Bayesian Epistemology
We can think of belief as an all-or-nothing affair. For example, I believe that I am alive, and I don’t believe that I am a historian of the Mongol Empire. However, often we want to make distinctions between how strongly we believe or disbelieve something.
I strongly believe that I am alive, am fairly confident that I will stay alive until my next conference presentation, less confident that the presentation will go well, and strongly disbelieve that its topic will concern the rise and fall of the Mongol Empire. The idea that beliefs can come in different strengths is a central idea behind Bayesian epistemology. Such strengths are called degrees of belief, or credences.
Bayesian epistemologists study norms governing degrees of beliefs, including how one’s degrees of belief ought to change in response to a varying body of evidence. Bayesian epistemology has a long history. Some of its core ideas can be identified in Bayes’ (1763) seminal paper in statistics (Earman 1992: ch.
1), with applications that are now very influential in many areas of philosophy and of science. The present entry focuses on the more traditional, general issues about Bayesian epistemology, and, along the way, interested readers will be referred to entries that discuss the more specific topics. A tutorial on Bayesian epistemology will be provided in the first section for beginners and those who want a quick overview.
📑 Contents
1. A Tutorial on Bayesian Epistemology
This section provides an introductory tutorial on Bayesian epistemology, with references to subsequent sections or related entries for details.
1.1 A Case Study
For a glimpse of what Bayesian epistemology is, let’s see what Bayesians have to say about this episode in scientific inquiry:
The above case makes a general point:
This intuition about how credences ought to change can be vindicated in Bayesian epistemology by appeal to two norms. But before turning to them, we need a setting. Divide the space of possibilities into four, according to whether hypothesis H is true or false and whether evidence E is true or false. Since H logically implies E, there are only three distinct possibilities on the table, which are depicted as the three dots in figure 1.
Those possibilities are mutually exclusive in the sense that no two of them can hold together; and they are jointly exhaustive in the sense that at least one of them must hold. A person can be more or less confident that a given possibility holds. Suppose that it makes sense to say of a person that she is, say, 80% confident that a certain possibility holds. In this case, say that this person’s degree of belief, or credence, in that possibility is equal to 0.8. A credence might be any other real number. (How to make sense of real-valued credences is a major topic for Bayesians, to be discussed in §1.6 and §1.7 below.)
Now I can sketch the two core norms in Bayesian epistemology. According to the first norm, called Probabilism, one’s credences in the three possibilities in figure 1 ought to fit together so nicely that they are non-negative and sum to 1. Such a distribution of credences can be represented by a bar chart, as depicted on the left of figure 2.
Now, suppose that a person with this credence distribution receives E as new evidence. It seems that as a result, there should be some change in credences. But how should they change? According to the second norm, called the Principle of Conditionalization, the possibility incompatible with E (i.e., the rightmost possibility) should have its credence dropped down to 0, and to satisfy Probabilism, the remaining credences should be scaled up—rescaled to sum to 1. So this person’s credence in hypothesis H has to rise in a way such as that depicted in figure 2.
Moreover, suppose that new evidence E is very surprising. It means that the person starts out being highly confident in the falsity of E, as depicted on the left of figure 3.
Then conditionalization on E requires a total credence collapse followed by a dramatic scaling-up of the other credences. In particular, the credence in H is raised significantly, unless it is zero to begin with. This vindicates the intuition reported in the case of Eddington’s Observation.