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The output of a fine-tuned model, denoted as $$F_{\tilde{\omega},\tilde{\theta}}(\cdot)$$, for a new input sequence $$\mathbf{x}_{\mathrm{new}}$$ is a probability distribution over the predefined classes. For a polarity classification task, this output is a vector containing the conditional probabilities of the input being 'positive', 'negative', or 'neutral'. This is represented by the formula: $$F_{\tilde{\omega},\tilde{\theta}}(\mathbf{x}_{\mathrm{new}}) = \begin{bmatrix} \Pr(\mathrm{positive}|\mathbf{x}_{\mathrm{new}}) & \Pr(\mathrm{negative}|\mathbf{x}_{\mathrm{new}}) & \Pr(\mathrm{neutral}|\mathbf{x}_{\mathrm{new}})\end{bmatrix}$$.

Output Formula for a Polarity Classification Model

Based on the fundamental properties of a probability distribution for a classification task, which sentence's output is problematic, and why? Explain the mathematical property that is being violated.

Evaluating Model Output for Polarity Classification

A language model fine-tuned for a three-class polarity classification task (positive, negative, neutral) produces an output vector for a new input. The sum of the values in this output vector must equal 1.0.

A language model is fine-tuned for a sentiment analysis task with three possible labels: 'positive', 'negative', and 'neutral'. When given a new sentence, the model produces the following output vector, where each value corresponds to the probability of the respective label in the given order: `[0.15, 0.80, 0.05]`. How should this output be interpreted to determine the final classification?

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