I disagree with the hearsay conclusion that you should update toward the truth of Z.
My first problem is that it’s a straw objection. The actual objection is that while X can be further questioned to inspect in detail whether their testimony is credible, Y cannot. This immensely weakens the link between X’s testimony and the truth of Z, and admitting such evidence would open up bad incentives outside the courtroom as well.
The next problem is that considering the chance that X and Y are truthful is only part of a Bayesian update procedure. If you have a strong prior of Y’s testimony being reliable but not that X’s is[1], you should update away from the truth of Z. If both X and Y were correct in their statements, then Y would be a much stronger witness and should have been called by the lawyer. Now you have evidence that Y’s testimony would have harmed the case for Z. It is straightforward but tedious to work through a Bayesian update for this. For example:
Suppose priors are P(X’s testimony is truthful) = 1⁄2, P(Y made a true statement about Z) = 9⁄10, independent of each other and Z. Let E be the event “the lawyer only calls X to give the testimony that Y said Z”. This event is incompatible with XYZ, since Y should have been called. It is also incompatible with XYZ’, XY’Z, and X’YZ since in these cases X would not testify that Y said Z (where primes ′ are used to indicate negation). So
If P(E|X’YZ’) >= 1⁄9, then P(Z’|E) > P(Z|E) and you should update away from Z being true. That is, suppose Z was actually false, Y correctly said that it was false, and X is not truthful in their testimony. What is the probability that the lawyer only calls X to testify that Y said Z? It seems to me quite a lot greater than 1⁄9. The lawyer has incentive to call X, who will testify in support of Z for their client. They will probably attempt to contact Y, but Y will very likely not testify in support of Z. There are possibilities for P(E’|X’YZ’), but they seem much less likely.
This is interesting, because it seems that you’ve proved the validity of the “Strong Adversarial Argument”, at least in a situation where we can say:
This event is incompatible with XYZ, since Y should have been called.
In other words, we can use the Adversarial Argument (in a normal Bayesian way, not as an acausal negotiation tactic) when we’re in a setting where the rule against hearsay is enforced. But what reason could we have had for adopting that rule in the first place? It could not have been because of the reasoning you’ve laid out here, which presupposes that the rule is already in force! The rule is epistemically self-fulfilling, but its initial justification would have seemed epistemically “irrational”.
So, why do we apply it in a courtroom setting but not in ordinary conversation? In short, because the stakes are higher and there’s a strong positive incentive to deceive.
This calculation just used the fact that Y would have been able to give stronger testimony than X, and that lawyers have incentives to present a strong case for their client where possible. In this scenario, the fact that Y was not called is evidence that Y’s testimony would have weakened the case for Z.
The actual objection against hearsay has nothing to do with this calculation at all, as I mentioned in my comment.
You can apply it in ordinary conversation too (to the extent that you apply Bayesian updates in ordinary conversation at all). It’s just that the update is stronger when the equivalent of E|XYZ is more unlikely, and in ordinary conversation it may not be very unlikely resulting in a weaker update.
I disagree with the hearsay conclusion that you should update toward the truth of Z.
My first problem is that it’s a straw objection. The actual objection is that while X can be further questioned to inspect in detail whether their testimony is credible, Y cannot. This immensely weakens the link between X’s testimony and the truth of Z, and admitting such evidence would open up bad incentives outside the courtroom as well.
The next problem is that considering the chance that X and Y are truthful is only part of a Bayesian update procedure. If you have a strong prior of Y’s testimony being reliable
but not that X’s is[1], you should update away from the truth of Z. If both X and Y were correct in their statements, then Y would be a much stronger witness and should have been called by the lawyer. Now you have evidence that Y’s testimony would have harmed the case for Z. It is straightforward but tedious to work through a Bayesian update for this. For example:Suppose priors are P(X’s testimony is truthful) = 1⁄2, P(Y made a true statement about Z) = 9⁄10, independent of each other and Z. Let E be the event “the lawyer only calls X to give the testimony that Y said Z”. This event is incompatible with XYZ, since Y should have been called. It is also incompatible with XYZ’, XY’Z, and X’YZ since in these cases X would not testify that Y said Z (where primes ′ are used to indicate negation). So
P(ZE) = P(X’Y’ZE) = P(E | X’Y’Z) P(X’Y’Z) < P(X’Y’Z) = 1⁄40,
P(Z’E) = P(XY’Z’E) + P(X’YZ’E) + P(X’Y’Z’E) > P(E|X’YZ’) P(X’YZ’) = P(E|X’YZ’) 9⁄40.
If P(E|X’YZ’) >= 1⁄9, then P(Z’|E) > P(Z|E) and you should update away from Z being true. That is, suppose Z was actually false, Y correctly said that it was false, and X is not truthful in their testimony. What is the probability that the lawyer only calls X to testify that Y said Z? It seems to me quite a lot greater than 1⁄9. The lawyer has incentive to call X, who will testify in support of Z for their client. They will probably attempt to contact Y, but Y will very likely not testify in support of Z. There are possibilities for P(E’|X’YZ’), but they seem much less likely.
So in this scenario you should update against Z.
It turns out that this part is not necessary. Almost all of the evidential weight comes from credibility of Y.
This is interesting, because it seems that you’ve proved the validity of the “Strong Adversarial Argument”, at least in a situation where we can say:
In other words, we can use the Adversarial Argument (in a normal Bayesian way, not as an acausal negotiation tactic) when we’re in a setting where the rule against hearsay is enforced. But what reason could we have had for adopting that rule in the first place? It could not have been because of the reasoning you’ve laid out here, which presupposes that the rule is already in force! The rule is epistemically self-fulfilling, but its initial justification would have seemed epistemically “irrational”.
So, why do we apply it in a courtroom setting but not in ordinary conversation? In short, because the stakes are higher and there’s a strong positive incentive to deceive.
This calculation just used the fact that Y would have been able to give stronger testimony than X, and that lawyers have incentives to present a strong case for their client where possible. In this scenario, the fact that Y was not called is evidence that Y’s testimony would have weakened the case for Z.
The actual objection against hearsay has nothing to do with this calculation at all, as I mentioned in my comment.
You can apply it in ordinary conversation too (to the extent that you apply Bayesian updates in ordinary conversation at all). It’s just that the update is stronger when the equivalent of E|XYZ is more unlikely, and in ordinary conversation it may not be very unlikely resulting in a weaker update.