A Primer on Bayesian Philosophy: an introduction

As I have only recently began to explore Bayesian philosophy, I am perhaps unqualified to be giving lessons. However, I have found that what little I understand has been quite beneficial, and I think I am with others' help capable of highlighting common errors in probabilistic and evidential thinking. As I find myself directing others to resources quite often, it would be handy to have a reference, both for myself and for others. So I intend this primer to be just that: a small, workable start useful for ordering future in-depth reading. The criterion of my success or failure in this endeavor is this: after working through these posts, can you or can you not begin to apply Bayesian tools or know where to look in order to learn them?

To some extent, Bayesian philosophy can be thought of as both extending deductive logic and as formalizing everyday reasoning. In dealing with an uncertain world and comparing uncertain hypotheses via evidence, there is a need for the sort of rigor which probability theory provides. Regardless of whether one accepts a `Dutch Book defense' of Bayesian epistemology, there are fantastic pragmatic reasons for understanding it. `To some extent' is an important caveat; thinking of probability solely as an extension of deductive logic can lead to errors. It also helps one be humble: it makes clear how difficult it can be to convict a coherent and consistent person of being `unreasonable'.

The difficulty with Bayesianism is that it takes serious work for the mathematically or philosophically uninitiated. For those lacking the math, this is obvious. But the vindication of this work is that as a skeptic and a person interested in science, you should put in the effort. If you lack the mathematical prerequisites for Bayesianism, you lack the mathematical prerequisites for everyday science and statistics. In so doing you will fail to understand the shortcomings of studies, among so many other things. I do not think I should have to convince an interested reader of the import of set theory and mathematical logic, or of axiomatic methodology. For those lacking the philosophy but in some possession of the math, you might come to feel betrayed by your probability textbook, as I have come to feel.

I thought of including a post introducing abstract mathematics, but this would be too much. I have to assume this on the reader, asking her to consult other resources. If you understand naive sets, de Morgan's laws, basic quantifiers, mathematical induction, and Cantor's diagonalization theorem, you are probably ready to tackle the first post. Do not feel the need to memorize a Junior-level course before entering; unless you are well-studied in mathematics, be content with regularly consulting the material as you go.

I organize my approach as follows:

1. Basic axiomatic probability
2. Bayesianism outlined
3. Arguments for and against Bayesianism
4. Interpretation and subjectivity, part 1 and part 2
5. Other outstanding issues and some resources
6. Applications: deductive logic extended?

For (1), the goal is to introduce readers to Kolmogorov's axioms and key theorems, including e.g. the Law of Total Probability and Bayes' Theorem. Naturally, those familiar with the maths can skip this section. (2) extends the discussion in (1), listing probabilistic assumptions vital to Bayesianism, for example conditionalization and rigidity. (3) will focus on the standard defense of Bayesian epistemology: the Dutch Book argument. (4) explores the subjective/objective divides in probabilistic philosophy, esp. frequentism, propensity theories, and equivocation. (5) allows me to retain some comprehensiveness while keeping other sections clean; the title is self-explanatory. The title of (6) is similarly explanatory. Once you have an idea of formalism, seeing it in practice is vital to absorbing it, just as those who are attempting to learn mathematics should focus on the exercise section of the text.

Again, my approach will be simplistic and designed to prompt further reading. As many of the topics discussed are matters of professional dispute, I can do no other.