My Research in Belief Revision Theory

Belief Revision Theory

When I was investigating problems of belief revision in dynamic games (see my research in epistemic game theory) I became more and more interested in the problem of belief revision itself. The scientific discipline that studies the problem of belief revision is called belief revision theory.

Belief revision theory is a highly interdisciplinary subject, as it is studied by computer scientists, people in artificial intelligence, philosophers, psychologists, logicians, decision theorists and game theorists, and probably some other people that I forgot to mention. The main question in belief revision theory is the following: Suppose you hold a certain belief about the world surrounding you, but at a given moment you observe an event that contradicts this belief. How should you then revise your belief about the world? 

Minimal Belief Revision

A generally accepted idea in belief revision theory is that of minimal belief revision: If you observe an event that contradicts the previous belief you had about the world, then you should find a new belief about the world that

1. explains the event you just observed, and
2. is "as close as possible" to the previous belief you had.

The intuition behind this principle is that your previous belief was based on extensive reasoning about the world surrounding you. Therefore, if your belief turns out to be not entirely correct, then you should change your belief, but not more than necessary.

The idea of minimal belief revision has been applied extensively to models of non-probabilistic beliefs. By the latter we mean situations where the belief of a person is given by a number of statements, or formulae, about the world. For instance: "I believe there are three balls in this box", or, "I believe it will rain this afternoon".

Minimal Probabilistic Belief Revision

A probabilistic belief is, for instance: "I believe that, with probability 0.3, there are three balls in the box, and with probability 0.7 there are four balls in the box". Another example would be: "I believe that, with probability 0.6, it will rain this afternoon".  

In my opinion, the idea of minimal belief revision also makes perfect sense in models of probabilistic beliefs. Consider, for instance, the following thought experiment:

 Suppose you may glimpse for a few seconds at a transparent urn which is filled with black balls and white balls. Afterwards, you must form a probabilistic belief about the number of black balls and white balls. More precisely, you are given three possible options:

∙ 30 black balls and 30 white balls (state a);
∙ 30 black balls and 20 white balls (state b);
∙ 20 black balls and 20 white balls (state c);

One of these options is true, you are said, but you do not know which one since the observation only gave you a vague impression of the number of balls of each color. Your task is to assign probabilities to each of these three options. You recall to have seen approximately the same number of black and white balls, but time was to short to count the balls you have seen. Based on these considerations, you deem the states a and c equally likely, but at the same time you deem the event "a or c" twice as likely as the state b. Hence, you assign probability 1/3 to each of the states.
Now, you are told that state c is false. Your new task is to assign probabilities to the two remaining states a and b. How would you do this?

Bayesian updating suggests that we should not change our belief about the relative likelihood of the two remaining states, and hence our revised belief by Bayesian updating would assign probability 1/2 to states a and b. In other words, if we hear that c is false and use Bayesian updating, then we loose much of our confidence in the event that the number of black and white balls is equal.

This seems counterintuitive, however, since you were rather confident that you have seen an equal number of black and white balls, but much less confident about the precise number of balls. Therefore, the information that c is false should not be a reason to drastically change your belief about the proportion of black and white balls. As an extreme case, one could decide to maintain the belief that the event "a or c" is twice as likely as state b, which would yield a revised belief in which you assign probability 2/3 to a and probability 1/3 to b.

Imaging

In fact, the last way of revising your belief is an example of imaging, as proposed by the philosopher David Lewis. The idea behind imaging is that, upon observing that some state x is impossible, you transfer the probability initially assigned to x completely towards the remaining state you deem most similar to x. In the example above, you deem state c much more similar to a than to b, since c and a share the same proportion of black and white balls, and the observation has given you a pretty good idea of this proportion. Therefore, if we hear that c is false and use imaging, then we transfer the probability 1/3 initially assigned to c completely towards the most similar state, a, and hence our revised belief would assign probability 2/3 to a and probability 1/3 to b.

The main difference between Bayesian updating and imaging in this context is thus that imaging makes explicit use of the perceived similarity between states when revising the belief, whereas Bayesian updating ignores this perceived similarity. Although the theory of imaging makes intuitive sense in contexts where similarity between states plays a prominent role, as the example above hopefully illustrates, it has received little attention, especially among decision theorists, game theorists and economists.

Peter Gärdenfors extends Lewis' idea by allowing to transfer a part λ of the probability 1/3, initially assigned to c, towards state a and the remaining part 1-λ to state b. Important is that these fractions λ and 1-λ should be independent of the initial belief. That is, if the initial belief would not be (1/3,1/3,1/3) but (1/8,3/8,1/2), then the person should transfer the fraction λ of the probability 1/2 to a and the fraction 1-λ of 1/2 to b, where λ is the same fraction as above. Gärdenfors' model is also known as general imaging.

A possible interpretation of λ is that it represents the similarity between c and a, as compared to the similarity between c and b. For instance, if λ is very close to 1, this indicates that the person deems c much more similar to a than to b, and for that reason shifts almost all the weight initially assigned to c towards a. If λ is close to 1/2, this indicates that the person deems c almost as similar to a as to b, and therefore redistributes the weight initially assigned to c almost equally among a and b. Lewis' model of imaging is thus a special case of Gärdenfors' model, by choosing λ equal to 0 or 1.

My Theoretical Paper on Belief Revision

In my theoretical paper, called A Model of Minimal Probabilistic Belief Revision, I focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules.

The three axioms I use are called linearity, transitivity and information order independence.

Linearity is defined as follows: Consider three different states a,b and c and a person who initially assigns probability p to c. If this person finds out later that the true state must be in {a,b}, he should shift some fraction λp towards a, and the remaining fraction (1-λ)p towards b. Linearity states that these fractions λ and 1-λ should be independent of the initial belief, since λ reflects the perceived similarity between c and a, as compared to the perceived similarity between c and b. Among the three axioms, linearity is the one that rules out Bayesian updating as a possible candidate. It is easily seen that Bayesian updating violates linearity, but satisfies the other two axioms to be defined below.

Transitivity states that the "equally similar to" relation between beliefs should be transitive. That is, whenever a person deems the belief β equally similar to the opinionated beliefs [a] and [b], and deems β equally similar to [b] and [c], then he should deem β equally similar to [a] and [c]. In terms of belief revision, this means that if the revised belief upon observing {a,b} would assign equal probabilities to a and b, and the revised belief upon observing {b,c} would assign equal probabilities to b and c, then the revised belief upon observing {a,c} should assign equal probabilities to a and c.

Information-order independence states that the revised belief should not depend on the order in which information is received. For instance, revising the belief upon observing the event {a,b} at once should give the same result as first observing the event {a,b,c}, and then observing the event {a,b}. This axiom thus guarantees that iterated belief revision does not lead to problems.

The main theorem is as follows: A probabilistic belief revision function satisfies linearity, transitivity and information order independence if and only if the belief revision function can be described by the following procedure:  (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief.

The paper will be published in Theory and Decision.

My Experimental Work on Belief Revision

At this moment, I am working on two experiments, both about probabilistic belief revision. The objective in both experiments is to see how people, in practice, revise their probabilistic beliefs. In particular, we wish to test my theoretical model of minimal probabilistic belief revision in the laboratory. 

In the first experiment, together with Werner Güth and Anthony Ziegelmeyer (both from the Max Planck Institute in Jena, Germany), we test the linearity axiom, as described in my theoretical paper above. Since the linearity axiom is precisely the axiom that separates Bayes' rule from my model of minimal probabilistic belief revision, we can test whether the belief revision of people in the laboratory is closer to Bayes' rule, or closer to minimal probabilistic belief revision.

In my second experiment, together with Emin Karagozoglu (Maastricht University), we test whether people, in practice, take into account the similarity between states when revising their probabilistic belief. In my model of minimal probabilistic belief revision, a person, when observing that some state is no longer possible, will shift a large fraction of its probability towards a still possible state that is similar to the state that has been ruled out. In contrast, he will only shift a low (or zero) fraction of its probability towards a still possible state that is perceived to be very different from the state that has been ruled out. So, in my model of minimal probabilistic belief revision, the perceived similarity between states plays an important role.

In contrast, Bayes' rule completely ignores the perceived similarity between states. So, in the second experiment we can again test whether people, in practice, behave closer to Bayes' rule, or closer to minimal probabilistic belief revision. However, we now focus on similarity, and not on the linearity axiom.