Statistical metaphors

Tom Bourbon [940831.1456]

[From Rick Marken (940830.1420)] who wrote in reply to:

Martin Taylor (940830 13:40) who had earlier written about passive

statistical analyses by nervous systems. Here, Martin was replying to Rick's
question:

Martin:

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.

Rick:

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?

And so far as we know, _all_ of the sampling and calculations necessary to
produce these estimates are performed (more or less) knowingly by an entire
human being, or by a surrogate for a human being. Discussions of statistical
computations by small parts of human beings are metaphorical, in much the
same way that discussions in which nervous systems are said to transmit,
receive, or process information are metaphorical. Individual sensory
neurons do not sample selected parameters of the environment and then
calculate descriptive statistics (most-representative scores, ranges,
correlations, etc) or perform inferential statistical procedures (tests for
differences and their levels of significance, etc), or do they?

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.

To me, that is the _direct_ implication of the popular metaphor in which
sensory neurons perform statistical analyses on "features" of the
environment. If the metaphor is intended to be strict, then those
statistical procedures would be performed upon, and their results would then
comprise estimates of, something "real" in the environment. To determine the
adequacy of the statistical analyses, a person would compare the original
environmental parameters with the statistical estimates of those parameters.

When the metaphor is applied to nervous systems, all of these analyses and
comparisons are said to occur within individual neurons, but in fact they
require an entire person with access to all of the required data. The
limits on the plausibility of the statistical metaphor are similar to those
when the information metaphor is applied, as Rick also said:

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.

Nice understatement, Rick.

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.

Yes. A perceptual signal is a perceptual signal is an analog of
-- something? Speaking strictly, a perceptual signal _is_; it is not
"about."

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?

Interesting questions. Martin? :slight_smile:

Later,

Tom