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The foundation of statistics is strongly connected to philosophy, mathematics and economics. And today, we can say this foundation consists of sand.
Statistics put at the appropriate level of generality, assumes a person receiving a series of quantitative inputs and using only this finite amount of uncertain information to decide a course of action. Since a finite amount of information does not contain an abstract model of any sort, we must admit that this approach is by itself incomplete, as admitted by modern statistics.
The most serious problem we face here is the difference between probability and “uncertainty” (as defined by JM Keynes and others). Uncertainty is defined as a situation where the probability itself of an event is know and possibility cannot be known – the chance the Dow Jones Industrial Average will be between 10,000 and 20,000 in twenty years (or IF there will be a DJIA in twenty years).
If we are talking about a mathematical model, we must call this “uncertainty” conceptual hogwash. To say that a probability is not known in no ways negates its existence. Moreover, we can certainly determine the result of different strategies for determining the success of different strategy for acting uncertain information.
We can simplify our approach to the single case of considering an apparently rare phenomenon. Our observer O receives data showing a steady stream of input X. O must decide at a given point whether to bet on the further occurrence of X. If we assume the formalism of Bayes, we determine a small “maximum likelihood” of X in our next “round”.
This seems fine. Yet given that this incomplete, what are the alternatives? In possibility is that our situation we selected to be unlikely by we, ourselves. This in a sense contradicts the idea that this situation is intended to be useful since it is typical.
The other point where typicality breaks is the situation where the sequence of information is “correlated”. But correlated to what? In our decision-making question, the correlation must be to the qualities of O himself/herself. The simplest example is when an agent knows O’s decision rule and creates a situation which causes O to act, afterwards taking advantage of O’s action. This would correspond most dramatically to bandit who fires a shot, waits a minute and then fires a further shot to catch anyone poking their head out to see the disturbance.
So we can say that “typically”, those experiences where a Bayesian approach is not valid are those experiences where the O and the data are mutually generated by a single process. We can call this, somewhat grandly, the condition of being enmeshed historically. And tautologically, this implies that sufficient knowledge of the historical stages wherein O and data are generated will result in a calculable probability.
In the abstract, this is not an aesthetically satisfying formulation. Our “historical” formulation is simply all that is not Bayesian. If we, ourselves, look historically at the problems which give rise to this, however, this is quite useful as an approach. We should remember that, formally, we are assuming that the entire “real” world can be embedded in an abstract mathematical model. At the same time, historically, we are seeing this abstract mathematical model as a creation of our “real world”.
In particular, statistics is applied for particular purposes. The situation of those studying the foundation of statistics and specifying arbitrary information and arbitrary observes is not itself arbitrary.
The situation of investor in uncertainty is key.
A market is seen as a distributor of scarce information. Yet its ability to contribute that information to the system as a whole is fairly limited.
We postulate a productive system requiring information. We posit a way in which this information is action investment information.
Say, the question is choosing between oil fields. The problem is that one person having certain information that a given oil field is productive and investing based on that information wouldn’t help an oil company decide which field to drill on. The price of the field wouldn’t determine the decision.
Of course, a supply and demand model for