[currymj]

This is actually a case that shows the limits of Bayesian thinking.

The power of probability is that it can work in two directions. You can use it to make predictions, from causes to effects, from past to future. Or you can use it to reason diagnostically, from effects to causes, like deducing what must have happened in the past to produce the current observation. Thinking probabilistically, these two cases are treated the same: they're both just conditioning on evidence, which is really elegant.

The problem is that when the two cases really need to be treated differently, probability can't distinguish between them. For example, asking about the probability of hypothetical situations, or predicting the results of interventions. You need to know which variables are causes and which are effects, but this is outside the scope of probability.

Simpson's paradox is something that only shows up when the variables involved have certain cause-effect structures. If you think in terms of these structures, it stops being counterintuitive.