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An alternative approach to model combination involves averaging predictions at the token level. Instead of combining fully generated outputs, this method calculates the probability of each potential next token at every step of the generation process by averaging the probability distributions from multiple models or prompts.

Model Averaging for Token-Level Prediction

In prompt ensembling, token-level model averaging determines the predicted token $$\hat{y}_j$$ at the $$j$$-th step of the model combination. The token is selected by maximizing the sum of log-probabilities across all $$K$$ prompts. This decision rule is expressed by the formula: $$\hat{y}_{j} = \arg\max_{y_j} \sum_{k=1}^{K} \log \Pr(y_j|\mathbf{x}_k, \hat{y}_1, ..., \hat{y}_{j-1})$$. Here, the probability of predicting the token $$y_j$$ is conditioned on the $$k$$-th prompt's input $$\mathbf{x}_k$$ and all previously generated tokens $$\hat{y}_1$$ through $$\hat{y}_{j-1}$$.

Formula for Token-Level Model Averaging in Prompt Ensembling

A team is developing a system to generate a single, high-quality sentence by leveraging two different language models. They are considering two approaches:

*   **Approach 1:** Generate a complete sentence from Model A and a complete sentence from Model B. Then, analyze these two final sentences to create a single, blended output.
*   **Approach 2:** To decide on the very first word, get the probability scores for all possible words from both Model A and Model B, average these scores, and pick the word with the highest average. Repeat this step-by-step process for each subsequent word until a full sentence is formed.

Based on these descriptions, what is the fundamental difference in *what* is being combined and *when* the combination occurs in Approach 2 compared to Approach 1?

Analysis of Text Generation Combination Methods

Based on the scenario provided, which method (A or B) directly implements the principle of averaging predictions at the token level? Justify your choice by explaining the fundamental difference in how the two methods combine model outputs to produce the final text.

Choosing a Generation Combination Strategy

Imagine two language models are tasked with completing the sentence: 'The weather today is exceptionally...'. At this specific step, they must choose the very next word. Their internal calculations produce the following probability scores for the top three candidate words:

*   **Model 1:** `warm` (0.6), `sunny` (0.3), `bright` (0.1)
*   **Model 2:** `warm` (0.2), `sunny` (0.7), `bright` (0.1)

If a system combines these models by averaging their token-level probability distributions to make a d

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