's wait and
see...h'mmm?).

So, when you say that the perceptual input function computes a statistic I
hear you saying that perceptions are _estimates_ of what is really out
there.

Now, that DOES depend on the use of the perceptual function. In its own
ECU, it provides a perceptual signal that is controlled. It isn't an
estimate of any population at all. It's a Popeye--I yam what I yam. But
as an element of another controlled perception, who knows what it might
be when seen by an outside analyst? Paul Revere controls for the correct
placement of the Minutemen to repel the British. He does not perceive the
location of the British directly, but uses his direct perceptions as
estimators for a perception of the probable location of the British.
It's what is "out there" that might kill you, whether you perceive it
accurately or not.

Of course, this is exactly the information theory view of perception;
the senses transmit information to the brain about what is actually out
there. The brain is a statistician using perceptual statistics to estimate
real world parameters. This is quite a different picture of perception
than that given by PCT. In PCT, perception is a functional analog of an
aspect of the external world that is defined by the perceptual function.

Other than in the Paul Revere kind of situation (which might not be uncommon),
there's no perception of an estimate except possibly in the mind of an
external analyst. Let's once more repeat after each other (since we seem
to like parroting it back and forth): we perceive what we have perceptual
functions for; the values of our perceptual signals are the content of
perception; we do not perceive the meanings of our perceptual signals except
insofar as they provide input to other perceptual functions. Anything
else?

Nevertheless, there IS the underlying presumption that there exists something
"out there," and when we draw control loops, no matter how simple, part
of the loop goes through the "great outthereness." What we perceive is
not "out there." It is "in here" and is different in kind from what is
"out there." In that sense, an external analyst who had PIFs for your
perceptions and for things "out there" could see that your perceptual signals
are in fact "_estimates_ of what is really out there." But the control
system "knows" none of that. Popeye.

ยทยทยท

===============

What is an "alerting problem"? How do you "alert" a control system?

Ref: multiple postings over the past year or two. Repetition of such
information is not anticipated unless new factors are introduced to the
discussion. But when you re-read them, please TRY to keep straight the
difference between a system that is an elementary control unit and a system
that is a complete functioning hierarchy with more input sensors than
output effectors.

Martin