It’s not entirely clear what the “probability” numbers here actually are, and whether two columns are needed.
The simplest is just to use odds ratios. It doesn’t matter if the likelihoods you give are correct, so long as they are proportional to each other. It would look like follows:
Prior:
1:1
Evidence:
1:99
1:1
1:9
1:19
1:9
1:9
product:
1:1371249
So there’s 1371249 to one odds in favor of the defense.
There are a few problems with this:
First, the prior shouldn’t be 1:1. You’d probably use something about how likely it is for someone that close to commit the crime. If you really want to be rigorous, use 1:6840507000 as the prior and use the relation and all that as evidence, but that’s probably overkill.
Second, there are biases that will cause problems exponentially with evidence. You will tend to notice evidence more on one side, and you will tend to think evidence is more likely for one side than it really is. This would be hard to account for. The best I can figure is to keep track of an error term that grows exponentially with each step. You might end up with
(1/64 to 64):(1371249/64 to 1371249*64) = 1:(334.777588 to 5616635904)
From there, you could integrate in some way I can’t think of right now, and you’d probably get something on the order of 2000:1 in favor of the defendant.
The simplest is just to use odds ratios. It doesn’t matter if the likelihoods you give are correct, so long as they are proportional to each other. It would look like follows:
Prior: 1:1 Evidence:
1:99
1:1
1:9
1:19
1:9
1:9
product:
1:1371249
So there’s 1371249 to one odds in favor of the defense.
There are a few problems with this:
First, the prior shouldn’t be 1:1. You’d probably use something about how likely it is for someone that close to commit the crime. If you really want to be rigorous, use 1:6840507000 as the prior and use the relation and all that as evidence, but that’s probably overkill.
Second, there are biases that will cause problems exponentially with evidence. You will tend to notice evidence more on one side, and you will tend to think evidence is more likely for one side than it really is. This would be hard to account for. The best I can figure is to keep track of an error term that grows exponentially with each step. You might end up with
(1/64 to 64):(1371249/64 to 1371249*64) = 1:(334.777588 to 5616635904)
From there, you could integrate in some way I can’t think of right now, and you’d probably get something on the order of 2000:1 in favor of the defendant.