# How does the logarithmic scoring rule work given that it's undefined for zero?

[I'm not a mathematician, so please forgive any misuse of terminology]

One way of understanding scoring rules is that they measure the 'distance' between the truth value of a statement, and the probability we assign to that statement. So, if 'it's raining' is true, that statement gets assigned 1. We find the difference between 1 and the probability we've assigned it. Let's suppose the difference is 0.3. We then plug that value into the scoring rule. If it's the Brier score, we'd square 0.3 to get 0.09. This is all well and good, but what happens if a) we'd assigned probability 1 to the true statement and b) we were using the logarithmic scoring rule. Since 1-1=0, and 0 is undefined for natural logarithms ... well, what does happen?

Relatedly, are there other scoring rules which have this feature? The only other scoring rules I'm familiar with are Brier and Spherical distance, neither of which do. But presumably there are other scoring rules besides these? (i.e., which aren't just linear transformations of them).

• The accuracy score would be $-\infty$ if you predicted a probability of 1.0 for an event that did not occur. This is not a convention; it's the math. There is no fix for that when you use "the" logarithmic scoring rule. There are many other rules available. Search for "proper scoring rule". Commented Jan 27, 2018 at 17:42