Often an interesting true value such as a stock price, sports score, or current temperature is only available via the observations of noisy and potentially conflicting sources. Several techniques have been proposed to reconcile these conflicts by computing a weighted consensus based on source reliabilities, but these techniques focus on static values. When the real-world entity evolves over time, the noisy sources can delay, or even miss, reporting some of the real-world updates. This temporal aspect introduces two key challenges for consensus-based approaches: (i) due to delays, the mapping between a source's noisy observation and the real-world update it observes is unknown, and (ii) missed updates may translate to missing values for the consensus problem, even if the mapping is known. To overcome these challenges, we propose a formal approach that models the history of updates of the real-world entity as a hidden semi-Markovian process (HSMM). The noisy sources are modeled as observations of the hidden state, but the mapping between a hidden state (i.e. real-world update) and the observation (i.e. source value) is unknown. We propose algorithms based on Gibbs Sampling and EM to jointly infer both the history of real-world updates as well as the unknown mapping between them and the source values. We demonstrate using experiments on real-world datasets how our history-based techniques improve upon history-agnostic consensus-based approaches.
