Construction of the Optimal Sequence in Greedy Search

When generating text one token at a time, a greedy algorithm aims to select the token y_i at step i that maximizes the log-probability of the entire sequence up to that point, log Pr(y_1...y_i | x). This optimization problem can be simplified to choosing the token that maximizes only the conditional log-probability of the current token, log Pr(y_i | x, y_{<i}). Why is this simplification mathematically valid for finding the best current token y_i?

The simplification of the greedy search objective relies on a specific mathematical derivation. Arrange the following steps to correctly represent the logical flow of this derivation, which shows how maximizing the log-probability of the entire sequence up to the current step (log Pr(y_1...y_i | x)) is equivalent to maximizing the log-probability of just the current token (log Pr(y_i | x, y_{<i})).

Explaining Greedy Search Optimization