Who is Afraid of Minimal Revision?

TL;DR

This paper investigates the minimal-change (minimal revision) approach to belief revision as a learning mechanism. It shows that, despite limited learning power in general, minimal revision universally identifies all finitely identifiable spaces and, on finite spaces, learns from both positive and negative data (under suitable space properties). It provides a precise characterization of which prior plausibility orders enable learning via mini on finite spaces and extends the analysis to conditioning and lexicographic upgrade via generalized tell-tale maps. The results delineate when minimal revision is effective and clarify how more radical revision methods relate to prior structures, while also showing that errors in observations can destroy these guarantees. Overall, the work maps the boundaries of mini’s applicability and offers foundational tools for choosing priors in belief-update-based learning systems.

Abstract

The principle of minimal change in belief revision theory requires that, when accepting new information, one keeps one's belief state as close to the initial belief state as possible. This is precisely what the method known as minimal revision does. However, unlike less conservative belief revision methods, minimal revision falls short in learning power: It cannot learn everything that can be learned by other learning methods. We begin by showing that, despite this limitation, minimal revision is still a successful learning method in a wide range of situations. Firstly, it can learn any problem that is finitely identifiable. Secondly, it can learn with positive and negative data, as long as one considers finitely many possibilities. We then characterize the prior plausibility assignments (over finitely many possibilities) that enable one to learn via minimal revision, and do the same for conditioning and lexicographic upgrade. Finally, we show that not all of our results still hold when learning from possibly erroneous information.

Figures (3)

• Figure 1: On the left: An epistemic space with $S=\{u,s,t\}$, $O=\{p,q\}$. On the right: a plausibility order on the same space with $t\prec u\prec s$ and the revised order $u\prec t\prec s$ after observing $p$ (represented by the $p$-labelled arrow). This is an example of (one-step) minimal revision (Def. \ref{['def:min_update']}). An arrow from one world to another indicates that the latter is more plausible than the former. We omit reflexive arrows.
• Figure 2: Example of three epistemic spaces that are finitely identifiable. On the left: $S=\{s,u,t\}$, $O=\{p,q,r\}$ . At the center: $S=\{s,u,t,w\}$, $O=\{p,\bar{p},q,\bar{q}\}$. On the right: $S=\{s_i|i\in N\}$, $O=\{p_i:i\in \mathbb{N}\}$, and for all $s_i$, $O_i=\{p_j\in O: j=i\textrm{ or }j=i+1\}$.
• Figure 3: Example of a plausibility order that is not appropriate to learn a space via mini. The epistemic space is given by $S=\{s,u,t\}$ and $O=\{p,q,r\}$; the prior plausibility order (leftmost space) is $t\prec s$, $t\prec u$, $u\simeq s$. The labelled arrows represent two revision steps, first when observing $q$, and then $p$.

Theorems & Definitions (56)

• Definition 1: baltag2019truth
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• Definition 10: baltag2019truth, originally in GOLD1967447