[From Bill Powers (2009.01.21.2218 MST)]

It seems to me that statistics (Bayesian or otherwise) is not relevant to

the kinds of models PCTers have been working on, for the simple reason

that the models are expressed in terms of systems made of differential

equations, not propositional logic. That statement may merely reveal my

lack of mathematical sophistication, but if that’s the case, someone else

is going to have to demonstrate the relevance of statistics to PCT,

because I can’t do it.

The only reason I can see for using ideas like correlation and

probability in our modeling is to try to suggest, in a semi-meaningful

way, how accurately the control-system models represent behavior, when

we’re communicating with people who use statistics as their main means of

evaluating theories. It’s highly unlikely that real behavioral data

conform to the assumptions that underlie concepts like correlation or

probable error, such as having a distribution that is normal, Poisson, or

something else with known properties. Much simpler kinds of analysis are

sufficient to show when one control model predicts better than another in

the realms we explore; if we had only ourselves to satisfy, why would we

ever bother to calculate a correlation or a probability? You can see that

the correlations are going to be almost perfect just by looking at the

data plots. The probability of such fits by chance is too close to zero

to measure.

In the tracking experiment, for example, the “damping”

coefficient can be set to zero, effectively leaving it out of the model,

and we can then assess how the fit is improved by adjusting it for the

least RMS difference between model behavior and real behavior. The same

can be done for the delay. The differences in RMS error we see are in the

third decimal place for the damping, and the second for the delay, with

total RMS error being 1% to 5% of the range of the real data. The

correlations we would compute here would be from 0.99 on up, with no

justification at all for using correlation since the differences we’re

talking about are not even random. If a difference can be reduced by

changing a single parameter, we’re looking at systematic effects, not

random effects. Even by eye one can see that the residuals are not

distributed unsystematically.

Our problem, I think, is that normal psychological experiments produce

data that are barely visible as a bias in the predominently random

fluctuations. In a control-system experiment, it’s just the opposite:

often, you have to look closely to see any irregularities. Here is the

fit of model mouse position to real mouse position over a one minute run

(done just now) with medium difficulty factor:

The darker trace is the model prediction; the light green trace is the

real mouse position. I would guess that p is much less than 1E-10. The

RMS error is a little over one per cent of the total range. The black

trace shows the actual residual difference between the two mouse traces.

It is nothing like normally distributed.

Obviously, the uses for advanced statistical treatments of this kind of

data are minimal. As long as we continue to do the right kinds of

experiments, that will continue to be the case – and why do any other

kind? It’s not as though we have explained such a large proportion of

known phenomena that we have to start searching with a magnifying glass

for something to study. See my Essay on the Obvious.

Best,

Bill P.