This is intended as a caution for your own thinking, not a Fully General Counterargument against conclusions you don’t like.
It’s a very lossy model to account for the arguer more than the arguments. It’s valid and important to remember that most fact aggregators are selective in what they say, with biases that don’t match the underlying truth (which they often don’t have access to either). But it’s not intended to be a formal literal “zero evidence”. It’s intended only as a reminder not to take the arguments at face value.
Note that in the example EY used, it IS literally zero evidence—the clever arguer doesn’t actually have any additional access to truth, or any evidence that the observer doesn’t already have. In this case, your inequality is false—there’s no difference between H and ~H in terms of probability of E.
The comment at the bottom there is important:
It’s a very lossy model to account for the arguer more than the arguments. It’s valid and important to remember that most fact aggregators are selective in what they say, with biases that don’t match the underlying truth (which they often don’t have access to either). But it’s not intended to be a formal literal “zero evidence”. It’s intended only as a reminder not to take the arguments at face value.
Note that in the example EY used, it IS literally zero evidence—the clever arguer doesn’t actually have any additional access to truth, or any evidence that the observer doesn’t already have. In this case, your inequality is false—there’s no difference between H and ~H in terms of probability of E.