Generalized Imaging
It is not yet universally agreed what revision means in a probabilistic setting. One school of thought says that probabilistic expansion is equivalent to Bayesian conditioning. This is evidenced by Bayesian conditioning () being defined only when , thus making expansion equivalent to revision. In other words, one could define expansion (restricted revision) to be
To accommodate cases where , that is, where contradicts the agent’s current beliefs and its beliefs need to be revised in the stronger sense, we shall make use of imaging. Imaging was introduced by Lewis (1976) as a means of revising a probability function. It has also been discussed in the work of, for instance, Gärdenfors (1988); Dubois and Prade (1993); Chhogyal et al. (2014); Rens and Meyer (2015). Informally, Lewis’s original solution for accommodating contradicting evidence is to move the probability of each world to its closest, world. Lewis made the strong assumption that every world has a unique closest world. More general versions of imaging allows worlds to have several, equally proximate, closest worlds.
Gärdenfors (1988) calls one of his generalizations of Lewis’s imaging general imaging. Our method is also a generalization. We thus refer to his as Gärdenfors’s general imaging and to our method as generalized imaging to distinguish them. It should be noted that all three these imaging methods are general revision methods and can be used in place of Bayesian conditioning for expansion. “Thus imaging is a more general method of describing belief changes than conditionalization,” (Gärdenfors, 1988, p. 112).
Let be the set of worlds closest to with respect to pseudodistance . Formally,
where is some pseudodistance measure between worlds (e.g., Hamming or Dalal distance).
Example 1.
Let the vocabulary be . Let be . Suppose is Hamming distance. Then
Definition 2 ().
Then generalized imaging (denoted ) is defined as
In words, is the new belief state produced by taking the generalized image of with respect to . Notice how the probability mass of nonworlds is shifted to their closest worlds. If a nonworld with probability has closest worlds (equally distant), then each of these closest worlds gets mass from .
We define so that we can write , where is a revision operator.
Revision via and boundary belief states
Perhaps the most obvious way to revise a given belief base (BB) is to revise every individual belief state in and then induce a new BB from the set of revised belief states. Formally, given observation , first determine a new belief state for every via the defined revision operation:
If there is more than only a single belief state in , then contains an infinite number of belief states. Then how can one compute ? And how would one subsequently determine from ?
In the rest of this section we shall present a finite method of determining . What makes this method possible is the insight that can be represented by a finite set of ‘boundary’ belief states – those belief states which, in a sense, represent the limits or the convex hull of . We shall prove that the set of revised boundary belief states defines . Inducing from is then relatively easy, as will be seen.
Let be every permutation on the ordering of worlds in . For instance, if , then , , , , . Given an ordering , let be the th element of ; for instance, . Suppose we are given a BB . We now define a function which, given a permutation of worlds, returns a belief state where worlds earlier in the ordering are assigned maximal probabilities according to the boundary values enforced by .
Definition 3.
is the such that for , , if , then .
Example 3.
Suppose the vocabulary is and . Then, for instance, , , , , , , , , , .
Definition 4.
We define the boundary belief states of BB as the set
Note that .
Example 4.
Suppose the vocabulary is and . Then
Next, the revision operation is applied to every belief state in . Let .
Example 5.
Suppose the vocabulary is and . Let be . Then
(Two revision operations produce .)
To induce the new BB from , the following procedure is executed. For every possible world, the procedure adds a sentence enforcing the upper (resp., lower) probability limit of the world, with respect to all the revised boundary belief states. Trivial limits are excepted.
For every , , where , except when , and , where , except when .
The intention is that the procedure specifies to represent the upper and lower probability envelopes of the set of revised boundary belief states – thus defines the entire revised belief state space (cf. Theorem 1).
Example 6.
Continuing Example 5, using the translation procedure just above, we see that , , , .
Note that if we let , , then .
Example 7.
Suppose the vocabulary is and . Let be . Then
, , , .
Note that if we let , , then .
Let be a partition of such that is a block in iff . Denote an element of block as , and the block of which is an element as . Let , in other words, the superscript in indicates the size of . Let .
Observation 1.
Let be positive integers such that iff . Let be values in such that . Associate with every a maximum value it is allowed to take: . For every , we define the assignment value
Determine first , then and so on. Then
whenever for some .
For instance, let , , , . Let , , , . Then , , , and
But
And
Lemma 1 essentially says that the belief state in which causes a revised belief state to have a maximal value at world (w.r.t. all belief states in ), will be in .
Lemma 1.
For all , is in .
Proof.
Note that
can be written in the form
Observe that there must be a such that . Then by the definition of the set of boundary belief states (Def. 4), will assign maximal probability mass to , then to and so on.
That is, by Observation 1, for some , for all . Therefore, is in . ∎
Let
Lemma 2 states that for every world, the upper/lower probability of the world with respect to is equal to the upper/lower probability of the world with respect to . The proof requires Observation 1 and Lemma 1.
Lemma 2.
For all , and .
Proof.
Note that if , then and .
We now consider the cases where .
iff
iff
if
, where
and
Note that
can be written in the form
Then by Observation 1, is in . And also by Lemma 1, the belief state in identified by must be the one which maximizes
where . That is, .
With a symmetrical argument, it can be shown that . ∎
In intuitive language, the following theorem says that the BB determined through the method of revising boundary belief states captures exactly the same beliefs and ignorance as the belief states in which have been revised. This correspondence relies on the fact that the upper and lower probability envelopes of can be induce from , which is what Lemma 2 states.
Theorem 1.
Let . Let be the BB induced from . Then .
Revising via a Representative Belief State
Another approach to the revision of a belief base (BB) is to determine a representative of (call it ), change the representative belief state via the the defined revision operation and then induce a new BB from the revised representative belief state. Selecting a representative probability function from a family of such functions is not new (Goldszmidt, Morris, and Pearl, 1990; Paris, 1994, e.g.). More formally, given observation , first determine , then compute its revision , and finally induce from .
We shall represent (and thus ) by the single ‘least biased’ belief state, that is, the belief state in with highest entropy:
Definition 5 (Shannon Entropy).
where is a belief state.
Definition 6 (Maximum Entropy).
Traditionally, given some set of distributions , the most entropic distribution in is defined as
Suppose . Then the belief state satisfying the constraints posed by for which is maximized is , , , .
The above distribution can be found directly by applying the principle of maximum entropy: The true belief state is estimated to be the one consistent with known constraints, but is otherwise as unbiased as possible, or “Given no other knowledge, assume that everything is as random as possible. That is, the probabilities are distributed as uniformly as possible consistent with the available information,”
(Poole and Mackworth, 2010). Obviously world 00 must be assigned probability 0.1. And the remaining 0.9 probability mass should be uniformly spread across the other three worlds.Applying to on evidence results in , , , .
Example 8.
Suppose the vocabulary is , and is . Then , , , . Applying to on results in , , , . can be translated into as .
Still using , notice that . But how different are and , , , ? Perhaps one should ask, how different is from the representative of : The least biased belief state satisfying is . That is, How different are and ?
In the case of , we could compare , , , with , , , . Or if we take the least biased belief state satisfying , we can compare , , , with , , , .
It has been extensively argued (Jaynes, 1978; Shore and Johnson, 1980; Paris and Vencovská, 1997) that maximum entropy is a reasonable inference mechanism, if not the most reasonable one (w.r.t. probability constraints). And in the sense that the boundary belief states method requires no compression / information loss, it also seems like a very reasonable inference mechanism for revising BBs as defined here. Resolving this misalignment in the results of the two methods is an obvious task for future research.
Future Directions
Some important aspects still missing from our framework are the representation of conditional probabilistic information such as is done in the work of KernIsberner, and the association of information with its level of entrenchment. On the latter point, when one talks about probabilities or likelihoods, if one were to take a frequentist perspective, information observed more (less) often should become more (less) entrenched. Or, without considering observation frequencies, an agent could be designed to have, say, one or two sets of deeply entrenched background knowledge (e.g., domain constraints) which does not change or is more immune to change than ‘regular’ knowledge.
Given that we have found that the belief base resulting from revising via the boundarybeliefstates approach differs from the belief base resulting from revising via the representativebeliefstate approach, the question arises, When is it appropriate to use a representative belief state defined as the most entropic belief state of a given set ? This is an important question, especially due to the popularity of employing the Maximum Entropy principle in cases of undespecified probabilistic knowledge (Jaynes, 1978; Goldszmidt, Morris, and Pearl, 1990; Hunter, 1991; Voorbraak, 1999; KernIsberner, 2001; KernIsberner and Rödder, 2004) and the principle’s wellbehavedness (Shore and Johnson, 1980; Paris, 1994; KernIsberner, 1998).
Katsuno and Mendelzon (1991) modified the eight AGM belief revision postulates (Alchourrón, Gärdenfors, and Makinson, 1985) to the following six (written in the notation of this paper), where is some revision operator.^{3}^{3}3In these postulates, it is sometimes necessary to write an observation as a BB, i.e., as – in the present framework, observations are regarded as certain.

.

If is satisfiable, then .

If is satisfiable, then is also satisfiable.

If , then .

.

If is satisfiable, then .
Testing the various revision operations against these postulates is left for a sequel paper.
An extended version of maximum entropy is minimum crossentropy (MCE) (Kullback, 1968; Csiszár, 1975):
Definition 7 (Minimum CrossEntropy).
The ‘directed divergence’ of distribution from distribution is defined as
is undefined when while ; when , , because . Given new evidence , the distribution satisfying diverging least from current belief state is
Definition 8 ().
Then MCE inference (denoted ()) is defined as
In the following example, we interpret revision as MCE inference.
Example 9.
Suppose the vocabulary is and . Let be . Then
, .
Note that if we let , then .
Recall from Example 6 that included . Hence, in this particular case, combining the boundary belief states approach with results in a less informative revised belief base than when is used. The reason for the loss of information might be due to and
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