Spohn's Theory Research Articles

Causal compositional models in valuation-based systems with examples in specific theories

We show that Pearl's causal networks can be described using causal compositional models (CCMs) in the valuation-based systems (VBS) framework. One major advantage of using the VBS framework is that as VBS is a generalization of several uncertainty theories (e.g., probability theory, a version of possibility theory where combination is the product t -norm, Spohn's epistemic belief theory, and Dempster–Shafer belief function theory), CCMs, initially described in probability theory, are now described in all uncertainty calculi that fit in the VBS framework. We describe conditioning and interventions in CCMs. Another advantage of using CCMs in the VBS framework is that both conditioning and intervention can be easily described in an elegant and unifying algebraic way for the same CCM without having to do any graphical manipulations of the causal network. We describe how conditioning and intervention can be computed for a simple example with a hidden (unobservable) variable. Also, we illustrate the algebraic results using numerical examples in some of the specific uncertainty calculi mentioned above. • We show that Pearl's causal networks can be described using causal compositional models in the valuation-based systems. • We introduce a novel notion of dominance that is necessary to describe the composition of valuations. • We present an algebraic example of hidden variables elimination in the VBS framework. • We provide numerical examples in probability, possibility, and Spohn's theories.

Conditional independence and natural conditional functions

The concept of conditional independence (CI) within the framework of natural conditional functions (NCFs) is studied. An NCF is a function asribing natural numbers to possible states of the world; it is the central concept of Spohn's theory of deterministic epistemology. Basic properties of CI within this framework are recalled, and further results analogous to the results concerning probabilistic CI are proved. Firstly, the intersection of two CI-models is shown to be a CI-model. Using this, it is proved that CI-models for NCFs have no finite complete axiomatic characterization (by means of a simple deductive system describing relationships among CI-statements). The last part is devoted to the marginal problem for NCFs. It is shown that (pairwise) consonancy is equivalent to consistency iff the running intersection property holds.

On Spohn's rule for revision of beliefs

The main ingredients of Spohn's theory of epistemic beliefs are (1) a functional representation of an epistemic state called a disbelief function and (2) a rule for revising this function in light of new information. The main contribution of this paper is as follows. First, we provide a new axiomatic definition of an epistemic state and study some of its properties. Second, we study some properties of an alternative functional representation of an epistemic state called a Spohnian belief function. Third, we state a rule for combining disbelief functions that is mathematically equivalent to Spohn's belief revision rule. Whereas Spohn's rule is defined in terms of the initial epistemic state and some features of the final epistemic state, the rule of combination is defined in terms of the initial epistemic state and the incremental epistemic state representing the information gained. Fourth, we state a rule of subtraction that allows one to recover the addendum epistemic state from the initial and final epistemic states. Fifth, we study some properties of our rule of combination. One distinct advantage of our rule of combination is that besides belief revision, it can be used to describe an initial epistemic state for many variables when this information is given as several independent epistemic states each involving few variables. Another advantage of our reformulation is that we can show that Spohn's theory of epistemic beliefs shares the essential abstract features of probability theory and the Dempster-Shafer theory of belief functions. One implication of this is that we have a ready-made algorithm for propagating disbelief functions using only local computation.