Yes, I’m aware of the Bernoulli model—my point is that the vector →Y is itself the outcome of that experiment; I suppose you can call it Z though it makes the notation a bit confusing. The general point is that yes, you update your belief about X based on a series of outcomes on Y. In fact I think in general γ′(x)=P[X=x|→Y] works just fine.
Yes, I’m aware of the Bernoulli model—my point is that the vector →Y is itself the outcome of that experiment; I suppose you can call it Z though it makes the notation a bit confusing. The general point is that yes, you update your belief about X based on a series of outcomes on Y. In fact I think in general γ′(x)=P[X=x|→Y] works just fine.