Wait no, I’m stupid. What you do corresponds to argminRKL(P||R)+KL(Q||R), which is more like the union of hypotheses. You’d have to do something like argmaxQGh∼P[Gw∼Q[h(w)]] to get the intersection, I think. I should probably think through the math again more carefully when I have more time.
Note that this is just the arithmetic mean of the probability distributions. Which is indeed what you want if you believe that P is right with probability 50% and Q is right with probability 50%, and I agree that this is what Scott does.
At the same time, I wonder—is there some sort of frame on the problem that makes logarithmic pooling sensible? Perhaps (inspired by the earlier post on Nash bargaining) something like a “bargain” between the two hypotheses, where a hypothesis’ “utility” for an outcome is the probability that the hypothesis assigns to it.
The place where I came up with it was in thinking about models that focus on independent dynamics and might even have different ontologies. For instance, maybe to set environmental policy, you want to combine climate models with economics models. The intersection expression seemed like a plausible method for that. Though I didn’t look into it in detail.
Wait no, I’m stupid. What you do corresponds to argminRKL(P||R)+KL(Q||R), which is more like the union of hypotheses. You’d have to do something like argmaxQGh∼P[Gw∼Q[h(w)]] to get the intersection, I think. I should probably think through the math again more carefully when I have more time.
Note that this is just the arithmetic mean of the probability distributions. Which is indeed what you want if you believe that P is right with probability 50% and Q is right with probability 50%, and I agree that this is what Scott does.
At the same time, I wonder—is there some sort of frame on the problem that makes logarithmic pooling sensible? Perhaps (inspired by the earlier post on Nash bargaining) something like a “bargain” between the two hypotheses, where a hypothesis’ “utility” for an outcome is the probability that the hypothesis assigns to it.
The place where I came up with it was in thinking about models that focus on independent dynamics and might even have different ontologies. For instance, maybe to set environmental policy, you want to combine climate models with economics models. The intersection expression seemed like a plausible method for that. Though I didn’t look into it in detail.