I’ve been digging into the math behind LLM behavior recently, and one conclusion that keeps coming up is:

hallucination isn’t just a bug — it’s a consequence of the objective function.

At a high level, LLMs are trained to model:

P(x_t | x_<t)

using maximum likelihood. That means:

they optimize for probability , not truth

the learned distribution reflects the training data (which is incomplete + inconsistent)

softmax forces a normalized distribution → the model must always pick something

So when the model is uncertain, it doesn’t abstain — it still generates a high-probability continuation, which can look confident but be wrong.

From a more formal angle, hallucination can be seen as a combination of:

distribution approximation error (P_theta ≠ P*)

information loss (finite model capacity vs dataset entropy)

ambiguity in language (multiple valid continuations)

objective mismatch (likelihood vs factual correctness)

Even with perfect optimization, these don’t fully go away.

I wrote up a math-first…