Maximum Likelihood Estimation for Sequential Data

In training a model on a dataset (D) of sequences (\mathbf{x}), a primary goal is to find parameters that maximize the total log-probability of the observed sequences. This objective can be expressed in two equivalent ways:

Form 1: $\arg \max_{\theta} \sum_{\mathbf{x} \in D} \log \Pr_{\theta}(\mathbf{x})$

Form 2: $\arg \max_{\theta} \sum_{\mathbf{x} \in D} \sum_{i=0}^{m-1} \log \Pr_{\theta}(x_{i+1} | x_0, ..., x_i)$

What fundamental principle of probability justifies the mathematical equivalence between Form 1 and Form 2?

Verifying Log-Probability Equivalence

Analysis of a Language Model Training Objective

The mathematical equivalence between maximizing the log-probability of an entire sequence, log Pr(x), and maximizing the sum of its conditional log-probabilities, Σ log Pr(x_i | x_<i), is established because the logarithm function transforms a sum of probabilities into a product of log-probabilities.