SPSG #2: Two Probabilities and Frequency Too

This is a history of philosophy of science presentation, and it sets up the terminology used throughout the rest of the book: If the first chapter is about "safety", this one is about "probability". There are two very different concepts of probability, both of which are very old and generally familiar. From an etymological standpoint, the legal concept of probability dates back to Roman law. In common parlance, the evidential form of probability is being used whenever a proposition is said to be "probably true". Even though it wasn't called probability until Pascal and friends gave it that name in the 17th century, the concept of chance and its relationship to frequency of occurrence has been around since Aristotle. In common parlance, the probability of chance is being used whenever it is said that an event will "probably happen". In scientific and technical literature, it is the second "statistical" meaning of probability that is used almost exclusively, and that is almost certainly because it is more objective in an empirical sense. However, the other subjective form of probability, which is called "theoretical" or "evidential" probability in the book to emphasize its role in science, can still be found in scientific discussion all the time, but it usually appears in the words rather than the numbers.

There are two other important concepts introduced in this chapter as well. First, as a result of the statistical definition of probability, uncertainty and frequency of occurrence are often treated as identical concepts. That is incorrect both from a grammatical standpoint, and because even without the appendage of "probability", statistical theories concerning frequency of occurrence are important in many scientific disciplines. For example, in public health, regulatory issues often revolve around evidential probabilities of statistical theories. Second, abstract representations of probability are also sometimes referred to as probability themselves without reference to usage, which leads to yet another opportunity for semantic confusion. As they are taught in introductory courses concerned with probability and statistics, mathematical probability distributions that can be used to represent either frequency of occurrence or the probability of chance are well known. However, even though theoretical probability is clearly a matter of degree, it is much harder to quantify. A probability tree can be used to represent theoretical probability and provide a quantitative interpretation as well. The basic concept is very simple: The probability of all alternative theories or hypotheses under consideration sums to one. Scientific evidence may then be weighed in order to determine which hypotheses, and to what degree, are more probable than the others. Causal relationships are the most common issue in which scientific issues involving theoretical probability arise. For example, whether or not a particular chemical causes cancer, and if so how, involves theoretical probability. Although they have evolved somewhat, guidelines for establishing causal relationships in science have been around for centuries. They have been around in law for even longer than that.

Notes

Perhaps this should be the first chapter, but in the interest of not straining credulity at the outset I have put it second.

After having been at the FDA for several years, I ran into my first model uncertainty problem.  I traipsed over to the Office of Mathematics to inquire about the proper methodology for calculating the probability of a model being true. I asked a statistician who had been at the FDA for thirty years, and the answer I got astounded me:

You are not allowed to ask that question 

I obtained a second opinion from a younger statistician,  just trying to find out about how to go about identifying the best model among several and got an answer that I found to be no more satisfactory:

Find a biologist and beat it out of them

As a biologist who did not want to be beaten, I soon embarked upon a philosophy of science reading binge that lasted several years in the mid 90's.  The main thing I learned is that there is another probability that is quite different from the one the statisticians were using.  However, in the end I was unable learn very much about it from the philosophers.  This chapter is a compilation of the few gems that I managed to gather from my survey.  Perhaps not surprisingly, some of the most important insights came from practicing scientists and risk analysts. In any case, I managed to answer my own question, at least to my own satisfaction. Although beating the probabilities out of the biologists can be a fair characterization of the method, the beating can be avoided with a willing confession. The description of that solution begins in this chapter and finishes in chapter 9.

However, I ran into other difficulties subsequently.  It seems that not everyone wanted the problem to be solved at all.  The rest of the book is about that.