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# Incorporating Uncertain Evidence Into Arithmetic Circuits Representing Probability Distributions

*Proceedings of The 3rd International Workshop on Advanced Methodologies for Bayesian Networks*, PMLR 73:105-116, 2017.

#### Abstract

Arithmetic circuits have been used as tractable representations of probability distributions, either generated from models such as Bayesian networks, sum-product networks and Probability Sentential Decision Diagrams, or directly from data. An interesting question is how we can incorporate uncertain evidence, which specifies that the marginal probabilities of a variable has to undergo certain changes, directly into an arithmetic circuit and then perform reasoning on it to compute the probability distribution after incorporating this uncertain evidence. In this paper, we show that we can incorporate uncertain evidence on a variable by setting indicators of this variable in the arithmetic circuit to non-negative values based on the likelihood ratios in Pearl's method of virtual evidence and the current marginal probabilities of this variable. For tractable computation of these marginal probabilities, the arithmetic circuit has to satisfy the properties of decomposability and smoothness, and we show that an algorithm using a downward pass can compute these marginal probabilities for all single variables. We show a procedure of how to incorporate virtual evidence, including multiple pieces of virtual evidence.