Um, let’s see if I get this (thinking to myself but posting here if anyone happens to find this useful—or even intelligible)...
claiming you know about X without X affecting you, you affecting X, or X and your belief having a common cause, violates the Markov condition on causal graphs
The causal Markov condition is that a phenomenon is independent of its noneffects, given its direct causes. It is equivalent to the ordinary Markov condition for Bayesian nets (any node in a network is conditionally independent of its nondescendents, given its parents) when the structure of a Bayesian network accurately depicts causality.
So, this condition induces certain (conditional) independencies between nodes in a causal graph (that can be found using the D-separation trick), and when we find two such nodes, they must also be uncorrelated (this follows from probabilistic independence being a stronger property than uncorrelatedness).
If one therefore claims there’s a persistent correlation between X and belief about X, this means there’s got to be some active path in Bayesian network for probabilistic influence to flow between them—otherwise, X and Belief(X) would be D-separated and thereby independent and uncorrelated. Insisting there’s no such path (e.g. no chain of directed links) leads to violation of Markov condition, since it maintains there’s probabilistic dependence between two nodes in a graph that cannot be accounted for by the causal links currently in the graph.
Um, let’s see if I get this (thinking to myself but posting here if anyone happens to find this useful—or even intelligible)...
The causal Markov condition is that a phenomenon is independent of its noneffects, given its direct causes. It is equivalent to the ordinary Markov condition for Bayesian nets (any node in a network is conditionally independent of its nondescendents, given its parents) when the structure of a Bayesian network accurately depicts causality.
So, this condition induces certain (conditional) independencies between nodes in a causal graph (that can be found using the D-separation trick), and when we find two such nodes, they must also be uncorrelated (this follows from probabilistic independence being a stronger property than uncorrelatedness).
If one therefore claims there’s a persistent correlation between X and belief about X, this means there’s got to be some active path in Bayesian network for probabilistic influence to flow between them—otherwise, X and Belief(X) would be D-separated and thereby independent and uncorrelated. Insisting there’s no such path (e.g. no chain of directed links) leads to violation of Markov condition, since it maintains there’s probabilistic dependence between two nodes in a graph that cannot be accounted for by the causal links currently in the graph.