The efficient estimation of missing information in causal inverted multiway trees

On Filling-in Missing Conditional Probabilities in Causal Networks

“…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.…”

“…Even assuming uniqueness however, the number of constraints of type (2) will in practice tend to be prohibitively large, rendering actual calculation of the missing values infeasible. A possible alternative might be to drop the constraints of type (2) and simply take the maximum entropy solution of the existing constraints of type (3). [In real, tractable, cases the m i would not be so large as cause a repeat of these same computational headaches.]…”

“…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.…”

“…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.…”