The Nature of Probability

"To throw the dice is to face that which is given by the gods, by powers higher than human. It is to face reality at its most mysterious, like standing unflinching before the thunderstorm."

"Yet the human spirit is restless and nature forever compliant, willing to answer as yet undreamed questions, capable of opening up vast new vistas, revealing still undisclosed parts of her being."

(This one's for the newbie, because I think you are in danger of 'getting it'. ;)

"Probability is a way of expressing knowledge or belief that an event will occur or has occurred...

The word probability does not have a consistent direct definition. In fact, there are two broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:

Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.[1]

Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, given the evidence." (q)

I would submit that there is a third category of probability interpretation. Like Bayesian probability, it deals with the subjective aspect of knowledge, and it updates as the relative 'force' of a particular piece of knowledge or observation changes in the cognitive space.

But this third category of probability interpretation is entirely concerned with predicting state selection. (By the time I'm done with this post I'll have a catchy name for it.) Not traditional state selection though, which is still presumed to reflect a collapse of the wave function in an objective reality. This third category of probability (subjective state selection probability?) is not concerned with using subjective knowledge to measure the potential for a particular decision or action, but rather with using subjective elements of conscious experience to predict future elements of that same cognitive space. To emphasize how such a definition of probability would be superior to other definitions in predicting outcomes, it becomes necessary to compare them in a common context - hence my emphasis on situations involving 'randomness'.

When faced with a coin flip, classical probability will attempt to predict the outcome of the flip (heads or tails), using only information about any potential bias the coin may have. To be honest, I'm not entirely sure what Bayesians would be attempting to do in this situation. Quantify how they should update their beliefs regarding the bias of the coin?

Those of us using cognitive differential probability will attempt to quantify aspects of our cognition with respect to the as-yet-unobserved outcome in an attempt to predict and/or modify the biases governing the selection of the outcome state. Here's where things get different.

Cognitive differential probability (are we liking that name?) recognizes that different amounts of knowledge about the coin, as well as different attachments to the outcome of the coin flip and other elements of anticipation regarding the outcome of the coin flip, constitute biases in the process of selecting the final state of the flipped coin. Early on in the game I talked about the differences in these knowledge structures. The biases we're talking about change with each additional observation, but they also change in response to various elements of pure cognition. (I know - you either believe it's possible, or you don't.)

I think Bayesian's recognize the constantly-updating weight of subjective beliefs, but I don't think they are using it to predict outcomes in this way. Nor do I think that they model 'beliefs' as comprehensively as I propose to model cognitive space. So where does this leave us?

Once I can figure out how to model degree of overlap in the various elements of cognitive space/representation, modified bi-directionally across time, with suitable boundary conditions to prevent infinite regression yet still accurately predict outcome selection... Once I can do that for a single observer, I'll be back to haunt you. ;)