Hierarchies, degree of freedom

[Martin Taylor 920824 12:00]
(Bill/Mary Powers 920822)

Since Bill copied part of his mail to me to the net, I'll copy part of my
response, also slightly edited:



I can't agree that a change in combination method alters the NUMBER of
available degrees of freedom. It only changes the shape of the (hyper)surface
that defines variation in value for any one degree of freedom.

The number of degrees of freedom available is computable at any instant, but
things become harder if you include time variation, because then the
effective degrees of freedom depend on the tmeporal characteristics of the
feedback loop. Even in a linear system that can be hard (the first time
I came across fractional df was in Blackman & Tukey's Measurement of Power
Spectra, 1958). How many df per second there are depends not only on the
bandwidth, but also on the relative power in each delta frequency band.
When the system is nonlinear, the equivalent spectrum (if you can talk about
such a thing) may be very dependent on amplitude, and in a logical system it
can depend on all sorts of (apparently) trivial contextual factors. So when
you talk about the increase of df for sequence variables, you are talking about
summing over the duration of the sequence the df inherent in all the control
loops at any one level over that time period.

The one thing you cannot do is increase the available degrees of freedom.
Every loop has a bottleneck somewhere. A control structure cannot control
more df than the bottleneck permits. All I need for my argument is that
the bottleneck is not in the sensory input structure, and may be in the
muscular (and glandular, etc.) output systems. If it is in the outer world
behaviour, that's fine. It just must not be in the perceptual input system.

Configurations, no matter how they are defined, cannot produce degrees of
freedom. If there are nine notes from which configurations can be produced,
there can be no more than nine independently variable ways to produce
different configurations. Of course, if each note is discriminable as on or
off, there are 2^^9 different configurations, but that's not to say there are
2^^9 degrees of freedom for configuration, because you can't have so many
all independently variable. If you turn one note from on to off, you change
the configuration you had, as well as the one you now have. They work
together as aspects of the same single degree of freedom. How they work
together is contingent on the states of the other 8 notes, but only on that.
There are still no more than 9 df in the set of configurations.

As soon as you get to transitions, you get into the time-variation area I
was talking about. For the 9 notes, and 2 time samples sufficiently separate,
you get 18 df. No problem. But here the sensory systems overwhelm the
muscular bottleneck even more strongly. Sensory bandwidths are on the order
of tens to thousands of Hz, whereas muscular bandwidths are on the order
of single-digit Hz or less. So my argument becomes much stronger.

So I think that the degrees-of-freedom argument based on repeated
application of weighted summations to previous linear combinations of
weighted summations does not apply to the levels of perception as I
have defined them. This is because with each new level, as I have
defined levels, there is a change of logical type; no new level of
perception is commensurate with any lower level.

As you can see from the above, the df argument does not come from weighted
sums. I quite accept the change of type at each level, though there may
be some argument about whether a change of type means a change of mechanism.
But that's for later.