We introduce a new class of graphical representations, expected utility networks (EUNs), and discuss some of its properties and potential applications to artificial intelligence and economic theory. In EUNs not only probabilities, but also utilities enjoy a modular representation. EUNs are undirected graphs with two types of arc, representing probability and utility dependencies respectively. The representation of utilities is based on a novel notion of conditional utility independence, which we introduce and discuss in the context of other existing proposals. Just as probabilistic inference involves the computation of conditional probabilities, strategic inference involves the computation of conditional expected utilities for alternative plans of action. We define a new notion of conditional expected utility (EU) independence, and show that in EUNs node separation with respect to the probability and utility subgraphs implies conditional EU independence.
We introduce a new interpretation of two related notions - conditional utility and utility independence. Unlike the traditional interpretation, the new interpretation renders the notions the direct analogues of their probabilistic counterparts. To capture these notions formally, we appeal to the notion of utility distribution, introduced in previous paper. We show that utility distributions, which have a structure that is identical to that of probability distributions, can be viewed as a special case of an additive multiattribute utility functions, and show how this special case permits us to capture the novel senses of conditional utility and utility independence. Finally, we present the notion of utility networks, which do for utilities what Bayesian networks do for probabilities. Specifically, utility networks exploit the new interpretation of conditional utility and utility independence to compactly represent a utility distribution.
Probabilistic models can define relationships between variables and be used to calculate probabilities. For example, fully conditional models may require an enormous amount of data to cover all possible cases, and probabilities may be intractable to calculate in practice. Simplifying assumptions such as the conditional independence of all random variables can be effective, such as in the case of Naive Bayes, although it is a drastically simplifying step. An alternative is to develop a model that preserves known conditional dependence between random variables and conditional independence in all other cases. Bayesian networks are a probabilistic graphical model that explicitly capture the known conditional dependence with directed edges in a graph model.
Naive Bayes Classifiers are probabilistic models that are used for the classification task. It is based on the Bayes theorem with an assumption of independence among predictors. In the real-world, the independence assumption may or may not be true, but still, Naive Bayes performs well. Naive It is called naive because it assumes that all features in the dataset are mutually independent. Bayes, It is based on Bayes Theorem.
Relevance-based explanation is a scheme in which partial assignments to Bayesian belief network variables are explanations (abductive conclusions). We allow variables to remain unassigned in explanations as long as they are irrelevant to the explanation, where irrelevance is defined in terms of statistical independence. When multiple-valued variables exist in the system, especially when subsets of values correspond to natural types of events, the over specification problem, alleviated by independence-based explanation, resurfaces. As a solution to that, as well as for addressing the question of explanation specificity, it is desirable to collapse such a subset of values into a single value on the fly. The equivalent method, which is adopted here, is to generalize the notion of assignments to allow disjunctive assignments. We proceed to define generalized independence based explanations as maximum posterior probability independence based generalized assignments (GIB-MAPs). GIB assignments are shown to have certain properties that ease the design of algorithms for computing GIB-MAPs. One such algorithm is discussed here, as well as suggestions for how other algorithms may be adapted to compute GIB-MAPs. GIB-MAP explanations still suffer from instability, a problem which may be addressed using ?approximate? conditional independence as a condition for irrelevance.