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In nano-domain logic circuits, errors generated are transient in nature and will arise due to the uncertainty or the unreliability of the computing element itself. This type of errors - which we refer to as dynamic errors - are to be distinguished from traditional faults and radiation related errors. Due to these highly likely dynamic errors, it is more appropriate to model nano-domain computing as probabilistic rather than deterministic. We propose a probabilistic error model based on Bayesian networks to estimate this expected output error probability, given dynamic error probabilities in each device since this estimate is crucial for nano-domain circuit designers to be able to compare and rank designs based on the expected output error. We estimate the overall output error probability by comparing the outputs of a dynamic error-encoded model with an ideal logic model. We prove that this probabilistic framework is a compact and minimal representation of the overall effect of dynamic errors in a circuit. We use both exact and approximate Bayesian inference schemes for propagation of probabilities. The exact inference shows better time performance than the state-of-the art by exploiting conditional independencies exhibited in the underlying probabilistic framework. However, exact inference is worst case NP-hard and can handle only small circuits. Hence, we use two approximate inference schemes for medium size benchmarks. We demonstrate the efficiency and accuracy of these approximate inference schemes by comparing estimated results with logic simulation results. We have performed our experiments on LGSynth'93 and ISCAS'85 benchmark circuits. We explore our probabilistic model to calculate: 1) error sensitivity of individual gates in a circuit; 2) compute overall exact error probabilities for small circuits; 3) compute approximate error probabilities for medium sized benchmarks using two stochastic sampling schemes; 4) compare and vet design with resp- - ect to dynamic errors; 5) characterize the input space for desired output characteristics by utilizing the unique backtracking capability of Bayesian networks (inverse problem); and 6) to apply selective redundancy to highly sensitive nodes for error tolerant designs.