Log-scoring really means you’re just multiplying together the probabilities you assigned to every thing that did turn out true.
If every time you say an event has probability p, and it happens, you assign yourself log p points, you might as well just take all the p’s you’re taking logs of and multiply them together first, and then take the log after.
The logarithm is what creates addition (which is how you accumulate points in any points-based system) from multiplication (the thing probabilities want to do with each other).
I guess the reason to take the actual logarithm at the end, rather than just keeping the product of p’s, is to make the expected scores incentivize accurate predicted probabilities. But you don’t need that if all you want to do is compare who performed better at predicting a series of truths.
Log-scoring really means you’re just multiplying together the probabilities you assigned to every thing that did turn out true.
If every time you say an event has probability p, and it happens, you assign yourself log p points, you might as well just take all the p’s you’re taking logs of and multiply them together first, and then take the log after.
The logarithm is what creates addition (which is how you accumulate points in any points-based system) from multiplication (the thing probabilities want to do with each other).
I guess the reason to take the actual logarithm at the end, rather than just keeping the product of p’s, is to make the expected scores incentivize accurate predicted probabilities. But you don’t need that if all you want to do is compare who performed better at predicting a series of truths.