Efficient computation of marginal probabilities in multivalued causal inverted multiway trees given incomplete information

Abstract

wŽ . x Maung and Paris Internat J Intell Syst 1990, 5 5 , 595᎐603 have shown that, in the general case, solving causal networks using maximum entropy techniques is NP complete. This paper considers multivalued causal inverted multiway trees, a nontrivial class of causal networks, in which any event can be influenced by any number of other events but itself only influences at most one event. We show that for this class of causal networks, maximum entropy can be used to find minimally prejudiced estimates for missing information. The techniques required for the current problem are substantially different from those used in the case of causal multiway trees in that nonlinear constraints arising from independence have to be incorporated. In addition, a new algebraic method is presented which isolates an unknown Lagrange multiplier by using the quotient of two pairs of state probabilities. Equating the joint probability distributions given by the Bayesian and maximum entropy models enables the Lagrange multipliers of the latter to be determined. An efficient iterative tree traversal algorithm which converges to the minimally prejudiced estimates for the missing information is described. When this information is added to that already provided, any existing method for updating the causal network can be used. ᮊ

On Filling-in Missing Conditional Probabilities in Causal Networks

“…This alternative, of treating all probability functions satisfying the constraints as equally likely to be the true probability function, is clearly very much in the flavor of Bayesian methods and immediately suggests approximating, or estimating, the true probability by taking the 'average' of all probability function 5 As a general 'principle' we have little sympathy for 'indifference' in general, unless, as in the Renaming Principle, see [12], it can be justified in terms of invariance under symmetries of the language. In neither this case, nor in the case of CM ∞ which we shall shortly be considering, are any such supporting arguments apparent.…”

“…In papers [1], [3], [5], [6], [21], Rhodes, Garside and Holmes described efficient algorithms for filling in missing conditional probabilities in various classes of causal networks by the maximum entropy method. Whilst their main interest, apparently, was the formulation of algorithms a key step in their methods is to first isolate comparatively simple, computationally manageable, subsets of the set of probabilistic independence constraints associate with such causal networks whose maximum entropy solution (hereafter shortened to maxent solution) is the unique solution of the full set of constraints.…”

“…This observation leads to the following corollary which appears in the work of Rhodes-Garside-Holmes (see specifically [3], [5]) under the implicit assumption of uniqueness.…”

“…This counter-example also deals a blow to the algorithms proposed in [3] and [5] for finding maxent solutions. In short the method is to first guess the unknowns to be 1/2 and then sequentially tune each of them in turn to the value which maximizes the entropy when all the others are fixed at their current values.…”

“…Our second criticism of using this justification for applying maxent in the context of causal networks is somewhat less formal and is aimed at what we believe to be the underlying assumption justifying 'indifference' across data sets 5 , namely that it is the data set that is primary and determines the distribution.…”