A Study on Generalising Bayesian Inference to Evidential Reasoning

In this paper the relationship between Bayes' rule and the Evidential Reasoning (ER) rule is explored. The ER rule has been uncovered recently for inference with multiple pieces of uncertain evidence profiled as a belief distribution and takes Dempster's rule in the evidence theory as a special case. After a brief introduction to the ER rule the conditions under which Bayes' rule becomes a special case of the ER rule are established. The main findings include that the normalisation of likelihoods in Bayesian paradigm results in the degrees of belief in the ER paradigm. This leads to ER-based probabilistic (likelihood) inference with evidence profiled in the same format of belief distribution. Numerical examples are examined to demonstrate the findings and their potential applications in probabilistic inference. It is also demonstrated that the findings enable the generalisation of Bayesian inference to evidential reasoning with inaccurate probability information with weight and reliability.