outubro 28, 2010

outubro 25, 2010


In probability theory there is a very clever trick for handling a problem that becomes too difficult. We just solve it anyway by:
  1. making it still harder;
  2. redefining what we mean by ‘solving’ it, so that it becomes something we can do;
  3. inventing a dignified and technical-sounding word to describe this procedure, which has the psychological effect of concealing the real nature of what we have done, and making it appear respectable.
In the case of sampling with replacement, we apply this strategy as follows.
  1. Suppose that, after tossing the ball in, we shake up the urn. However complicated the problem was initially, it now becomes many orders of magnitude more complicated, because the solution now depends on every detail of the precise way we shake it, in addition to all the factors mentioned above.
  2. We now assert that the shaking has somehow made all these details irrelevant, so that the problem reverts back to the simple one where the Bernoulli urn rule applies.
  3. We invent the dignified-soundingword randomization to describe what we have done. This term is, evidently, a euphemism, whose real meaning is: deliberately throwing away relevant information when it becomes too complicated for us to handle.
We have described this procedure in laconic terms, because an antidote is needed for the impression created by some writers on probability theory, who attach a kind of mystical significance to it. For some, declaring a problem to be ‘randomized’ is an incantation with the same purpose and effect as those uttered by an exorcist to drive out evil spirits; i.e. it cleanses their subsequent calculations and renders them immune to criticism.We agnostics often envy the True Believer, who thus acquires so easily that sense of security which is forever denied to us.

However, in defense of this procedure, we have to admit that it often leads to a useful approximation to the correct solution; i.e. the complicated details, while undeniably relevant in principle, might nevertheless have little numerical effect on the answers to certain particularly simple questions, such as the probability for drawing r red balls in n trials when n is sufficiently small. But from the standpoint of principle, an element of vagueness necessarily enters at this point; for, while we may feel intuitively that this leads to a good approximation, we have no proof of this, much less a reliable estimate of the accuracy of the approximation, which presumably improves with more shaking. The vagueness is evident particularly in the fact that different people have widely divergent views about how much shaking is required to justify step (2). Witness the minor furor surrounding a US Government-sponsored and nationally televized game of chance some years ago, when someone objected that the procedure for drawing numbers from a fish bowl to determine the order of call-up of young men for Military Service was ‘unfair’ because the bowl hadn’t been shaken enough to make the drawing ‘truly random’, whatever that means. Yet if anyone had asked the objector: ‘To whom is it unfair?’ he could not have given any answer except, ‘To those whose numbers are on top; I don’t know who they are.’ But after any amount of further shaking, this will still be true! So what does the shaking accomplish?

Shaking does not make the result ‘random’, because that term is basically meaningless as an attribute of the real world; it has no clear definition applicable in the real world. The belief that ‘randomness’ is some kind of real property existing in Nature is a form of the mind projection fallacy which says, in effect, ‘I don’t know the detailed causes – therefore – Nature does not know them.’ What shaking accomplishes is very different. It does not affect Nature’s workings in any way; it only ensures that no human is able to exert any wilful influence on the result. Therefore, nobody can be charged with ‘fixing’ the outcome.

At this point, you may accuse us of nitpicking, because you know that after all this sermonizing, we are just going to go ahead and use the randomized solution like everybody else does. Note, however, that our objection is not to the procedure itself, provided that we acknowledge honestly what we are doing; i.e. instead of solving the real problem, we are making a practical compromise and being, of necessity, content with an approximate solution. That is something we have to do in all areas of applied mathematics, and there is no reason to expect probability theory to be any different. E.T. Jaynes, Probability Theory, The Logic of Science

outubro 21, 2010

outubro 18, 2010


Temos disponível (somos?), no cérebro, um modelo do mundo, uma aproximação cujos detalhes a nossa cultura dá especial valor na interacção com a realidade externa. Apesar dos compromissos e da negociação, para este modelo funcionar fabricamos ilusões e arbitrários vários. Por exemplo, os objectos não têm cor nem cheiro, uma música ou uma voz são ondulações de ar, mas temos palavras, tradições e estéticas para lidar com eles. O 'agora', o 'aqui' e o 'eu' são igualmente peças desse modelo mas não têm um correspondente físico objectivo. Porém, incrivelmente, no seio deste difuso e imenso oceano de padrões e informação, conseguimos navegar com o possível de um rumo.

outubro 14, 2010


A experiência passada altera, ajusta o nosso comportamento. Podemos agir de forma distinta sobre dois casos similares apenas porque integrámos, na nossa mente, as consequências do primeiro. A lei escrita não se comporta assim. É apenas revista após a passagem de vários anos (ou como reacção ao inesperado de uma catástrofe). Isto é uma desvantagem porque dá à lei uma inércia que certos aspectos do mundo não toleram. Mas também é uma vantagem porque, assim, é mais imune a excessos emocionais, traumas, modas passageiras. Neste hesitar de balança talvez resida um argumento em favor da jurisprudência assente numa deontologia concisa, exigente e cega porque justa.

outubro 09, 2010

Não fazemos porque não sabemos

Many people are fond of saying, ‘They will never make a machine to replace the human mind – it does many things which no machine could ever do.’ A beautiful answer to this was given by J. von Neumann in a talk on computers given in Princeton in 1948, which the writer was privileged to attend. In reply to the canonical question from the audience (‘But of course, a mere machine can’t really think, can it?’), he said:
You insist that there is something a machine cannot do. If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that!
In principle, the only operations which a machine cannot perform for us are those which we cannot describe in detail, or which could not be completed in a finite number of steps. E.T.Jaynes, Probability Theory, The Logic of Science

outubro 06, 2010