Friday, September 6, 2013

Determinism, Laplace's Demon, and the Statistical Divide

The concept of ex post determinism, which I have referred to here as scientific determinism, is not a new concept by any stretch. Various forms of determinism have been proposed and debated over the last several centuries, and while the boundaries I have laid out may differ, the idea that events are deterministic is well trodden ground.

Perhaps one of the most extreme, or pure, forms of determinism is represented by Laplace's demon. "Laplace's demon" is the name given to the ideas posited in a passage late in Laplace's life on probability (available here). In simple terms, Laplace expressed the idea of pure determinism: If the position and speed of all particles in the universe are known at a time t=0, then the state of the world at a later time t=1 can be perfectly calculated. Laplace, an early statistician, perhaps most clearly demonstrates the divide between frequentist (or probabilist) and Bayesian modes of thought: You may notice that Laplace's demon seems to have nothing at all to do with statistics as we know it today.

Generally speaking: A probabilist is concerned only with outcomes. The confidence with which he can make a scientific statement is derived from an analysis of the outcomes; the distribution of the outcomes greatly concerns him, as it is only what we have observed that we can predict, and observations in an imprecise world necessarily have variance. Probabilistic studies make no claim to the root cause of the process that they help describe, in and of themselves (after all, correlation does not imply causation). It is left to the researcher to interpret probabilistic results and to hypothesize as to their meanings and implications for the underlying process.  

Bayesian thought, on the other hand, expressly prohibits true knowledge of the underlying process in any situation where probability exists. Unless the probability of a proposition being true is 1, Bayes' theorem always admits that there is some chance that the proposition is false. Further, any proposition which starts from an uncertain state cannot be made certain, unless evidence of a nature so strong to provide absolute certainty is presented such that all prior evidence on the matter is declared invalid. Put another way, the evidence would need to provide absolute certainty (probability of 1) that the proposition were true, and it would need to invalidate all prior evidence, so that prior evidence were given a weight of zero and the new evidence a weight of 1. This new evidence would constitute true knowledge - but, as we have seen, true knowledge is not something easy to come by. More or less any data will lead the Bayesian to some level of confidence in the proposition he is interested in.

Thus, both Bayesians and probabilists are concerned with confidence, and therefore with probability. Lest I seem as though I have repeated myself in the preceding two paragraphs, however, consider how a probabilist and a Bayesian might describe their findings. Once the data has been analyzed, it is common for researchers to boldly claim that they have a "statistically significant" conclusion - but still, if this conclusion is significant to a 95% confidence level, there remains not a 5% likelihood that the claim being tested ("X results in Y") is untrue, but rather, as much as 5% of actual events which occurred violated the claim. Can we, from this, claim that we have 95% confidence that X results in Y? 

To the Bayesian, this depends on how you weigh the evidence. Equal weight may lead him to answer "yes"; however, the distribution of outcomes plays a large role. Additionally, our weighing of evidence does not occur in a lab; there may be long periods of time between old evidence and new, or we may have reason to believe that conditions have changed such that the new evidence is more relevant. In short, we cannot say that 95% confidence in a proposition is the same to a Bayesian as it is to a probabilist.

More importantly, the difference can be reflected in the philosophy behind the two schools of statistical thought. Particularly irresponsibly probabilists - far too many of which take up tenure in positions of research - may not be overly concerned in a proposition at all: They may build distributions, or perform simulations, based on data without applying any scientific theory or thought to the data; or, worse yet, they may conclude something about the results without having had a proposition to test against in the first place. This may, in some instances, remove biases and allow the data to "speak for itself"; yet, it may also lead to dangerous conclusions which are not supported by the evidence that exists outside the dataset. Probabilists may also believe that processes are inherently random, and that the only practical solution is to estimate the amount of variation and to take it under consideration when assessing the likelihood of outcomes.

By contrast, all evidence is necessarily applied to propositions in a purely Bayesian thought framework. Evidence will have very little weight towards some propositions, and a great deal towards others. But these propositions are the deterministic truths that Laplace's demon sought; without them, there is nothing to apply evidence towards. From the set of all possible propositions (an infinite set), some subset (also infinite) will prove to be true in retrospect. Laplace's demon is simply the concept that perfect information will necessarily lead to perfect foresight.

Claims that Laplace's demon have been discredited always rest upon our limitations of knowledge. These arguments take the demon too literally. Our knowledge being inherently limited down to the most fundamental level, surely we will never be able to calculate the universe with certainty. The limitations of our mind, and subsequently, of the computing power constructed by humanity, may never provide for this. Does this, however, invalidate the concept of determinism itself? Have we not, time and again, underestimated what science we can rely upon, or what computing power we might achieve?

Laplace's demon, as a concept, stands as a sort of ultimate goal for science: something to strive for, if never to achieve. It also represents a great epistemological dilemma: Our Bayesian minds can never obtain true knowledge, yet the universe is theoretically knowable with absolute certainty

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