Chapter 9: Belief Revision

Overview

I was always puzzled by the fact that people have a great deal of trouble and pain when and if they are forced or feel forced to change a belief or circumstance which they hold dear. I found what I believe is the answer when I read that a Canadian neurosurgeon discovered some truths about the human mind which revealed the intensity of this problem. He conducted some experiments which proved that when a person is forced to change a basic belief or viewpoint, the brain undergoes a series of nervous sensations equivalent to the most agonizing torture.

—Sidney Madwed

Suppose that an agent believes φ₁, …, φ_n ("belief" here is taken in the sense of Section 8.1) and then learns or observes ψ. How should she revise her beliefs? If ψ is consistent with φ₁ ∧ … ∧ φ_n, then it seems reasonable for her to just add ψ to her stock of beliefs. This is just the situation considered in Section 3.1. But what if ψ is, say, φ₁? It does not seem reasonable to just add φ₁ to her stock of beliefs, for then her beliefs become inconsistent. Nor is it just a simple matter of discarding φ₁ and adding φ₁. Discarding φ₁ may not be enough, for (at least) two reasons:

1. Suppose that φ₁ is φ₂ ∧ φ₃. If the agent's beliefs are closed under implication (as I will be assuming they are), then both φ₂ and φ₃ must be in her stock of beliefs. Discarding φ₁ and adding φ₁ still leaves an inconsistent set. At least one of φ₂ or φ₃ will also have to be discarded to regain consistency, but which one?

2. Even if the result of discarding φ₁ and adding φ₁ is consistent, it may not be an appropriate belief set. For example, suppose that φ₄ is φ₁ ∨ p. Since φ₄ is a logical consequence of φ₁, it seems reasonable to assume that φ₄ is in the agent's belief set (before learning φ₁). But suppose that the only reason that the agent believed φ₄ originally was that she believed φ₁. Discarding φ₁ removes the justification for φ₄. Shouldn't it be removed too? Note that if φ₄ remains among the agent's beliefs, then the fact that both φ₁ and φ₄ are included in the agent's beliefs suggests that p should be too. But there is nothing special about p here; it could be any formula. It certainly does not seem reasonable to have a procedure that allows an arbitrary formula to be among the agent's beliefs after learning φ₁.

Chapter 3 has a great deal of discussion as to how an agent's beliefs should be updated in the light of new information. Surely some of that discussion should be relevant here. In fact, it is highly relevant. Characterizing an agent's beliefs in terms of formulas (which is the standard approach in the literature) obscures what is really going on here. I argue in this chapter that belief revision can be completely understood in terms of conditioning. Using an (unconditional) probability measure as a representation of belief (where belief is taken to mean "probability 1") will not quite work, since the main issue here is what happens if an event that previously was not believed is observed. This amounts to conditioning on an event of probability 0. However, if beliefs are represented using a qualitative plausibility measure (i.e., using any of the representations discussed in Chapter 8, including using a conditional probability measure), then the appropriate notion of conditioning for that representation does indeed capture the standard properties of belief revision. Different decisions as to how to revise turn out to correspond to different prior plausibilities.

Most of the effort in this chapter involves showing that the standard approaches to belief revision in the literature can be understood in terms of conditioning, and that doing so leads to further insights into the belief revision process. Making the connection also brings into sharper focus some of the issues discussed in Sections 3.1 and 6.7.1, such as the need to assume perfect recall and that the way that an agent obtains new information does not itself give information.