This is where elements of evidence (for example data or measurements) are gathered from entirely distinct sources, or where they refer to aspects of the same data source that are not in any way related. For example, we might perform a user study on a new piece of educational software in two schools in completely different cities. It is far easier to reason about independent evidence than it is when there is some form of connection between the evidence. For example, if the new software was tested on two classes in the same school, the second class might have heard about it from their friends in the first class and so their satisfaction scores might merely reflect the views of the first class.
This can be a particular problem in Bayesian statistics if the prior probability is based on multiple evidence, some of which have been influenced by a common source. It is possible to attempt to model this common factor, but that can be complex.
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