This is correct in the special case where all the information that your experts are basing their conclusions off of is independent. Slightly modifying your example, if the prior is 1:2 and you have n experts giving odds of 1:1, they each have a likelihood ratio of 2:1, so you get 1:2 * (2:1)^n = 2^(n-1):1. However, if they’ve all updated based on looking at the results from the same experiment, you’re double-counting the evidence; intuitively, you actually want to assign an odds ratio of 1:1.
The right thing to do here is to calculate for each subset of the experts what information they share, but you probably don’t have that information and so you’d have to estimate it, which I’d have to think a lot about in order to do well. Hopefully, the assumption of independence is approximately true in your data and you can just go with the naive method.
This is correct in the special case where all the information that your experts are basing their conclusions off of is independent. Slightly modifying your example, if the prior is 1:2 and you have n experts giving odds of 1:1, they each have a likelihood ratio of 2:1, so you get 1:2 * (2:1)^n = 2^(n-1):1. However, if they’ve all updated based on looking at the results from the same experiment, you’re double-counting the evidence; intuitively, you actually want to assign an odds ratio of 1:1.
The right thing to do here is to calculate for each subset of the experts what information they share, but you probably don’t have that information and so you’d have to estimate it, which I’d have to think a lot about in order to do well. Hopefully, the assumption of independence is approximately true in your data and you can just go with the naive method.