Arnold Neumaier
Aug16-04, 12:55 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nIgor Khavkine wrote on p.2 of\n\n> http://www.physics.utoronto.ca/~igor/densityop.pdf\n\n\'\'statistical mechanics may be applied to a system\nwith any number of degrees of freedom as long as\nthere is not enough information available to apply\nthe laws of the underlying precise mechanical theory.\'\'\n\n\n\nIt is offen erroneously assumed that incomplete knowledge can\nalways be described by statistics. But this is by no means the case.\n\nIf you know about a number x only that it is in [0,1], you cannot apply\nstatistics since you know nothing at all about the distribution\n(except for its support). In particular, it would be a mistake to assume\nthat the distribution is uniform (ignorance interpretation)\n-- the noninformative prior of the Bayesian school, which makes this\nassumption, may be seriously flawed, since it is perfectly consistent with\nthe knowledge that in fact always x=0.75, except that you don\'t know it,\nor that x oscillates regularly, or.... The ignorance is in this case\nsimply deterministic lack of information.\n\nIn general, all one can deduce from information that takes the form of\ndeterministic bounds on a vector x of variables and/or on expressions in x\nis bounds on derived quantities y=f(x) one would like to compute from it.\nThis leads to global optimization problems, where f(x) is minimized or\nmaximized subject to the known constraints. See\nhttp://www.mat.univie.ac.at/~neum/glopt/intro.html\n\nStochastic lack of knowledge is of a different kind - it assumes that\nthe _maximal_attainable_ knowledge about the system, at the given level\nof description, is a probability distribution, and that this probability\ndistribution is indeed known.\n\nThere are also combinations of both, where one knows the maximal knowledge\nabout a system should be stochastic, but one lacks complete information\non the distribution. This is handled by the emerging field of \'imprecise\nprobability\', although there is not yet a generally accepted way for\nanalyzing such situations, and different schools with quite different\nbasic approaches compete. See, e.g,\nhttp://class.ee.iastate.edu/berleant/home/ServeInfo/Interval/intprob.html\n\nTheoretical physics is always concerned about describing the maximal\nattainable knowledge about a system (at a given level of description),\nirrespective of what anyone actually knows about it. In this way, and only\nin this way, it is possible to get close to the objectivity that science\nalways is striving for.\n\n\nArnold Neumaier\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Igor Khavkine wrote on p.2 of
> http://www.physics.utoronto.ca/~igor/densityop.pdf
''statistical mechanics may be applied to a system
with any number of degrees of freedom as long as
there is not enough information available to apply
the laws of the underlying precise mechanical theory.''
It is offen erroneously assumed that incomplete knowledge can
always be described by statistics. But this is by no means the case.
If you know about a number x only that it is in [0,1], you cannot apply
statistics since you know nothing at all about the distribution
(except for its support). In particular, it would be a mistake to assume
that the distribution is uniform (ignorance interpretation)
-- the noninformative prior of the Bayesian school, which makes this
assumption, may be seriously flawed, since it is perfectly consistent with
the knowledge that in fact always x=0.75, except that you don't know it,
or that x oscillates regularly, or.... The ignorance is in this case
simply deterministic lack of information.
In general, all one can deduce from information that takes the form of
deterministic bounds on a vector x of variables and/or on expressions in x
is bounds on derived quantities y=f(x) one would like to compute from it.
This leads to global optimization problems, where f(x) is minimized or
maximized subject to the known constraints. See
http://www.mat.univie.ac.at/~neum/glopt/intro.html
Stochastic lack of knowledge is of a different kind - it assumes that
the _maximal_attainable_ knowledge about the system, at the given level
of description, is a probability distribution, and that this probability
distribution is indeed known.
There are also combinations of both, where one knows the maximal knowledge
about a system should be stochastic, but one lacks complete information
on the distribution. This is handled by the emerging field of 'imprecise
probability', although there is not yet a generally accepted way for
analyzing such situations, and different schools with quite different
basic approaches compete. See, e.g,
http://class.ee.iastate.edu/berleant/home/ServeInfo/Interval/intprob.html
Theoretical physics is always concerned about describing the maximal
attainable knowledge about a system (at a given level of description),
irrespective of what anyone actually knows about it. In this way, and only
in this way, it is possible to get close to the objectivity that science
always is striving for.
Arnold Neumaier
> http://www.physics.utoronto.ca/~igor/densityop.pdf
''statistical mechanics may be applied to a system
with any number of degrees of freedom as long as
there is not enough information available to apply
the laws of the underlying precise mechanical theory.''
It is offen erroneously assumed that incomplete knowledge can
always be described by statistics. But this is by no means the case.
If you know about a number x only that it is in [0,1], you cannot apply
statistics since you know nothing at all about the distribution
(except for its support). In particular, it would be a mistake to assume
that the distribution is uniform (ignorance interpretation)
-- the noninformative prior of the Bayesian school, which makes this
assumption, may be seriously flawed, since it is perfectly consistent with
the knowledge that in fact always x=0.75, except that you don't know it,
or that x oscillates regularly, or.... The ignorance is in this case
simply deterministic lack of information.
In general, all one can deduce from information that takes the form of
deterministic bounds on a vector x of variables and/or on expressions in x
is bounds on derived quantities y=f(x) one would like to compute from it.
This leads to global optimization problems, where f(x) is minimized or
maximized subject to the known constraints. See
http://www.mat.univie.ac.at/~neum/glopt/intro.html
Stochastic lack of knowledge is of a different kind - it assumes that
the _maximal_attainable_ knowledge about the system, at the given level
of description, is a probability distribution, and that this probability
distribution is indeed known.
There are also combinations of both, where one knows the maximal knowledge
about a system should be stochastic, but one lacks complete information
on the distribution. This is handled by the emerging field of 'imprecise
probability', although there is not yet a generally accepted way for
analyzing such situations, and different schools with quite different
basic approaches compete. See, e.g,
http://class.ee.iastate.edu/berleant/home/ServeInfo/Interval/intprob.html
Theoretical physics is always concerned about describing the maximal
attainable knowledge about a system (at a given level of description),
irrespective of what anyone actually knows about it. In this way, and only
in this way, it is possible to get close to the objectivity that science
always is striving for.
Arnold Neumaier