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Real World Evidence (RWE) and its uses are playing a growing role in medical research and inference. Prominently, the 21st Century Cures Act-approved in 2016 by the US Congress-permits the introduction of RWE for the purpose of risk-benefit assessments of medical interventions. However, appraising the quality of RWE and determining its inferential strength are, more often than not, thorny problems, because evidence production methodologies may suffer from multiple imperfections. The problem arises to aggregate multiple appraised imperfections and perform inference with RWE. In this article, we thus develop an evidence appraisal aggregation algorithm called EA3. Our algorithm employs the softmax function-a generalisation of the logistic function to multiple dimensions-which is popular in several fields: statistics, mathematical physics and artificial intelligence. We prove that EA3 has a number of desirable properties for appraising RWE and we show how the aggregated evidence appraisals computed by EA3 can support causal inferences based on RWE within a Bayesian decision making framework. We also discuss features and limitations of our approach and how to overcome some shortcomings. We conclude with a look ahead at the use of RWE., Competing Interests: The authors have declared that no competing interests exist.