When a student challenges a teacher on a topic and that teacher answers with “Bayes Theorem,” then this can be seen as showing that “A is B,” and “B is C.” On the other hand, when the student challenges the teacher on “Bayes Theorems,” and he says “It’s not true because it doesn’t work,” then this can be seen as showing that “A is C,” and “C is D.” The second conclusion, “A is B,” is the most correct answer. The third, “B is C,” is incorrect. Thus, if one wishes to prove something, one needs to use one or more of the tools provided by Bayes Theorems.
First, a priori Bayesians must demonstrate that there is some probability that the conclusion “A” is true, and that the other conclusion “B” is false. This proof needs to be very strong. There is no reason why it cannot be built from a few other premises, as long as they are consistent with each other. A priori Bayesians might also wish to apply statistical methods to show that their theory has strong probability evidence.
Second, for Bayesians, the prior probability that A is true, and the posterior probability that B is true, are independent. This means that the results of both A and B cannot be influenced in any way by the existence or non-existence of some other truth. This makes the A-B comparison a kind of test or proof, as it were. Students will test or prove that A is true by discovering independent information about whether or not A is true, and how much strength that evidence has. They will also look for independent confirmation of B.
Third, to show that A is true, the student must convince the Axiom of Proxies. That is, the student must show how an argument from ignorance can be applied to solve a problem, and how many false premises lead to true conclusions. The Axiom of Proxies shows that the existence of many independent variables produces a conclusion about probability. If A is true, then B must be false, but A and B must both be true. The student therefore needs to convince the Axiom of Proxies that A and B are both true by discovering independent information about both A and B.
In order to do this, the student uses techniques such as induction, which he or she learns in advanced algebra classes. Induction is used to show that the conclusion reached by any procedure is actually derived from the prior beliefs and opinions of the student. It is usually followed by more detailed arguments based on other concepts and laws of physics. The student uses this method to deduce that the rate of cooling in summer, the rate of heating in winter, and the age of the earth are all measured in different ways by using appropriate methods, and hence that the temperature at any point is independent of its place in space and time. The Bayes’ Theorem can be proved by inductive reasoning, and can thus be used as an argument to support any conclusion the student wishes to draw about any existing facts. It also provides a neat and tidy way to sum up all the results obtained using various deductive methods and so provides a very nice example of how mathematics can be made more precise and rigorous by combining techniques derived independently by different people.
A final step in the process leads the student to reject the Axiom of Proxies outright, and hence to embrace the strong conclusions implied by his or her reasoning. The Bayes’ Theorem shows that there are many hypothesis (or even more than one) that can bring about a conclusion about probability. As such, it gives rise to a number of testable conclusions, each of which can be falsified by further investigation. And as such, the Bayes’ Thesis can play an important role in the study of statistics, and it can show the limitations of statistical sampling, proving theorems true even in the face of contradictory evidence.
Although the Bayes’ Thesis has been used in mathematics for ages, it has only recently received the status of a formal mathematical theses. This is perhaps owing to the fact that mathematicians had already been tempted to use methods originating from linguistics, when constructing theorems and arguments. The Bayes’ Thesis thus provides a perfect example of how different areas of mathematics can be used to form conclusions about probability. Its success lies in the elegance of its formulation, and also in the elegance of its logical structure, which has made it extremely versatile. Thanks to recent advances in mathematics, the Bayes’ Thesis can now be used with great success in many areas of research.
