Four Walls metaphor for probability

Of the great moments in the history of classical probability theory, three stand out.  First comes the idea of a relative frequency, which in essence is a discovery about the stability of certain kinds of measurement outcomes. That relative frequency compared with other relative frequencies, accounting for a disjoint and complete set of possible outcomes.  This idea must have been at least comprehensible, if not actually thought about, since the very first time a human could throw a lump of shit at a wall - the likelihood of it sticking to any brick is proportional to that brick's (relative) wall area.  In other words, even if no-one explicitly had that thought, I can easily imagine that if I suggested to people in prehistoric times to bet on which brick the shit would land, their choices would be partly determined by (all other things being equal) the brick's presenting surface area.

The second great moment in the intellectual history of classical probability is the creation of an artificially regular event space, the so-called equi-probable event space.  

This simplifying adjustment to the idea of a stable and complete set of relative frequencies allows analysts to physically count tiles to work out likelihoods or (proto-)probabilities - it being a ratio of one count over another, larger one.  Each of the $n$ equi-probable tiles has a probability of $\frac{1}{n}$ and of course $\sum_1^n{\frac{1}{n}} =1$

Explicitly counting only gets you so far before you might start making errors.  Imagine immense walls of tens of thousands of mosaic tiles.  As problem complexity increases, counting needs to be industrialised.  Enter the third great idea to become relevant - namely the application of counting and permutation rules as a way of formalising and regularising the process of counting enormous fractional event spaces.  The formulae for these counting power tools are $C_{n,k} = \frac{n!}{k!(n-k)!}$ and $P_{n,k} = \frac{n!}{(n-k)!}$ and the 'cheat sheet' is known as Pascal's triangle.

A fourth moment (not so much of classical probability but of modern decision theory and psychology), is the ongoing sequence of discoveries of the biasses and flaws, the self-deceptions and regularly occurring mistakes humans make in estimating subjective probabilities themselves.  This brings in the work of Kahneman and Tversky and the subject of behavioural finance,  and aims to discover the perceptual and cognitive bias which real humans inject into the mathematically rather more rational classical probabilistic modelling approach which has been achieved by our cultures so far in history.

Two final pleasing elements of the wall metaphor.

One.  The wall has two relevant dimensions, which reminds me of the law of multiplication of outcomes for independent events.  If an experiment has $n_1$ possible outcomes and a second, independent experiment has $n_2$ possible outcomes, then the joint experiment has $n_1 \times n_2$ outcomes.  Just like the length-wise and breadth-wise brick counts of a wall.  This generalises up nicely to many dimensions: $\prod^{k} \frac{1}{n_k}$

Two.  Walls are human constructs which help make buildings, another wonder of human culture.  As both of these technologies evolved, so too did the edifices they enabled become more marvellous.

Arabian Financial Modelling

The Greeks gave the world modern rationalist, logic-based Platonic thinking - idealised models of real world phenomena.  What they didn't give us is a usable number system.   Ninth century India did that, together with thinking on algebra and fractional manipulation, and an emphasis on practical commerce-based puzzle-solving.  It was the dominance of the Arabs in southern Europe which allowed both worlds to come together forcefully.   There's a lot of be said for looting the libraries of Alexandria.  This culture has played a powerful role as the brains behind financial structuring and modelling ever since.  Their presence is all around in quant desks of major investment banks to this day.  Culture clash can have a lot of upside.

Transcendent analysis beats innate bias

In an earlier post I said that probability was really a common ground for three moments in the life of humanity.  First our own subjective ability to judge how likely or not certain events are.  It is now well known that we humans are particularly bad at some kinds of judgement on outcomes.  I'm very much looking forward to reading The Drunkard's Walk, on my desk now, which deals with this subject, emphasising the Kahneman and Tversky discoveries in economics and psychology.  Keynes himself in his doctoral dissertation, suggests we don't have the capacity for making fine-grained probabilistic calls, but instead are capable of broadly ranking outcomes in terms of their likelihood.  According to the final chapter of The Lady Tasting Tea, the philosopher  Patrick Suppes attended a Tversky presentation highlighting just how incoherent we can be when it comes to subjective probability estimation and came up with a version of 'approximate probability' which not only was consistent with what Tversky found in his psychological experimentation but which was also consistent with Kolmogorov's axioms of probability.

Second is the moment we construct randomisation machines, whose behaviour was sufficiently outside our own heads to allow experimenters and theoreticians to begin an analysis of probability independent of, perhaps in spite of how limited or biassed our own subjective abilityis  to reason about uncertainty.

Third is the moment Kolmogorov mainlines these probabilistic analyses into the core of mathematics, via a creative re-interpretation of the frequentist approach as a number of manipulations of set theory.

Historically, however, step two came before step one.  But the order of their discovery does not imply that the Kahneman and Tversky created a refinement of the classical theory of Pascal and Fermat, or later of Kolmogorov.  No, they merely discovered what has been true about human brains for probably tens of thousands of years - they've merely probed our bias and limits.  The right formal analysis of a financial structure or a game of chance is a better analysis than one which is sensitive to our own limits.  Take the famous two games of the Chevalier De Mere (Antoine Gombaud) the very same games which he engaged Fermat and Pascal to find out how to split the pot fairly if the game is broken up early by mutual consent.

In game 1, he bet he could get a 6 on four rolls of the dice.  Since the probability of this happening is more likely than not, it is a good game to play.  Probably at some point during the end of the life of this game, it became sufficiently widely known that the probability of a 6 in 4 rolls is more likely than not that no-one would play with him.  So he moved on to game 2 - rolling two dice 24 times, the challenge being to get a double-six.  It turns out that this is slightly less likely than $\frac{1}{2}$.  De Mere played this game regularly  and made many losses.  Knowing peoples' psychological biasses and limits doesn't really help you much here; being a human being with all of our above-mentioned weaknesses in judging uncertainty is also not very helpful.  Knowing the analysis most certainly is helpful.  Disseminating this knowledge in a sense creates a more rational player.  A similar point is maintained by Perry Mehrling about Fischer Black, the so-called CAPM-man.  

My point is the rational analytical approach adds value here in spite of it not initially reflecting psychological reality - culturally, if the analytical approach pays off, that idea  can usurp our more primitive subjective reasonings.   Finally if the game you're playing isn't just 'out there' like De Mere's games, but instead involved a model of other players (e.g. poker), then psychological insight into human biasses can become just as invaluable as knowing the card probabilities.

Grumpy Old Man

I've been reading The Age of Absurdity.  It is only moderately amusing, slightly enlightening, fleetingly useful.  He starts off trying to tackle the problem of happiness from a modern intellectual perspective, but in the end becomes a grumpy old man airing his laundry list of grievances about the modern world.  Still, he's clearly a decent human being who I personally can relate to. Much better to read Montaigne, La Rochefoucauld or Lichtenberg or go and see a great actor perform Krapp's Last Tape.  The three reviews quoted on the back page of the paperback edition are united in their belief that this book will make you laugh, possibly out loud, possibly so heartily that your muscles will ache.  No it will not.

The Dice don't play God

For centuries humans have been interpreting random events as the direction of some spiritual being.  From consulting the oracle, the I Ching, astrology, Cleromancy we've used randomisation machines like astralagi, dice and cards to give ourselves advice or sanction certain types of behaviour.  On the surface, this is such a strange thing to do.  Why did we do it?  What did we do before then?  What do we currently do?

I can understand the role randomisation machines performed in games of chance and gambling.  They represented an ideal of fairness (I say an ideal because I'm aware that randomisation machines could be loaded and indeed were.)  That is to say, they were made machines which weren't sensitive to the desires or deceits of any player.  Their behaviour could be relied on.  They provided as objective reference point around which players could gamble.  Perhaps this aspect of their behaviour - being beyond human control - was what  allowed people to make the leap to attributing the generating impulse to some higher spiritual being.  The randomisation machine is an trans-human referee.  I'm also reminded of Comte's description of early attempts by humans to understand their world beginning with the theological, passing through the metaphysical, then ending up at the positive.

Well, certainly all of the practices of middle-Eastern and European cultures since the invention of agriculture are still going on in one form or another today in some cultures, often in a dramatically diluted form.  Think of Nancy and Ronnie Reagan and Carroll Righer the astrologer.  Or indeed the daily astrology pages of many national newspapers.  In astrology the movement of planets also serve this trans-human purpose, though the machine in question certainly wasn't random.  Perhaps early humans' use of the planetary movements isn't too unlike our own use of pseudo-random number generators - essentially deterministic machines which exhibit behaviours which certainly seem random enough to people.  You can have a pseudo-random number generator with a short repeat cycle (Linear congruential) or one with a larger repeat cycle - the Mersenne Twister was only invented as recently as 1997.  Strange to think that the inventors of the twister and the developers of astrology could very well have been aiming at a similar kind of  reference machine.

The impulse to allow randomisation machines to control our destiny is clearly still strong, and has survived the religious interpretation, evidenced by the cult classic, 'The Dice Man'.