CONCLUSION

The central contribution of this paper is the application of Walley's definitions of irrelevance and independence to the study of locally defined Quasi-Bayesian networks. The main technical contributions are novel algorithms for inference with natural extensions; research must now be conducted to limit the combinatorial explosion that occurs in the formulation of linear fractional programs for inferences with natural extensions. The paper also ties type-1 extensions to d-separation; this result provides a formal basis for the conceptual and computational attractiveness of type-1 extensions.

This paper focused on the calculation of upper bounds for the posterior probability of the event $\{ X_q = X_{qj} \}$. Other problems can be solved using the same algorithms. For example, calculation of inferences for non-atomic events A is immediate by enlarging the summations that must be computed in the inference procedures. Algorithms presented in this paper also apply to calculation of lower and upper expectation, by enlarging summations and objective functions in linear programs.

The results presented in this paper pose an intellectually challenging question: Should we consider irrelevance or independence as a basic notion in the treatment of uncertainty? Both notions agree in standard probability theory, but they disagree in Quasi-Bayesian theory. Irrelevance is a more basic notion, as it can be used to define independence, and irrelevance judgements are less forceful than independence ones but still quite powerful. Should irrelevance be a more fundamental notion? This question can only be answered as research and applications are developed using Quasi-Bayesian models.