Comment by inigyou

No, they really do one at a time. You're incorrect on that.

Mathematically, a long chain of conditional probabilities is equivalent to a single probability over the whole range. But computationally, for that to work out, the computation for the first probability needs to somehow consider all the downstream probabilities depending on it, which obviously isn't how autoregressive language models work. They can pack in as much downstream computation as their neural architecture allows for, which is quite a lot.

Suppose in some context you have three equally plausible conpletions after "Be": "tween a rock and a hard place", "twixed he stood there" and "lieve he can fly". To model this probability distribution of the whole sentence, the next token "tw" needs to appear at 2/3 probability and "lie" at 1/3. After "tw" would be a 1/2 chance of "ix" and a 1/2 chance of "een"; after "lie" would be a 100% chance of "ve " and in any case the rest of the sentence after that would be 100%.

The model needs to somehow "think ahead" to know those are the possible completions. For example if "lieve he can swim like a dolphin" was another equally plausible completion, that first token would need to be 50/50 instead of 67/33. So the computation of the first token somehow needs to encode the fact that the guy thinks he can fly but not swim, even though it doesn't become relevant in the output until several tokens later.

In practice this probably happens to some degree but definitely doesn't happen perfectly. To perfectly model the first token's probability distribution, it would have to include knowledge of the entire distribution of all possible outputs, which is just not happening. So it approximates. Surprisingly, the approximation is good enough to produce language.

You can see this breaking down in the seahorse emoji incident from last year. When you ask the model if there's a seahorse emoji, it first completes "Yes," as if a few tokens later it's about to produce a seahorse emoji. But when it actually gets to the token that would produce a seahorse emoji, it can't because there isn't one. But it's already outputted "Yes, the seahorse emoji is" and can't just go back and change that to "No, there's no seahorse emoji." Some models would try a few times and then say there isn't one or a system error seems to be making them unable to produce one, other models (including then-current ChatGPT) would loop forever with ensuing hilarity.