News Flash: 1st May 2016: Substantial Breakthough in this area -- Using the concept of Metaphor proposed by Colin Turbayne leads to tremendously powerfully insights as well as a completely new definition of probability.
Please see REVISITING FOUNDATIONS OF PROBABILITY.
NEW REVISION NAMED: Subjective Probability Does Not Exist.
Submitted to International Econometric Review
MAIN QUESTION: How do we interpret the sentence "The probability that the coin comes up heads is one half"
RELATED MATERIAL ON CAUSALITY & INSTRUMENTAL VARIABLES
NEW IDEAS:
1: Likelihood Principle: Condition on What happened. Two things can happen. One is Highly Unlikely, Other is Highly LIkely, under NULL. Unlikely casts doubt on the null. likely does not. This violates likelihood principle. Check current status of controversey.
PROBABILITY deals with what might have happened, and also with the potential possiiblities that might happen. THese two are different. What might happen is pre-experiment. What might have happened is post-experiement. pre and post are different knowledge states. POST experiment, knowledge that we are on an unlikely branch under the null hypothesis discredits the null. (is this Bayesian principle?).
2. Solpcism: My knowledge of the world cannot be distinguished from the world. Thing-in-itself is inherently unknowable. Floating Brain can never be logically refuted. THIS MEANS that we have to go beyond logic and certainty. In addition, our daily lives requires conjectures about how others feel, which is again beyond the reach of logic. Massive confusion created by taking appearance of reality for reality itself. However, pre-random-event I know that there exist possibilities. post-random event, these possibilites are extinguished. So even if I dont know what happened, I am in a different knowledge state from pre-random event.
Post-Positivist Probability
Dr. Asad Zaman
International Institute of Islamic Economics,
International Islamic University of Islamabad
Revised: 21 December 2008. comments welcome; email to: asadzaman@alum.mit.edu
Abstract: Positivist philosophy provided the foundation for all social sciences as they were developed in the twentieth century. Now that positivism has been abandoned by philosophers, it is essential to replace methodology founded on positivist precepts. We show that the foundations of probability (both Bayesian and Frequentist) are wedded to positivist assumptions, which have been shown not to be workable. We suggest an alternative non-positivist foundation for probability theory, which avoids many of the problems faced by the current methodologies.
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My contention is that logical positivism BLOCKED all natural interpretations. natural interpretations have to do with counterfactuals and what might have happened, and this is meaningless according to positivist philosophy.
Both of the two interpretations (frequentist & Bayesian) which were developed are positivist == and were developed by positivists, but both are incorrect for reasons outlined in the draft.
Now the goal is to present a viable alternative, which is natural interpetation of the above statement. After going through many possibilities the one which seems best to me is the following:
This is inspired by Colin Turbayne: The Myth of Metaphor (which is a very worth reading book, highly illuminating). Turbayne argues that models are like metaphors. A comparison of two very different objects -- like John is a Lion. This means that some properties of the two are the same (courage, for example). At the same time it means that some properties (maybe most) are radically different.
Now I believe that the natural domain of probability is an IMAGINARY world -- lets say an imaginary ideal box with lottery tickets from which we make a perfectly random draw with equal chances for all tickets.
When we say that Probability of heads is one half, we mean that there is a resemblance between draws from a box with two tickets entitled Heads, Tails and the Coin Flip. It still needs to be specified WHAT are the aspects which resemble, and it should be obvious that there are many things WHICH DO NOT resemble.
Anyhow, I think this is an entirely new interpretation of probability, different from any that I have seen. It is a natural interpretation but again access to it has been blocked by the positivist mindset -- which rejects imaginary worlds as being non existent.
With this interpretation, one would have to ask: What are the characteristics which resemble? Alternatively, and equivalently, for what purposes is this model useful? For some issues dealing with coin flips, the model will be useful, For many issues (such as the shape and physical structure of the coin) the model is not useful. However the shape of the coin may determine whether or not the model would be useful.
This leads to the interesting possiblity that the model be useful even if it not "true" -- for example, if numbers being generated by a good pseudo random generator. Then it is a deterministic sequence, but still, very hard to figure out for anyone who does not know the background algorithm and the seed. For practical purposes the model would be useful in predicting patterns that we could expect to see.