Last time we took Brier distance beyond two dimensions. We showed that it’s “proper” in any finite number of dimensions. Today we’ll show that Euclidean distance is “improper” in any finite number dimensions.
When I first sat down to write this post, I had in mind a straightforward generalization of our previous result for Euclidean distance in two dimensions. And I figured it would be easy to prove.
Not so....
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Last time we saw why accuracy-mavens prefer Brier distance to Euclidean distance. But we did everything in two dimensions. That’s fine for a coin toss, with only two possibilities. But what if there are three doors and one of them has a prize behind it??
Don’t panic! Today we’re going to verify that Brier distance is still a proper way of measuring inaccuracy, even when there are more than two possibilities....
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Last time we saw that Euclidean distance is an “unstable” way of measuring inaccuracy. Given one assignment of probabilities, you’ll expect some other assignment to be more accurate (unless the first assignment is either perfectly certain or perfectly uncertain).
That’s why accuraticians don’t use good ol’ Euclidean distance.
Instead they use… well, there are lots of alternatives. But the closest thing to a standard one is Brier distance: the square of Euclidean distance....
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If you’ve bumped into the accuracy framework before, you’ve probably seen a diagram like this one:
The vertices $(1,0)$ and $(0,1)$ represent two possibilities, whether a coin lands heads or tails in this example.
According to the laws of probability, the probability of heads and of tails must add up to $1$, like $.3 + .7$ or $.5 + .5$. So the diagonal line connecting the two vertices covers all the possible probability assignments… $(0,1)$, $(....
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