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Notes -
Chatbot Arena is awesome; what are the usage limits there?
I tried my applied math questions out on Claude 3 Opus; unlike Sonnet, it didn't make any sign errors on the easier of the two questions. It did miss an important part of the answer on the harder question ... but honestly, this is a question for which I'd pull out a reference rather than rederive the whole thing from scratch in my head, so I think my only complaint here is the overconfidence. It's not nearly as bad in that regard as Sonnet was (arguing with me for a prompt or two before admitting its sign error), but in cases where there's any grey area I'd still vastly prefer answers of the form "I think it's X, but I might be missing something" over "It's definitely X, no doubt about it!" where only 90% are actually correct.
In hindsight this should have been an obvious problem with training LLMs on published text, huh? "I dunno" and "I'm not sure but" and "I think maybe" are the domain of ephemeral chat logs; by the time you're ready to publish, even to a serious web page, you've hopefully figured out the right answer and you don't waste your audience's time with the missteps along the way ... which means that a language model trained on what you've published doesn't have nearly as much "experience" with what to do when there's a chance of a misstep.
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