Probabilistic belief change is an operation that takes a probability distribution representing a belief state along with an input sentence representing some information to be accommodated or removed, and maps it to a new probability distribution. In order to choose from many such mappings possible, techniques from information theory such as the principle of minimum cross-entropy have previously been used. Central to this principle is the Kullback-Leibler (KL) divergence. In this short study, we focus on the contraction of a belief state P by a belief a, which is the process of turning the belief a into a non-belief. The contracted belief state \(P^-_a\) can be represented as a mixture of two states: the original belief state P, and the resultant state \(P^*_{\lnot a}\) of revising P by \(\lnot a\). Crucial to this mixture is the mixing factor \(\epsilon \) which determines the proportion of P and \(P^*_{\lnot a}\) that are to be used in this process. We show that once \(\epsilon \) is determined, the KL divergence of \(P^-_a\) from P is given by a function whose only argument is \(\epsilon \). We suggest that \(\epsilon \) is not only a mixing factor but also captures relevant aspects of P and \(P^*_{\lnot a}\) required for computing the KL divergence.
[1]
Abhaya C. Nayak,et al.
Probabilistic Belief Contraction
,
2012,
Minds and Machines.
[2]
Nico Potyka,et al.
Changes of Relational Probabilistic Belief States and Their Computation under Optimum Entropy Semantics
,
2013,
KI.
[3]
Abdul Sattar,et al.
Probabilistic Belief Contraction Using Argumentation
,
2015,
IJCAI.
[4]
R. A. Leibler,et al.
On Information and Sufficiency
,
1951
.
[5]
Gabriele Kern-Isberner,et al.
Linking Iterated Belief Change Operations to Nonmonotonic Reasoning
,
2008,
KR.
[6]
E. Jaynes.
Information Theory and Statistical Mechanics
,
1957
.
[7]
Peter Gärdenfors,et al.
On the logic of theory change: Partial meet contraction and revision functions
,
1985,
Journal of Symbolic Logic.