Hi Digby

The short answer to your least squares question is

'because Matheron was a least squares person' ;-)

There are three basic schools of statistics:

(1) least squares (sometimes known as frequentist)

which probably includes the majority of

non-statisticians doing statistics. The least squares

approach might be paraphrased as "closest to the real

answer on average". It is based on the concept of

approximately Normal 'errors'

(2) maximum likelihood. This could be paraphrased as

'that solution from which the samples are most likely

to have come'. This yields much more general but not

always unbiassed answers. For example, the maximum

likelihood estimator of the variance is divided by 'n'

not 'n-1' as in least squares. Maximum likelihood

demands a pretty fair knowledge of the underlying

distribution of the samples, not the errors as such.

(3) bayesian estimation. Based somewhat on a maximum

likelihood method, you can build in your own prior

knowledge about the situation to affect the final

answers. (1) and (2) above rely solely on hard data.

This is, of course, a massive over-simplification but

serves, I think, to emphasise why there are many

different answers to the same problem and why you have

to define what you mean by "best" before you get the

"best answer". I try to cover these concepts in

non-rigorous terms in all my teaching.

It is also true that the mathematics is much simpler

if you use variances than if you use any other

function of differences.

Isobel

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