Friday, March 18, 2011

Is Probability An Illusion? (Part 4 of 4)

Q: What are statisticians?
A: Mathematicians broken down by age and sex.

Q: What are Bayesian statisticians?
A: Mathematicians broken down by age, sex, and level of alcohol consumption.

(Above jokes and picture from the website "Bayesian Fun", click here)

What is the ontological status of probability? That is, to what extent does it really exist in the world? There are two broad schools of thought: Frequentism and Bayesianism.

For a frequentist, randomness is taken as an intrinsic part of reality, which probability quantifies. To say that event A has a probability of one half means that if the experiment was repeated many times then A would occur exactly one half of the time. In other words the probability of A is a measure of the frequency with which A happens, given the initial conditions. (This would only be an approximate after finitely many repetitions, but would be exact in the limit.)

As this shows, the principle does not apply very easily to one-off events, but is best suited to repetitive occurrences.

In contrast to frequentists, for a Bayesian, probability does not exist in the external world. It is purely a way for humans to quantify our degree of certainty on the basis of incomplete information. In other words, probability is a subjective concept. People will make different assessments of probability, based on the different data they have available.

So if a coin flip is initially judged to have a probability of one-half of resulting in a head, this is because we know little about it. More data about the weighing of the coin, its initial position, and the technique of the flipper would allow us to modify our probability. If we knew these things in great detail, we would be able to predict the outcome with some certainty. (The mathematician John Conway is reputed to have mastered the art of flipping coins to order.)

There is a consequence to the Bayesian view. To a Bayesian, all probability is conditional. Suppose you estimate the probability of A happening as P(A). (This is really P(A/C) where C represents your current knowledge, but we suppress this). This is your prior probability. When some new data (B) comes to light, you need to update this assessment. This means using conditional probability to calculate P(A/B), called your posterior probability.

As the fallacy of probability inversion shows, Bayesian interference can throw up counter-intuitive results. Bayesian thinkers deploy this technique to improve probability assessment in a broad range of subjects, from economics to artificial intelligence.

From: Mathematics 1001 by Dr. Richard Elwes


Pat B said...

The odds are 4:1 that it isn't... but that's only my a priori probability...On the other hand if it IS an illusion, then this is my illusiory probability.

Steven Colyer said...

"Time is an illusion, lunchtime doubly so."
... Douglas Adams

STILL the best quote about illusions, ever.

Stay tuned, this is but Part 4 or 4 ... working backwards ... up next? ... Part 3 ! :-)

Phil Warnell said...

Hi Steven,

From the Bohmian perspective probability represents simply a way to deal with nature's imposed ignorance in regards to the consequence of outcome and not that cause isn’t relatable to effect. The interesting thing about it is that at the quantum level the reason is very closely related to uncertainty as to be found in thermodynamics. It has long intrigued me that perhaps the same natural impetus that drives process and time might be the same which has us unable to have its action completely determinable and yet them still being actually determined.

Interestingly enough it was Bohm who addressed as to solve Einstein’s long held objection and yet found him unable to accept it. I’ve never been able to definitively determine why yet have long suspected that his conviction that the nature of causality was restricted by his own theories being the reason. The funny thing is most Bohmians themselves believe such actions must be resultant of superluminal action, while Bohm believed it more due to the holistic nature of reality related to its innate and immutable quality. In other words Humpty Dumpty falls off the wall not solely because of his own nature, yet resultant of the nature of all of reality’s in total. That is the wave is not an artefact of probability, yet the physical manifestation of it. So this has particles as the coins to be flipped and the wave as the flipper and what we call probability as our best guess as to what’s been decided.

“The end and the means towards it may come about by chance. We say, for instance, that a stranger has come by chance, paid the ransom, and gone away, when he does so as if he had come for that purpose, though it was not for that that he came. This is incidental, for chance is an incidental cause, as I remarked before. But when an event takes place always or for the most part, it is not incidental or by chance. In natural products the sequence is invariable, if there is no Impediment......... It is absurd to suppose that purpose is not present because we do not observe the agent deliberating. Art does not deliberate. If the ship-building art were in the wood, it would produce the same results by nature. If, therefore, purpose is present in art, it is present also in nature. The best illustration is a doctor doctoring himself: nature is like that. It is plain then that nature is a cause, a cause that operates for a purpose.”

-Aristotle, “Physics” (aprox. 350 B.C.)



Steven Colyer said...

Phil, you REALLY have to stop impressing me by quoting Aristotle, because you damn well know the man known as "THE Philosopher" during the middle ages is my downright favorite, BY FAR, for inventing Logic, even if he did screw up by thinking the Earth was flat. Oh well, low tech times mean low tech theories, dude was still WAY ahead of our times. :-) Or any times.

Anyway, I just read this re Bayesian Analysis by Eric Weisstein at Mathematica:

Bayesian analysis is somewhat controversial because the validity of the result depends on how valid the prior distribution is, and this cannot be assessed statistically.

Controversy in Mathematics?! Say it isn't so.

But it is so. I reckon.

Phil Warnell said...

Hi Steven,

Actually I’m happier you’d be impressed with Aristotle, as it’s indicative the foundational issues must deal with logic as its first concern. The thing is that with the dawn of quantum mechanics this has thought to be in some way as legitimate to ignore. The bottom line being is when one abandons causality it is invariably at the sacrifice of logic. Einstein realized this and it’s truly what lies at the very heart of the EPR paper.

The problem arose as he could not understand how spatially separated events could be still causally tied without breaking relativistic premise. Another way to express it being that concepts like destiny or Karma are words which explain the existence of those like probability and uncertainty, as expressions of nature’s imposed ignorance on us, while not ones it imposes on itself.

Now as to the question of free will, it comes down to if consciousness is in some way a connected expression of the whole or merely another of its servants. In regards to such questions I would prefer it be the former rather than the latter and yet resigned to the fact my preference doesn’t count, only that which can be shown to be consistent with nature. This is one of the central questions the foundationalists ask as to explore and the one I hope to live to see decided, as all others pale in significance.

“We have become Antipodean in our scientific expectations. You believe in the God that plays dice, and I in complete law and order in a world which objectively exists, and which I, in a wildly speculative way, am trying to capture. I firmly believe, but I hope that someone will discover, a more realistic way, or rather a more tangible basis than it has been my lot to find. Even the great initial success of quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility. No doubt the day will come when we will see whose instinctive attitude was the correct one.”

-Albert Einstein, a letter to Max Born September 7, 1944[Born-Einstein Letters],