From: Amara Graps (amara@amara.com)
Date: Fri May 24 2002 - 10:33:17 MDT
Pat Fallon:
>There are many sources of human knowledge, but none has authority. All
>theories are tentative. Popper argues that a propper scientific theory
>makes testable predictions; it is falsifiable.
_Falsificationism in Statistics_
Pg. 171-173, Howson and Urbach
<begin quote>
Theories seriously entertained by scientists at one time are often later
rejected when reviewed in the light of new experimental evidence. The
most straightforward form for such rejections is that of a logical
refutation, and provided one is prepared to concede certainty to the
refuting data, such refutations may be regarded as scientific modes of
inference which require no concession to Bayesian principles. Indeed
some philosophers, keen to avoid a subjective probabilistic assessment
of hypotheses, maintain that logical refutations assessment of
hypotheses, maintain that logical refutations are the only significant
type of inference in science. However, as we explained earlier, a large
part of modern science is concerned with statistical hypotheses, and
these are generally not refutable in this way. As an example of a
simple statistical hypothesis, take the theory that a particular penny
has an even chance of landing heads and tails, with separate tosses
being independent (the penny is then said to be a 'fair' coin). This
theory cannot be refuted by observing the outcomes of trials in which
the penny is tossed; no proportion of heads in any sequence, however
large, is precluded by the theory. Nevertheless, scientists do not
regard statistical theories as necessarily unscientific, nor have they
dispensed with procedures for rejecting them in the face of what they
take to be unfavourable evidence. What principles apply here and how can
they be justified?
One answer, in the falsificationist spirit, that is occasionally
canvassed claims that although statistical theories are not strictly
falsifiable, they are falsifiable in an extended sense of the term: as
Cournot (1843, p. 155) expressed the idea, events which are sufficiently
improbable "are rightly regarded as physically impossible". Popper had
the same notion: scientists, he said, should make "a methodological
decision to regard highly improbable events as ruled out -- as
prohibited" (1959a, pg. 191). And he talked of hypotheses as having been
"practically falsified" if they attached sufficiently low probabilities
to events that actually occurred. Watkins, endorsing the idea called it a
"non-arbitary way of reinterpreting probabilistic hypothesis so as to
render them falsifiable" (1984, pg. 244).
Popper defended his position with a surprisingly weak argument. He
claimed that extremely improbable events that did happen "would not be
physical effects, because, on account of their immense improbability,
*they are not reproducible at will* (1959a, pg. 203). This
unreproducibility of very improbable events, Popper reasoned, means that
a physicist "would never be able to decide what really happened in this
case, and whether he may not have made an observational mistake".
Poppers seems to be operating here with a rather eccentric definition of
'physical effect', which would exclude most natural phenomena from that
category, for most natural phenomena cannot be humanly controlled and so
are not reproducible at will. More importantly, Popper's claim that
unreproducible effects cannot be properly checked is clearly mistaken;
improbable and unreproducible events -- for example the sequence of
heads and tails produced when a coin is tossed ten thousand times are
not necessarily so fleeting as to prevent a close examination.
The Cournot-Popper view overlooks the fact that very improbable events
occur all the time. Indeed, it would be difficult to name a probability
so small that no event of some smaller probability had not already taken
place or is not taking place right now: events of miniscule probability
are ubiquitous. Even a probability of 10^(10^(12)), which Watkins
considered to be 'vanishingly small' and to amount to an impossibility
(1984, pg. 244) is nothing of the sort. The probability of the precise
distribution of genes in the five billion members of contemporary
humanity is incomparably smaller than this, relative to Mendel's laws of
inheritance, as is the probability that the atoms in the jug of water on
this table have a particular spatial distribution at a given time.
Popper attempted to give pragmatic effect to this thesis by propounding
a rule that would tell us in particular cases how small a probability
should be in order to be classed as a practical impossibility. We shall
follow Watkin's recent exposition of the rule.
<end quote>
[... typing fingers and time constraints ... Please see this book
for the pages of text and math and reasoning etc etc.]
REFERENCES
_Scientific Reasoning: The Bayesian Approach_ by Colin Howson and Peter
Urbach, 1989, Open Court Publishing.
Cournot, A.A. 1843, _Exposition de la Theorie des Chances et des
Probabilites_ Paris.
Popper, K.R. 1959a, _The Logic of Scientific Discovery_. London:
Hutchinson.
Watkins, J.W.N. 1984. _Science and Scepticism_. London: Hutchinson and
Princeton: Princeton University Press.
This URL might be useful too (Edwin Jaynes)
http://bayes.wustl.edu/
-- ******************************************************************** Amara Graps, PhD email: amara@amara.com Computational Physics vita: ftp://ftp.amara.com/pub/resume.txt Multiplex Answers URL: http://www.amara.com/ ******************************************************************** "We don't see things as they are, we see them as we are." --Anais Nin
This archive was generated by hypermail 2.1.5 : Sat Nov 02 2002 - 09:14:19 MST