Feb

17

I thought this was an interesting opinion piece from David Deutsch who has some creative ideas in physics theory:

"Probability is as useful to physics as flat-Earth theory"

Gibbons Burke writes:

String theory, or more particularly, M-theory, which represents a current SWAG (Scientific Wild-Assed Guess) at the grand-unifying-theory-of-everything, requires some eleven dimensions to make it all work out.

Our mortal finite deterministic mental capacities can wrap our space-time evolved brains around four or five, with instruments perhaps a few more.

Perhaps randomness is how we get a handle on behavior which defies rational explanation in our four-dimensional flatland of what seems to be the 'natural' material world; if there are eleven or more dimensions, then perhaps what seems random for us has rules beyond our ken which govern the dynamics of the other invisible, shall we say, 'super-natural', dimensions.

Ralph Vince writes: 

I think people are missing the point of the article Dylan puts here. The author of this simple piece is discussing things that are right in my ambit, what I call "Fallacies in the Limit." The fundamental notion of expectation (the probability-weighted mean outcome), foundational to so much in game theory, is sheer fallacy (what one "expects" is the median of the sorted, cumulative outcomes at the horizon, which is therefore a function of the horizon).

To see this, consider a not-so-fair coin that pays 1:-1 but falls in your favor with a probability of .51 The classical expectation is .02 per play, and after N plays, .5N is what you would expect to make or lose for player and house, as the math of this fallacious approach - and I say fallacious as it does not comport to real-life. That is, if I play it on million times, sequentially, I expect to make 20,000 and if a million guys play it against a house, simultaneously, (2% in the house's favor) the house expect to make 20,000

And I refer to the former as horizontal ergodicity (I go play it N times), the latter as vertical ergodicity (N guys come play it one time each). But in real-life, these are NOT equivalent, given the necessarily finite nature of all games, all participants, all opportunities.

To see this, let is return to our coin toss game, but inject a third possible outcome — the coin lands on its side with a probability of one-in-one-million and an outcome which costs us one million. Now the classical thinking person would never play such a game, the mathematical expectation (in classical terms) being:

.51 x 1 + .489999 x -1 + .000001 x - 1,000,000 = -.979999 per play.

A very negative game indeed. Yet, for the player whose horizon is 1 play, he expects to make 1 unit on that one play (if I rank all three possible outcomes at one play, and take the median, it i a gain of one unit. Similarly, if I rank all 9 possible outcomes after 2 plays, the player, by my calculations should expect to make a net gain of .0592146863 after 2 plays of this three-possible-outcome coin toss versus the classical expectation net loss of -2.939997 (A wager I would have gladly challenged Messrs. Pascal and Huygens with). To see this, consider the 9 possible outcomes of two plays of this game:

outcome

0.51                     0.51    1.02

0.51             -0.489999    0.020001

0.51             -1000000    -999999.49

-0.489999            0.51    0.020001

-0.489999    -0.489999    -0.979998

-0.489999    -1000000    -1000000.489999

-1000000            0.51    -999999.49

-1000000    -0.489999    -1000000.489999

-1000000    -1000000    -2000000

The outcomes are additive. Consider the corresponding probabilities for each branch:

product

0.51          0.51                 0.260100000000

0.51          0.489999          0.249899490000

0.51          0.000001          0.000000510000

0.489999   0.51                  0.249899490000

0.489999    0.489999          0.240099020001

0.489999    0.000001          0.000000489999

0.000001    0.51                 0.000000510000

0.000001   0.489999           0.000000489999

0.000001    0.000001          0.000000000001

The product at each branch is multiplicative. Combining the 9 outcomes, and their probabilities and sorting them, we have:

outcome             probability         cumulative prob
1.02              0.260100000000    1.000000000000
0.999999       0.249899490000    0.739900000000
0.020001       0.249899490000    0.490000510000
-0.979998      0.240099020001    0.240101020000
-999999.49    0.000000510000    0.000001999999
-999999.49    0.000000510000    0.000001489999
-1000000.489999    0.000000489999    0.000000979999
-1000000.489999    0.000000489999    0.000000490000
-2000000    0.000000000001    0.000000000001

And so we see the median, te cumulative probability of .5 (where half of the event space is above, half below — what we "expect") as (linearly interpolated between the outcomes of .999999 and .020001) of .0592146863 after two plays in this three-possible-outcome coin toss. This is the amount wherein half of the sample space is better, half is worse. This is what the individual, experiencing horizontal ergodicity to a (necessarily) finite horizon (2 plays in this example) expects to experience, the expectation of "the house" not withstanding.

And this is an example of "Fallacies of the Limit," regarding expectations, but capital market calculations are rife with these fallacies. Whether considering Mean-Variance, Markowitz-style portfolio allocations or Value at Risk, VAR calculations, both of which are single-event calculations extrapolated out for many, or infinite plays or periods (erroneously) and similarly in expected growth-optimal strategies which do not take the finite requirement of real-life into account.

Consider, say, the earlier mentioned, two-outcome case coin toss that pays 1:-1 with p = .51. Typical expected growth allocations would call for an expected growth-optimal wager of 2p-1, or 2 x .51 - 1 = .02, or to risk 2% of our capital on such an opportunity so as to be expected growth optimal. But this is never the correct amount — it is only correct in the limit as the number of plays, N - > infinity. In fact, at a horizon of one play our expected growth-optimal allocation in this instance is to risk 100%.

Finally, consider our three-outcome coin toss where it can land on it;s side. The Kelly Criterion for determining that fraction of our capital to allocate in expected growth-optimal maximization (which, according to Kelly, to risk that amount which maximizes the probability-weighted outcome) would be to risk 0% (since the probability-weighted outcome is negative in this opportunity).

However, we correctly us the outcomes and probabilities that occur along the path to the outcome illustrated in our example of a horizon of two plays of this three-outcome opportunity.

Russ Sears writes:

Ok after a closer look, the point the author is making is scientist assume probabilities are true/truth based on statistics. But statistics are not pure math, like probability, because they are not infinite. Therefore they can not detect the infinitely small or infinitely large.

But the author assumes that quantum scientist must have this fallacy and do not understand. Hence he proposes that thought experiments or philosophical assumptions of deterministic underpinnings of physics must hold and should carefully supercede statistical modeling. Hence denying the conscious mind any role is creating a physical world outside itself.

So basically the author accuses others of not understanding the difference between the superiority of probability over statistics. So he tries to use pure thought to get pure physics devoid of the necessity of consciousness to exist. Perhaps he does not confuse the terms himself. It would be better written however, if he used the terminology a 1st year probability and statistics student learns. 

Jim Sogi adds: 

I believe that the number and size of trades at a price, or the lack of density at that price lead to certain gravitational effects. The other somewhat unknown are the standing orders at those levels but the orders and trade density are related.


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