Abstract
There are several formalisms that enhance Bayesian networks by including relations amongst individuals as modeling primitives. For instance, Probabilistic Relational Models (PRMs) use diagrams and relational databases to represent repetitive Bayesian networks, while Relational Bayesian Networks (RBNs) employ first-order probability formulas with the same purpose. We examine the coherence checking problem for those formalisms; that is, the problem of guaranteeing that any grounding of a well-formed set of sentences does produce a valid Bayesian network. This is a novel version of de Finetti’s problem of coherence checking for probabilistic assessments. We show how to reduce the coherence checking problem in relational Bayesian networks to a validity problem in first-order logic augmented with a transitive closure operator and how to combine this logic-based approach with faster, but incomplete algorithms.
Highlights
Most statistical models are couched so as to guarantee that they specify a single probability measure
We show how to reduce the coherence checking problem in relational Bayesian networks to a validity problem in first-order logic augmented with a transitive closure operator and how to combine this logic-based approach with faster, but incomplete algorithms
Bayesian networks with relations and first-order formulas: more precisely, we introduce techniques that allow one to check whether a given relational Bayesian network, or a given probabilistic relational model is guaranteed to specify a probability distribution
Summary
Most statistical models are couched so as to guarantee that they specify a single probability measure. Note that as long as the graph consisting of parameterized random variables is acyclic, we know that every Bayesian network generated from the plate model is consistent. Using syntax that will be explained later (Section 2), one can describe Scenario 4 in Example 2 with the following RBN: burglary(x) = 0.001; alarm(x) = 0.9 * burglary(x) + 0.01 * (1-burglary(x)); calls(x) = NoisyOR { alarm(y) | y; neighbor(x,y) }; One problem that surfaces when we want to use an expressive formalism, such as RBNs or PRMs, is whether a particular model is guaranteed to always produce consistent Bayesian networks.
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