You might be right. It makes sense to me that P(X|A)=.5, P(X|B)=.5, independent(A,B) ⇒ P(X|A,B) = .75. But I can’t derive it.
Of course you can’t derive it—you gave a counterexample!
Jim gives a fallacy that it looks like people might actually commit (do they?), but I don’t think it’s what’s going on here. I think the issue is the meaning of “independent evidence.”
One kind of evidence is measurements. In that case the event of interest causes the measurement, plus there’s some noise. I think what we usually mean by “independent measurements” is that the noise for the one measurement is independent of the noise for the other measurement. How you combine the measurements depends on your noise model (as does even saying that the noises are independent). If your noise model is that there’s a large chance of a correct read and a small chance p of an incorrect read, then agreeing reads allow you to multiply the two p’s (if p is not small, what happens depends on the details of an incorrect read), which is roughly what you did, except that you confused the measurement of a probability with the probability of noise. You might be able to struggle through interpreting the measured p=.5 as a noise, but it would require a detailed noise model.
The lawn has opposite causal structure from the kinds of measurements/evidence above. Causal structure has to do with sides of |, so maybe when you unwind this, it turns into Jim’s fallacy, but I doubt it.
Of course you can’t derive it—you gave a counterexample!
Jim gives a fallacy that it looks like people might actually commit (do they?), but I don’t think it’s what’s going on here. I think the issue is the meaning of “independent evidence.”
One kind of evidence is measurements. In that case the event of interest causes the measurement, plus there’s some noise. I think what we usually mean by “independent measurements” is that the noise for the one measurement is independent of the noise for the other measurement. How you combine the measurements depends on your noise model (as does even saying that the noises are independent). If your noise model is that there’s a large chance of a correct read and a small chance p of an incorrect read, then agreeing reads allow you to multiply the two p’s (if p is not small, what happens depends on the details of an incorrect read), which is roughly what you did, except that you confused the measurement of a probability with the probability of noise. You might be able to struggle through interpreting the measured p=.5 as a noise, but it would require a detailed noise model.
The lawn has opposite causal structure from the kinds of measurements/evidence above. Causal structure has to do with sides of |, so maybe when you unwind this, it turns into Jim’s fallacy, but I doubt it.