Abstract
Elicitation, estimation and exact inference in Bayesian Networks (BNs) are often difficult because the dimension of each Conditional Probability Table (CPT) grows exponentially with the increase in the number of parent variables. The Noisy-MAX decomposition has been proposed to break down a large CPT into several smaller CPTs exploiting the assumption of causal independence, i.e., absence of causal interaction among parent variables. In this way, the number of conditional probabilities to be elicited or estimated and the computational burden of the joint tree algorithm for exact inference are reduced. Unfortunately, the Noisy-MAX decomposition is suited to graded variables only, i.e., ordinal variables with the lowest state as reference, but real-world applications of BNs may also involve a number of non-graded variables, like the ones with reference state in the middle of the sample space (double-graded variables) and with two or more unordered non-reference states (multi-valued nominal variables). In this paper, we propose the causal independence decomposition, which includes the Noisy-MAX and two generalizations suited to double-graded and multi-valued nominal variables. While the general definition of BN implicitly assumes the presence of all the possible causal interactions, our proposal is based on causal independence, and causal interaction is a feature that can be added upon need. The impact of our proposal is investigated on a published BN for the diagnosis of acute cardiopulmonary diseases.
Highlights
Bayesian Networks (BNs, Pearl, 1988) provide a formal framework to represent uncertain knowledge and to reason under uncertainty
We focus on the number of free parameters defining the Conditional Probability Table (CPT) in a BN, which determines the efficiency of elicitation and estimation, and on the size of each CPT, which determines the efficiency of exact inference
We have proposed an extension of the Noisy-MAX decomposition to non-graded variables, called Causal Independence Decomposition (CID)
Summary
Bayesian Networks (BNs, Pearl, 1988) provide a formal framework to represent uncertain knowledge and to reason under uncertainty. In the Noisy-OR, dichotomous parent variables are assumed to influence the value of a dichotomous response through independent latent causes. In the Noisy-MAX, latent causes have the same sample space of the response and each one may ‘activate’ one non-reference state of the response, which in turn takes value on the highest among the ‘active’ states. The Noisy-MAX decomposition simplifies elicitation from domain experts and estimation from collected data because, due to the assumption of causal independence, the number of free parameters is linear in the number of parent variables, instead of exponential. We propose the causal independence decomposition for BNs, which includes the Noisy-MAX and two generalizations suited to double-graded and multi-valued nominal variables.
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