27 Aug 2011 14:38:40 UTC

Topic 195923

(moderation:

Take this for what its worth.. A Worldview.

The Principle of Mediocrity (ArXiv)

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## The Principle of Mediocrity

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This is probably the closest 'multi-universe' theory to observational testing. To capture the ideas fully one must distinguish between 'large-but-finite' vs. 'infinite' ( the enumeration of quantum states, however enormous, will not be infinite provided that the enclosing volume is finite and Planck's constant is fixed & non-zero ). For a real hoot on infinity, it's logic and anti-intuitive behaviour/results look up "Hilbert's Hotel" : it is an amusing look at what it means to have a countably infinite set.

Indeed the history of science has mediocrity as a thread eg. Copernicus/Newton etc by displacing our human-centric thinking and challenging us to take a view outside of our particular place and time. Galileo, and Giordano Bruno before him, got into trouble because he mocked the church's franchise on position. However as the mythical and egocentric part of us wants us to be important and special then there's a tension to enclose us in a relevance that we may not actually have. I'd suspect that if we could talk to, say beavers, they would have a world view arranged to suit - different in detail than ours of course, but having the same tools of self centered-ness vs external view. There could well have been beaver popes knocking heads against beaver heretics. Who can say ???? :-)

My guess is that one doesn't survive very long ( enough to yield like minded descendants ) if some egocentricity doesn't come into play. You can see this over every gatepost and fence in 'civilisation', albeit wrapped ( often deeply ) in layers of justification to obscure it.

[ An especial example of subjective/objective extremes ( flagrant in Aussie politics right now ) is the dual use of the theme of 'democracy' - whatever that is - is it one vote per one person with majority ruling, or the freedom to do whatever I want and stuff the rest of you? Recently the very same individuals have used both arguments consecutively within the space of a fortnight ... with some disingenuous observers/commentators knowingly presenting such contradictions as only apparent. There's not much reward in special pleading if it's not contained by words to hide that ( eg. NIMBY ). But I digress :-) :-) ]

So I think one reason why this exposition of mediocrity has opposition is that deep down we all are the centre of our own personal universes. The trick is to recognise that others may also have that as well, mutatis mutandis.

Another good principle worth retaining is 'regression towards the mean'. Basically if you have some variable which has a range of values ( well, that's what 'variable' means otherwise it would be 'constant' ) then you can define measures of centre and spread by calculations upon some set metric. The mean and standard deviation, say. It is then likely that any random choice of value will not lie exactly on the mean ( as the mean, like any member of the set, is only one of usually very many ). So a random choice will lie off the mean. No surprises there. However suppose we try to correlate consecutive/separate choices of the variable's value, randomly chosen. So if my first choice A is say above the mean, then what is a second choice ( B ) likely to be compared to A? It's going to be most likely below A because A is above the mean in the first place, so by definition of the mean ( 'the middle ground' ) more randomly chosen values that could be a 'B' lie below 'A' than above 'A'. Similiar logic applies if A was initially below the mean, and remains true regardless of whether the mean and/or spread is actually known. Note that the values of A and B each stand alone as random choices, no bias whatsoever ( ie. 'random' means exactly that ). What gives the 'regression' is the arbitrary decision to compare separate choices of our variable, and that mental construct lies outside the set in question and in particular any detailed mechanisms for determining the values when sampled. Hence regression towards the mean will give the appearance of trend when none exists, and in particular cannot be used alone to distinguish sets with random tendencies versus those without. Hypotheses that do not acknowledge this effect are bound to contain hidden bias ( ie. lying by using statistics, an ancient game indeed ). If selection bias exists eg. removing data points that don't suit a view, then no conclusions what-so-ever can be drawn and especially note that systematic selection of data doesn't approximate a random choice hypothesis to some degree : it is not 'near' or 'far', it is unrelated and any 'trend' is a direct product of the selection process and not the data set itself.

Cheers, Mike.

( edit ) Well, beaver popes knocking heads off beaver heretics ....

( edit ) Also worthy of note is that probability is the ratio of two numbers, a division of the subset by the whole. Many arguments wax eloquent and long about the subset, but insufficiently of the whole. So the statement that 'the probability that this seed will produce a red flower is such-and-such' depends crucially upon whether the seed came from a field of red flowers vs a field of flowers with some other color. There is also a frequent, and deliberate, confusion between the mathematical definition of probability as discussed here and the everyday natural-language/cultural-context meaning of 'likelihood'.

( edit ) .... and at a way deeper level, regression to the mean only exists if there is a metric on a set that has ranking at all ( semantics of greater than, less than, or equal to ). There may be no such metric on a set, not just for failure to cleverly define/imagine/discover one, but because 'rank' has no meaning per se. Thus the possibility exists that a metric is mis-applied. I see this so often in medical practice : in particular younger students and doctors who can't usefully assess a situation unless they have a rankable numeric metric to guide them. Nature is rarely as simple so as to provide us with one, gee they're good to have for sure but that's the minority of instances. Often it comes down to 'common sense' rather than science eg. if I can't get them breathing again they will die .... the only number that matters here is 'the next five minutes'.

( edit ) Whoops, more thoughts. One defence against selection bias if one doesn't know the set measures of mean & deviation ( the set is too big or too unknowable in practice ) is sampling along the lines of 'Monte Carlo' techniques. Basically you close your eyes and reach out and pluck a value, and then repeat. Assuming you are truly relying on dumb luck then the collection obtained ought have some resemblance in character to the remainder of the set that wasn't plucked.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal