<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>On 2004-08-25, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n> Aaron Denney wrote:\n>> On 2004-08-19, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>\n>>>>Right. Probability assignments inherently have some subjectivity --\n>>>>what someone knows determines the assignment.\n>>>\n>>>Actually, what someone assumes as known. One can never know anything\n>>>about probabilities unless one has seen the whole ensemble. Thus one\n>>>makes models of the situation at hand, based on partial information,\n>>>and proceeds as if this model were correct. Calling this \'knowledge\'\n>>>is a misnomer.\n>>\n>> partial information isn\'t knowledge? It\'s not complete knowledge, but\n>> it is knowledge.\n>\n> it is knowledge about the sample but not knowledge about the ensemble.\n\nWho said anything about knowledge of the sample? Knowledge about the\npossibilities -- what space the ensemble covers, in your terms.\n\n>> You can specify with probability theory "I\'m not\n>> sure of this, but I believe it likely".\n>\n> One can believe anything. Belief needs no mathematics.\n\nIt doesn\'t need it, but it can use it. If you want a consistent\ncalculus of beliefs represented with real numbers between 0 and 1\nthat reduces to aristotelian logic in the limit that all of these\nnumbers go to zero and one (representing false and true), then\nyou\'ll get probability theory.\n\nIf you want to construct "no loss" bets, you also gets probability\ntheory. The axioms you use to derive probability theory don\'t\nmatter so much as what you get at the end (which we broadly agree\non -- what we don\'t agree on are applicability and how to construct\nprobabilities "from scratch"). Kolmogorov\'s formulation gives the\nsame rules, but one doesn\'t need to go through the rigamarole of\ndefining the proper sigma algebra in order to actually use probability.\n\n>>>>>Indeed, suppose we intend to throw a coin exactly once.\n>>>>>Person A claims \'the probability of the coin coming out head is 50%\'.\n>>>>>Person B claims \'the probability of the coin coming out head is 20%\'.\n>>>>>Person C claims \'the probability of the coin coming out head is 80%\'.\n>>>>>Now we throw the coin and find \'head\'. Who was right? It is undecidable.\n>>>>\n>>>>Any of them, or none of them, depending on what they knew about the\n>>>>prior conditions of tossing. "Appropriate" probability assignment would\n>>>>be better language than "correct". All of them could be correct if A\n>>>>knows only that it has heads and tails, and that both can come up, if B\n>>>>knows that the coin is heavier on the heads side by a certain amount,\n>>>>and C knows that the tosser is extremely practiced and can make it come\n>>>>out heads 80% of time.\n>>>\n>>>But for this analysis it is irrelavant what the result of the throw was.\n>>>Each party feels it was right; and it was. This is the hallmark of\n>>>unscientific theories...\n>>\n>> Yes. Probability isn\'t science, nor is it supposed to be. It\'s a tool\n>> that can be used in science. It\'s an extended logic, that reduces to\n>> standard aristotelian logic in the case that all probabilities are 0\n>> or 1.\n>\n> Probabilities are interwoven into the fabric of quantum mechanics,\n> do you want to claim that quantum mechanics is not science???\n\nOf course not. Real numbers are interwoven into the fabric of classical\nmechanics, but I\'m not going to call real numbers science either.\n\n>>>A: It is fallacious to assume a head probability of 50% based on knowing\n>>>only that a coin has a head and a tail. Ignorance cannot be replaced by\n>>>equiprobability.\n>>\n>> And yet, that seems to be what you do when you take an ensemble and\n>> assume that all realizations of it are equiprobable.\n>\n> In defining an ensemble, you can _define_ the weights of each member.\n> As I had asserted, the choice of ensemble is a subjective act that\n> determines what the probabilities mean. It encodes what the user is\n> prepared to assume about the given situation.\n\nGiven that I know nothing more than heads are possible, and tails are\npossible, any assignment (any weighting of ensemble members) other\nthan 50:50 is ... presumptuous, at best. It doesn\'t obey the inherent\nsymmetry that a single toss has. (Multiple tosses, don\'t have that same\nsymmetry -- e.g. 2 tosses with no information other than that gives you\na symmetry between HH and TT and between HT and TH, but not between\nthose two classes.)\n\n>>>Thus the person\'s\n>>>assignment of probabilities is spurious. To trust completely, I\'d need\n>>>to hear a claim such as \'p=0.8 with confidence level of 5 sigma based\n>>>on a binomial distribution\'. The I\'d confidently assert that the\n>>>\'probability for tossing this coin\' is close to 0.8. But I would\n>>>still not claim a probability for \'the next toss\', except perhaps\n>>>informally in the sense of \'pars pro toto\'.\n>>\n>> Suppose they told you all that, with thousands of trials. Still, anyone\n>> else I\'ve talked to would gladly take 51:49 odds either way. (That is,\n>> behaved as if they believed p > 0.49, and p < 0.51.\n>\n> Oh yes; I\'d estimate probabilities the way everyone does. But I wouldn\'t\n> trust them completely.\n\nNo one expects you to trust completely, but even without full trust, they\nare useful, even for single cases. Multiple aggregated samples let you\nput more trust in the outcomes -- a high spread is less likely, but you\ncan\'t trust those completely.\n\n>>>And this is possible if and only if they know the complete ensemble\n>>>(with exception of a set of measure zero). This is what I claime\n>>>all the time. If the ensemble is known only incompletely, any claim\n>>>of \'enough information\' is spurious.\n>>\n>> Complete ensemble, all the ways something can happen, sure. That\'s only\n>> one limit, and reasoning before that limit is still possible.\n>\n> Yes, but it may give wrong results. Based on probability theory,\n> it even _has_ to give wrong results sometimes, with probability one.\n\nSo does reasoning with the full ensemble though -- say that the full\nensemble gives you a P=0.999999 chance of success, so you confidently\nassert that. Repeat the experiment a few million times, and you will\nsee some "failures".\n\n>>>>>>>Thus probabilities are meaningful not for the single event but only\n>>>>>>>as a property of the ensemble under consideration.\n>>>>>>\n>>>>>>And yet people bet on individual events all the time.\n>>>>>\n>>>>>Oh yes. They estimate probabilities, based on their favorite ensemble.\n>>>>>But as you know, people often lose their bets!\n>>>>\n>>>>Sure. That doesn\'t mean they estimated the probability wrong, or are\n>>>>misusing probability theory. Refusing to let probability theory deal\n>>>>with single events reduces its applicability to almost nothing, and\n>>>>people do successfully use it for single events.\n>>>\n>>>No. People who bet based on probabilities, tend to bet many times,\n>>>not just a single time.\n>>\n>> They also bet on non-repeatable events, knowing full well they aren\'t\n>> repeatable -- it will not happen that an ensemble similar to the one\n>> they assign the event will usefully describe an event in the future.\n>\n> Yes, but then they act irrational.\n\nAre both sides of the taker of bets irrational? Or only one? Or\nneither? Casinos and bookies expect other events "in similar ensembles"\nto cover this one, the average bettor doesn\'t bet enough for that to\nhappen, and the odds are skewed so that the chances of being down\nincrease with the number of bets. But even the casinos and bookies\nwill cover onetime events.\n\n>>>>>>of that subset occuring, or the probability of those particular\n>>>>>>realizations. Of course they don\'t say that the event will or\n>>>>>>will not happen, unless the probability is zero or one.\n>>>>>\n>>>>>Yes; this is why they say nothing at all about the single case.\n>>>>\n>>>>Sure they, just not something definite. That\'s why there probabilities\n>>>>instead of certainties.\n>>>\n>>>They say nothing that can be verified or falsified. Thus nothing.\n>>\n>> Well, as probabilities can\'t be verified or falsified for even\n>> multiple cases, they must mean nothing as well? No? Then what\'s the\n>> dividing point? 2 samples? 10? 100? 1000000? 10^100?\n>>\n>> Sampling an ensemble 10000 times will _probably_ have a result _near_\n>> 10000p successes, but it is not guaranteed. It\'s still not checkable,\n>> and there is no magical threshold. Everything that works for 10000,\n>> still works for 1, just with much looser bounds.\n>\n> Yes. There is no dividing line. Probabilities are known only if one\n> defines the ensemble by specifying its distribution. But then is is\n> no longer exactly the ensemble of interest. There is an uncontrolled\n> approximation step in between, made somewhat credible by the laws of\n> large number and other tools of probability theory.\n\nOkay. Why does this "uncontrolled approximation step" have to be done\nby choosing ensembles, rather than other ways of specifying\nprobabilities?\n\n>>>>Now, you can do most of this reasoning about uncertainty with ensembles\n>>>>rather than states of knowledge, but it\'s much harder and more\n>>>>complicated.\n>>>\n>>>\'States of knowledge\' are in fact not at all states of \'knowledge\',\n>>>but of \'prejudice\'. of \'assumptions about which ensemble to consider\'.\n>>>Those who make the best assumptions will have the best results.\n>>\n>> The choice of ensemble is also prejudice.\n>\n> Yes. But once it is chosen, probabilities are objective.\n> This is like for statements about infinite sets, say.\n> One must choose the axioms, which is a subjective act; then\n> the consequences are objective facts.\n\nSame for just specifying the probabilities, and not specifying an\nensemble.\n\n>>>>You have to make sure that the ensembles you come up with\n>>>>are not only consistent with your state of knowledge, but also don\'t\n>>>>tell you anything more -- that they aren\'t biased.\n>>>\n>>>And one has to do the same with the putative \'states of knowledge\'\n>>\n>> Yes, and the ways of doing this are the maximum entropy derivable\n>> ignorance priors you ridiculed earlier.\n>\n> There are many different ways one can do that; it depends on what\n> partial information is available. Maximum entropy is appropriate\n> only when the prior information is in the form of expectations.\n\nIt can accomodate constraints of many forms, actually. Mean\nvalue == stable, observed values are only one way of doing so, that\nleads to a nice expression in terms of equal lagrange multipliers for\nexchanges.\n\n>>>This is different when one has a multitude of events. Then the family\n>>>of ensembles that match the information available to a high degreee of\n>>>confidence is quite small, and finding such a description is already a\n>>>scientific task.\n>>\n>> To a high degree of confidence is a probabilistic statement, about a\n>> single event: the results of all the events.\n>\n> If you compound everything together, yes. But if one has such a complex\n> single event, the proability interpretation is no longer adequate,\n> and an interpretation of ensembles in terms of\n> \'\'predictions of numbers being close to their averages\'\'\n> is the appropriate one. This is what happens in statistical mechanics.\n> Rather than just accepting a rare occurence (e.g., a brick going upwards\n> due to fluctuations) as something within one\'s probabilistic model,\n> one would try to explain it away by assuming a hidden, unobserved cause\n> (someone throwing it). This is the way science weeds out the exceptions\n> to statistical reasoning.\n\nOr, one could say that the probability of someone throwing it up is much\n(i.e. many many many orders of magnitude) higher than that of spontaneous\nfluctuations causing it to go up. Observing that it goes up,\nprobability theory says that most likely someone threw it up. Expanded\nmodel, yes, but no need to have a magical cut off value, below which\nwe no longer trust probability theory to work.\n\n>>>That\'s why statistical physics is not subjective anymore.\n>>\n>> Because it\'s on the firm Bayesian foundation of maximizing entropy\n>> subject to constraints?\n>\n> No; this is a spurious foundation; it was invented by Jaynes 50 years\n> after the various Gibbs ensembles were already known to work well.\n> That max entropy is spurious can be seen from the fact that if we\n> were to know <H^2> instead of <H>, we\'d not get the canonical ensemble\n> but one that is completely off physically.\n\nYes, but <H^2> _isn\'t_ a good description of the system, and we can\'t\nreally know it. Energy is a conserved quantity, not energy squared; we\ncan\'t see energy squared move between system and environment. It\'s not\nan extensive quantity. There is no mechanism of constraint; we can\'t\nmeasure it locally physically, and be assured that it won\'t change.\n(Neither is energy, really. We can measure it relative some set zero\npoint. Fortunately, all we need to do is measure differences. Changing\nthe zero point of energy changes the energy squared in weird ways.\n\n> Only if we know the _right_ sort of information, max entropy works,\n\nCorrect. We have to know what the real physical, _stable_ constraints\nare to do physics. This is not surprising. If we get an answer not\nin accord with experiment, either we\'re asserting something is a\nconstraint that isn\'t, or there is some hidden constraint we don\'t\nknow about. This can vary depending on the timescale.\n\n> and to know what is the right sort of information presupposes\n> the insights of Gibbs. So max entropy is only a justification after\n> the facts are already in.\n\n\n\n--\nAaron Denney\n-><-\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>On 2004-08-25, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
> Aaron Denney wrote:
>> On 2004-08-19, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>>>>Right. Probability assignments inherently have some subjectivity --
>>>>what someone knows determines the assignment.
>>>
>>>Actually, what someone assumes as known. One can never know anything
>>>about probabilities unless one has seen the whole ensemble. Thus one
>>>makes models of the situation at hand, based on partial information,
>>>and proceeds as if this model were correct. Calling this 'knowledge'
>>>is a misnomer.
>>
>> partial information isn't knowledge? It's not complete knowledge, but
>> it is knowledge.
>
> it is knowledge about the sample but not knowledge about the ensemble.
Who said anything about knowledge of the sample? Knowledge about the
possibilities -- what space the ensemble covers, in your terms.
>> You can specify with probability theory "I'm not
>> sure of this, but I believe it likely".
>
> One can believe anything. Belief needs no mathematics.
It doesn't need it, but it can use it. If you want a consistent
calculus of beliefs represented with real numbers between and 1
that reduces to aristotelian logic in the limit that all of these
numbers go to zero and one (representing false and true), then
you'll get probability theory.
If you want to construct "no loss" bets, you also gets probability
theory. The axioms you use to derive probability theory don't
matter so much as what you get at the end (which we broadly agree
on -- what we don't agree on are applicability and how to construct
probabilities "from scratch"). Kolmogorov's formulation gives the
same rules, but one doesn't need to go through the rigamarole of
defining the proper \sigma algebra in order to actually use probability.
>>>>>Indeed, suppose we intend to throw a coin exactly once.
>>>>>Person A claims 'the probability of the coin coming out head is 50%'.
>>>>>Person B claims 'the probability of the coin coming out head is 20%'.
>>>>>Person C claims 'the probability of the coin coming out head is 80%'.
>>>>>Now we throw the coin and find 'head'. Who was right? It is undecidable.
>>>>
>>>>Any of them, or none of them, depending on what they knew about the
>>>>prior conditions of tossing. "Appropriate" probability assignment would
>>>>be better language than "correct". All of them could be correct if A
>>>>knows only that it has heads and tails, and that both can come up, if B
>>>>knows that the coin is heavier on the heads side by a certain amount,
>>>>and C knows that the tosser is extremely practiced and can make it come
>>>>out heads 80% of time.
>>>
>>>But for this analysis it is irrelavant what the result of the throw was.
>>>Each party feels it was right; and it was. This is the hallmark of
>>>unscientific theories...
>>
>> Yes. Probability isn't science, nor is it supposed to be. It's a tool
>> that can be used in science. It's an extended logic, that reduces to
>> standard aristotelian logic in the case that all probabilities are
>> or 1.
>
> Probabilities are interwoven into the fabric of quantum mechanics,
> do you want to claim that quantum mechanics is not science???
Of course not. Real numbers are interwoven into the fabric of classical
mechanics, but I'm not going to call real numbers science either.
>>>A: It is fallacious to assume a head probability of 50% based on knowing
>>>only that a coin has a head and a tail. Ignorance cannot be replaced by
>>>equiprobability.
>>
>> And yet, that seems to be what you do when you take an ensemble and
>> assume that all realizations of it are equiprobable.
>
> In defining an ensemble, you can _define_ the weights of each member.
> As I had asserted, the choice of ensemble is a subjective act that
> determines what the probabilities mean. It encodes what the user is
> prepared to assume about the given situation.
Given that I know nothing more than heads are possible, and tails are
possible, any assignment (any weighting of ensemble members) other
than 50:50 is ... presumptuous, at best. It doesn't obey the inherent
symmetry that a single toss has. (Multiple tosses, don't have that same
symmetry -- e.g. 2 tosses with no information other than that gives you
a symmetry between HH and TT and between HT and TH, but not between
those two classes.)
>>>Thus the person's
>>>assignment of probabilities is spurious. To trust completely, I'd need
>>>to hear a claim such as 'p=0.8 with confidence level of 5 \sigma based
>>>on a binomial distribution'. The I'd confidently assert that the
>>>'probability for tossing this coin' is close to .8. But I would
>>>still not claim a probability for 'the next toss', except perhaps
>>>informally in the sense of 'pars pro toto'.
>>
>> Suppose they told you all that, with thousands of trials. Still, anyone
>> else I've talked to would gladly take 51:49 odds either way. (That is,
>> behaved as if they believed p > .49, and p < .51.
>
> Oh yes; I'd estimate probabilities the way everyone does. But I wouldn't
> trust them completely.
No one expects you to trust completely, but even without full trust, they
are useful, even for single cases. Multiple aggregated samples let you
put more trust in the outcomes -- a high spread is less likely, but you
can't trust those completely.
>>>And this is possible if and only if they know the complete ensemble
>>>(with exception of a set of measure zero). This is what I claime
>>>all the time. If the ensemble is known only incompletely, any claim
>>>of 'enough information' is spurious.
>>
>> Complete ensemble, all the ways something can happen, sure. That's only
>> one limit, and reasoning before that limit is still possible.
>
> Yes, but it may give wrong results. Based on probability theory,
> it even _has_ to give wrong results sometimes, with probability one.
So does reasoning with the full ensemble though -- say that the full
ensemble gives you a P=0.999999 chance of success, so you confidently
assert that. Repeat the experiment a few million times, and you will
see some "failures".
>>>>>>>Thus probabilities are meaningful not for the single event but only
>>>>>>>as a property of the ensemble under consideration.
>>>>>>
>>>>>>And yet people bet on individual events all the time.
>>>>>
>>>>>Oh yes. They estimate probabilities, based on their favorite ensemble.
>>>>>But as you know, people often lose their bets!
>>>>
>>>>Sure. That doesn't mean they estimated the probability wrong, or are
>>>>misusing probability theory. Refusing to let probability theory deal
>>>>with single events reduces its applicability to almost nothing, and
>>>>people do successfully use it for single events.
>>>
>>>No. People who bet based on probabilities, tend to bet many times,
>>>not just a single time.
>>
>> They also bet on non-repeatable events, knowing full well they aren't
>> repeatable -- it will not happen that an ensemble similar to the one
>> they assign the event will usefully describe an event in the future.
>
> Yes, but then they act irrational.
Are both sides of the taker of bets irrational? Or only one? Or
neither? Casinos and bookies expect other events "in similar ensembles"
to cover this one, the average bettor doesn't bet enough for that to
happen, and the odds are skewed so that the chances of being down
increase with the number of bets. But even the casinos and bookies
will cover onetime events.
>>>>>>of that subset occuring, or the probability of those particular
>>>>>>realizations. Of course they don't say that the event will or
>>>>>>will not happen, unless the probability is zero or one.
>>>>>
>>>>>Yes; this is why they say nothing at all about the single case.
>>>>
>>>>Sure they, just not something definite. That's why there probabilities
>>>>instead of certainties.
>>>
>>>They say nothing that can be verified or falsified. Thus nothing.
>>
>> Well, as probabilities can't be verified or falsified for even
>> multiple cases, they must mean nothing as well? No? Then what's the
>> dividing point? 2 samples? 10? 100? 1000000? 10^100?
>>
>> Sampling an ensemble 10000 times will _probably_ have a result _near_
>> 10000p successes, but it is not guaranteed. It's still not checkable,
>> and there is no magical threshold. Everything that works for 10000,
>> still works for 1, just with much looser bounds.
>
> Yes. There is no dividing line. Probabilities are known only if one
> defines the ensemble by specifying its distribution. But then is is
> no longer exactly the ensemble of interest. There is an uncontrolled
> approximation step in between, made somewhat credible by the laws of
> large number and other tools of probability theory.
Okay. Why does this "uncontrolled approximation step" have to be done
by choosing ensembles, rather than other ways of specifying
probabilities?
>>>>Now, you can do most of this reasoning about uncertainty with ensembles
>>>>rather than states of knowledge, but it's much harder and more
>>>>complicated.
>>>
>>>'States of knowledge' are in fact not at all states of 'knowledge',
>>>but of 'prejudice'. of 'assumptions about which ensemble to consider'.
>>>Those who make the best assumptions will have the best results.
>>
>> The choice of ensemble is also prejudice.
>
> Yes. But once it is chosen, probabilities are objective.
> This is like for statements about infinite sets, say.
> One must choose the axioms, which is a subjective act; then
> the consequences are objective facts.
Same for just specifying the probabilities, and not specifying an
ensemble.
>>>>You have to make sure that the ensembles you come up with
>>>>are not only consistent with your state of knowledge, but also don't
>>>>tell you anything more -- that they aren't biased.
>>>
>>>And one has to do the same with the putative 'states of knowledge'
>>
>> Yes, and the ways of doing this are the maximum entropy derivable
>> ignorance priors you ridiculed earlier.
>
> There are many different ways one can do that; it depends on what
> partial information is available. Maximum entropy is appropriate
> only when the prior information is in the form of expectations.
It can accomodate constraints of many forms, actually. Mean
value == stable, observed values are only one way of doing so, that
leads to a nice expression in terms of equal lagrange multipliers for
exchanges.
>>>This is different when one has a multitude of events. Then the family
>>>of ensembles that match the information available to a high degreee of
>>>confidence is quite small, and finding such a description is already a
>>>scientific task.
>>
>> To a high degree of confidence is a probabilistic statement, about a
>> single event: the results of all the events.
>
> If you compound everything together, yes. But if one has such a complex
> single event, the proability interpretation is no longer adequate,
> and an interpretation of ensembles in terms of
> ''predictions of numbers being close to their averages''
> is the appropriate one. This is what happens in statistical mechanics.
> Rather than just accepting a rare occurence (e.g., a brick going upwards
> due to fluctuations) as something within one's probabilistic model,
> one would try to explain it away by assuming a hidden, unobserved cause
> (someone throwing it). This is the way science weeds out the exceptions
> to statistical reasoning.
Or, one could say that the probability of someone throwing it up is much
(i.e. many many many orders of magnitude) higher than that of spontaneous
fluctuations causing it to go up. Observing that it goes up,
probability theory says that most likely someone threw it up. Expanded
model, yes, but no need to have a magical cut off value, below which
we no longer trust probability theory to work.
>>>That's why statistical physics is not subjective anymore.
>>
>> Because it's on the firm Bayesian foundation of maximizing entropy
>> subject to constraints?
>
> No; this is a spurious foundation; it was invented by Jaynes 50 years
> after the various Gibbs ensembles were already known to work well.
> That max entropy is spurious can be seen from the fact that if we
> were to know <H^2> instead of <H>, we'd not get the canonical ensemble
> but one that is completely off physically.
Yes, but <H^2> _isn't_ a good description of the system, and we can't
really know it. Energy is a conserved quantity, not energy squared; we
can't see energy squared move between system and environment. It's not
an extensive quantity. There is no mechanism of constraint; we can't
measure it locally physically, and be assured that it won't change.
(Neither is energy, really. We can measure it relative some set zero
point. Fortunately, all we need to do is measure differences. Changing
the zero point of energy changes the energy squared in weird ways.
> Only if we know the _right_ sort of information, max entropy works,
Correct. We have to know what the real physical, _stable_ constraints
are to do physics. This is not surprising. If we get an answer not
in accord with experiment, either we're asserting something is a
constraint that isn't, or there is some hidden constraint we don't
know about. This can vary depending on the timescale.
> and to know what is the right sort of information presupposes
> the insights of Gibbs. So max entropy is only a justification after
> the facts are already in.
--
Aaron Denney
-><-