Language Model Maps for Prompt-Response Distributions via Log-Likelihood Vectors

1. Problem Statement

As Large Language Models (LLMs) become increasingly diverse, there is a critical need for frameworks that can systematically compare their behaviors, particularly in conditional generation settings (i.e., $p(y|x)$, where $y$ is a response to a prompt $x$).

Existing methods for comparing LLMs face specific limitations:

• Model Internals: Comparing weights or activations is difficult across different architectures and representation spaces.
• Generated Texts: Methods based on embedding distances of generated outputs are susceptible to the randomness of the generation process and the variability of embedding spaces.
• Unconditional Log-Likelihoods: Previous work (e.g., Oyama et al., 2025b) used log-likelihood vectors based on unconditional distributions ($p(y)$). This fails to capture the specific input-output dependencies crucial for tasks like dialogue and question answering.

The paper addresses the gap in comparing conditional probability distributions of many models in a shared, interpretable space without relying on stochastic text generation.

2. Methodology

The authors propose a framework that represents language models as vectors in a Euclidean space based on their log-likelihoods over a predefined set of prompt-response pairs.

A. Conditional Log-Likelihood Vectors

For a dataset of $N$ prompt-response pairs $D = \{(x_s, y_s)\}_{s=1}^N$, a language model $p_i$ is represented by a vector $\ell_i \in \mathbb{R}^N$:
$$\ell_i = (\log p_i(y_1|x_1), \dots, \log p_i(y_N|x_N))^\top$$
To center the data, the mean log-likelihood is subtracted to create a centered vector $\xi_i$.

B. Theoretical Foundation: KL Divergence Approximation

The core theoretical contribution is showing that the squared Euclidean distance between these centered vectors approximates the Kullback-Leibler (KL) divergence between the conditional distributions of two models, averaged over the input distribution:
$$KL(p_i, p_j) \approx \frac{1}{2N} \| \xi_i - \xi_j \|^2$$
This allows the distance in the vector space to be interpreted as a measure of distributional difference.

C. PMI Vectors (Pointwise Mutual Information)

To isolate the dependency between the prompt and response from the unconditional distribution of the response, the authors introduce PMI vectors. This is defined as the difference between the conditional and unconditional log-likelihoods:
$$\Delta \ell_{is} = \log p_i(y_s|x_s) - \log p_i(y_s)$$
This vector emphasizes the strength of the relationship between input $x$ and output $y$, effectively removing the influence of the model's general text generation bias.

D. Prompt Shift and Additive Compositionality

The framework analyzes how models move in this vector space when the input prompt is modified (e.g., adding a Chain-of-Thought phrase or repeating the prompt). The authors investigate whether these shifts exhibit additive compositionality, similar to word embeddings (e.g., $v_{A+B} \approx v_A + v_B$).

3. Key Contributions

1. Conditional Model Maps: A new framework for mapping LLMs based on conditional log-likelihood vectors, where distances approximate KL divergence between conditional distributions.
2. Prompt Shift Analysis: Demonstration that prompt modifications induce systematic, predictable shifts in model positions within the vector space.
3. Additive Compositionality: Discovery that the effects of composite prompt operations (e.g., adding CoT + repeating prompt) are approximately additive in the log-likelihood space, enabling the prediction of composite prompt effects.
4. PMI Vector Integration: Introduction of PMI vectors to filter out unconditional distribution noise, showing they better reflect training-data-specific differences in certain contexts.

4. Experimental Results

The authors evaluated 552 publicly available LLMs (up to 70B parameters) using two datasets: Tulu-3 and Infinity-Instruct.

• Global Structure & Attributes:
  • The maps successfully cluster models by architecture family (e.g., Llama, Mistral, Qwen form distinct regions).
  • Spatial proximity correlates with performance (average scores on Open LLM Leaderboard v2).
• Prompt Shift Visualization:
  • Visualizing models under different prompt conditions (Base, CoT, Repeat, Repeat+CoT) revealed systematic movements.
  • Additivity: The shift vector for "Repeat + CoT" was found to be approximately the sum of the "Repeat" and "CoT" shift vectors (Median relative prediction error $\approx 0.062$). This suggests a linear structure in how prompts affect model behavior.
• PMI vs. Conditional Maps:
  • Conditional Maps: Reflect general model behavior and performance.
  • PMI Maps: Better highlight training data provenance. Models trained on a specific dataset (e.g., Tulu-3) cluster tightly when evaluated on that same dataset in the PMI space, whereas this clustering weakens in the unconditional space.
• Empirical Validation:
  • KL Approximation: The squared Euclidean distance showed a strong correlation ($\rho = 0.94$) with Monte Carlo estimates of the true KL divergence.
  • Semantic Distance: The log-likelihood-based distances showed a positive association with embedding-based semantic distances but offered greater stability by avoiding generation randomness.

5. Significance and Implications

• Predictive Power: The additive compositionality of prompt shifts suggests that researchers can predict the behavioral impact of complex, composite prompt engineering strategies by summing the effects of individual operations, potentially predicting performance changes without running full inference.
• Training Data Analysis: The PMI-based approach provides a novel tool for analyzing how specific training data influences model behavior, distinguishing between general model capabilities and dataset-specific biases.
• Efficiency: The method allows for the comparison of hundreds of models without the computational cost and stochastic noise of generating thousands of text samples for each model.
• Unified Space: It bridges the gap between unconditional model analysis and practical conditional generation tasks, offering a unified geometric space to study LLM evolution and distillation.

6. Limitations

• Scope: Currently limited to publicly available weights; closed models cannot be analyzed.
• Data Bias: Experiments focus on instruction-style datasets; generalization to code, multilingual, or long-context tasks is unverified.
• PMI Approximation: The PMI calculation uses an approximation for marginal probability that may not perfectly match the true definition, and its effectiveness varies by dataset.
• Geometric Assumptions: The KL approximation assumes models are close to the data-generating distribution; significant deviations may reduce accuracy.
• Visualization: Low-dimensional projections (t-SNE/PCA) are used for visualization, which may distort high-dimensional geometric relationships.