Independent opinions? On the causal foundations of belief formation and jury theorems

It is often claimed that opinions are more likely to be correct if they are held independently by many individuals. But what does it mean to hold independent opinions? To clarify this condition, we distinguish four notions of probabilistic opinion independence. Which notion applies depends on environmental factors such as commonly perceived evidence. More formally, it depends on the causal network that determines how people interact and form their opinions. In a general theorem, we identify conditions on this network that guarantee the four notions of opinion independence. Our results have implications for ‘wisdom of crowds’ arguments, as we illustrate with old and new jury theorems.