In this work, we analyze limited knowledge about likelihoods in traditional fusion as an uncertain Naïve Bayesian network whose conditional probabilities are known within posterior Dirichlet distributions reflected of limited training data. In prior work, we showed that inferences from these uncertain networks are confidently precise despite finite training data as the number of features goes to infinity for a uniform prior. The inference is usually correct except for pathological cases that diminish as the training data size goes to infinity. This work extends the analysis by studying how various priors affect asymptotic inference when the priors do or do not match the generative process for the conditional probabilities. Furthermore, the work analyzes how the prior affects the convergence rate of the inference.

We study the problem of Quickest Change Detection (QCD) in a complex networked system consisting of a set of heterogeneous agents that sequentially feed information to a central fusion center. At any unknown deterministic time, a persistent anomaly occurs, causing the distribution of observations from an unknown distinguishable subset of agents to simultaneously change from a nominal (pre-change) distribution to an anomalous (post-change) distribution, and the goal of the fusion center is to detect the change as quickly as possible subject to a false alarm constraint. Traditionally, various fusion rules have been proposed that assume that the distributions at each agent are either completely known or unknown and are locally solved using the Cumulative Sum (CuSum) and Generalized Likelihood Ratio (GLR) statistics, respectively. When an agent has access to training data, the Uncertain Likelihood Ratio (ULR) test generalizes distributional assumptions using uncertain distributions. However, the ULR has not been implemented for network change detection. This paper empirically studies incorporating the ULR statistics into the existing fusion rules for QCD and compares the average detection delay. Our results show that the ULR test can improve the average detection delay over the GLR tests using certain fusion techniques, while approaching the detection delay of the CuSum tests as the training data increases. Our results provide insights into future theoretical analysis to improve network QCD with imprecise knowledge of the distributions.

Likelihood fusion is a special case of Bayesian networks known as naïve Bayes. It is well known that as the number of observations goes to infinity with known likelihoods, the aleatoric uncertainty of the queried (or parent) variable goes to zero, and furthermore, the declared values are guaranteed to match the ground truth. This work considers the case that the conditional probabilities are learned with limited training data leading to uncertain likelihoods, and second-order probabilistic reasoning is incorporated to characterize the aleatoric and epistemic uncertainty. Remarkably, it is shown that both the aleatoric and epistemic uncertainty goes to zero despite limited knowledge of the likelihoods. The rate of convergence is dictated by a quasi-divergence value that is related to the Kullback-Liebler (KL) divergence. However, the quasi-divergence can be negative leading to false declarations. This paper investigates when false declarations can emerge and shows how such cases diminish as the amount of training data for the likelihoods increases.