MLM Training Objective using Cross-Entropy Loss

In the context of training a language model, the objective is often to find parameters that maximize the likelihood of the training data. Consider the following mathematical expression for this objective:

Objective = ∑_{x ∈ D} ∑_{i ∈ A(x)} log Pr(xᵢ | x̄)

Here, D is the dataset, x is an original text sequence, x̄ is a version of x with some tokens masked, A(x) is the set of indices that were masked in x, and xᵢ is the original token at a masked position i.

What does the inner summation, ∑_{i ∈ A(x)} log Pr(xᵢ | x̄), represent in this training process?

Calculating Contribution to MLM Training Objective

A language model is being trained with the objective of maximizing the log-probability of the original tokens at masked positions. For the original sentence 'The fox jumps over the dog', the model is given the masked input 'The fox [MASK] over the dog'. Which of the following model predictions for the [MASK] token would contribute the most to achieving the training objective for this specific instance?

Example of Masked Language Modeling Loss Calculation