Large language models (LLMs) increasingly produce natural language explanations, yet these explanations often lack faithfulness, and they do not reliably reflect the evidence the model uses to decide. We introduce FaithLM, a model-agnostic framework that evaluates and improves the faithfulness of LLM explanations without token masking or task-specific heuristics. FaithLM formalizes explanation faithfulness as an intervention property: a faithful explanation should yield a prediction shift when its content is contradicted. Theoretical analysis shows that the resulting contrary-hint score is a sound and discriminative estimator of faithfulness. Building on this principle, FaithLM iteratively refines both the elicitation prompt and the explanation to maximize the measured score. Experiments on three multi-domain datasets and multiple LLM backbones demonstrate that FaithLM consistently increases faithfulness and produces explanations more aligned with human rationales than strong self-explanation baselines. These findings highlight that intervention-based evaluation, coupled with iterative optimization, provides a principled route toward faithful and reliable LLM explanations.

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