Arnold Neumaier
Aug24-04, 04:52 AM
<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>Nick Maclaren wrote:\n> In article <4124B703.1090900@univie.ac.at>,\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:\n\n> |> > Large ensembles will not help. They need to be infinite.\n> |>\n> |> Strictly speaking, yes. But in the approximations physicists are\n> |> generally working, this amounts to the same. In the limit of arbitrarily\n> |> large samples, the statistical uncertainty goes to zero, under reasonable\n> |> assumptions.\n>\n> Not necessarily. There are realistic problems where the law of\n> large numbers does not apply.\n\nThat\'s why I had added the phrase \'under reasonable assumptions\',\nwhich excludes the situations you are talking about.\n\n\n> |> In a stochastic setting, _every_ realization of a stochastic process has\n> |> probability 0; exactly one of them actually happens - - - the certainty\n> |> status of a stochastic model for a single history seems comparatively\n> |> poor.\n>\n> Again, that is wrong. It doesn\'t apply to discrete measures, such\n> as when the spin of an electron can be either up or down.\n\nThis is not a counterexample. Taking for simplicity the stochastic process\ndefined by independent flips of a fair coin, a realization is an infinite\nbinary sequence, and each of these has probability zero.\n\nThe case of spin is more difficult to analyze because as stated, it is\nnot a well-defined stochastic process. If it is taken as a continuous\nmeasurement, the flip is at random times, and so even a single flip at\na definite time has probability zero.\n\nIf it is taken as a discrete process, we need to specify a measuring\nprotocol that applies at definite, equidistant times. Then it is likely\nthat there are some correlations, and probabilities even of finite\npieces of a particular realization are hard to get by. Nevertheless,\nunder reasonably random circumstances (for example, when measuring spins\nof independent electrons), the probability of the most likely sequence\nof N measurements decreases exponentially with N, and the probability of\na complete realization (infinite sequence) is again zero.\n\n\nArnold Neumaier\n\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>Nick Maclaren wrote:
> In article <4124B703.1090900@univie.ac.at>,
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:
> |> > Large ensembles will not help. They need to be infinite.
> |>
> |> Strictly speaking, yes. But in the approximations physicists are
> |> generally working, this amounts to the same. In the limit of arbitrarily
> |> large samples, the statistical uncertainty goes to zero, under reasonable
> |> assumptions.
>
> Not necessarily. There are realistic problems where the law of
> large numbers does not apply.
That's why I had added the phrase 'under reasonable assumptions',
which excludes the situations you are talking about.
> |> In a stochastic setting, _every_ realization of a stochastic process has
> |> probability 0; exactly one of them actually happens - - - the certainty
> |> status of a stochastic model for a single history seems comparatively
> |> poor.
>
> Again, that is wrong. It doesn't apply to discrete measures, such
> as when the spin of an electron can be either up or down.
This is not a counterexample. Taking for simplicity the stochastic process
defined by independent flips of a fair coin, a realization is an infinite
binary sequence, and each of these has probability zero.
The case of spin is more difficult to analyze because as stated, it is
not a well-defined stochastic process. If it is taken as a continuous
measurement, the flip is at random times, and so even a single flip at
a definite time has probability zero.
If it is taken as a discrete process, we need to specify a measuring
protocol that applies at definite, equidistant times. Then it is likely
that there are some correlations, and probabilities even of finite
pieces of a particular realization are hard to get by. Nevertheless,
under reasonably random circumstances (for example, when measuring spins
of independent electrons), the probability of the most likely sequence
of N measurements decreases exponentially with N, and the probability of
a complete realization (infinite sequence) is again zero.
Arnold Neumaier
> In article <4124B703.1090900@univie.ac.at>,
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:
> |> > Large ensembles will not help. They need to be infinite.
> |>
> |> Strictly speaking, yes. But in the approximations physicists are
> |> generally working, this amounts to the same. In the limit of arbitrarily
> |> large samples, the statistical uncertainty goes to zero, under reasonable
> |> assumptions.
>
> Not necessarily. There are realistic problems where the law of
> large numbers does not apply.
That's why I had added the phrase 'under reasonable assumptions',
which excludes the situations you are talking about.
> |> In a stochastic setting, _every_ realization of a stochastic process has
> |> probability 0; exactly one of them actually happens - - - the certainty
> |> status of a stochastic model for a single history seems comparatively
> |> poor.
>
> Again, that is wrong. It doesn't apply to discrete measures, such
> as when the spin of an electron can be either up or down.
This is not a counterexample. Taking for simplicity the stochastic process
defined by independent flips of a fair coin, a realization is an infinite
binary sequence, and each of these has probability zero.
The case of spin is more difficult to analyze because as stated, it is
not a well-defined stochastic process. If it is taken as a continuous
measurement, the flip is at random times, and so even a single flip at
a definite time has probability zero.
If it is taken as a discrete process, we need to specify a measuring
protocol that applies at definite, equidistant times. Then it is likely
that there are some correlations, and probabilities even of finite
pieces of a particular realization are hard to get by. Nevertheless,
under reasonably random circumstances (for example, when measuring spins
of independent electrons), the probability of the most likely sequence
of N measurements decreases exponentially with N, and the probability of
a complete realization (infinite sequence) is again zero.
Arnold Neumaier