Autoregressive sequence log-likelihood

Decomposes sequence likelihood into next-token prediction terms that an autoregressive model can train and evaluate directly.

Log probability of the sequence x one through T equals the sum from t equals one through T of the log probability of token x t given all earlier tokens.

Score a whole sequence by adding how well the model predicted every observed token from its earlier context.

The summation is the outer reduction; logarithms convert a joint probability product into additive token evidence.

This is the training and evaluation factorization behind next-token prediction; causal masking prevents a term from accessing future tokens.