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Benefiting from large-scale pretrained vision language models (VLMs), the performance of visual question answering (VQA) has approached human oracles. However, finetuning such models on limited data often suffers from overfitting and poor generalization issues, leading to a lack of model robustness. In this paper, we aim to improve input robustness from an information bottleneck perspective when adapting pretrained VLMs to the downstream VQA task. Input robustness refers to the ability of models to defend against visual and linguistic input variations, as well as shortcut learning involved in inputs. Generally, the representations obtained by pretrained VLMs inevitably contain irrelevant and redundant information for a specific downstream task, resulting in statistically spurious correlations and insensitivity to input variations. To encourage representations to converge to a minimal sufficient statistic in multimodal learning, we propose Correlation Information Bottleneck (CIB), which seeks a tradeoff between compression and redundancy in representations by minimizing the mutual information (MI) between inputs and representations while maximizing the MI between outputs and representations. Moreover, we derive a tight theoretical upper bound for the mutual information between multimodal inputs and representations, incorporating different internal correlations that guide models to learn more robust representations and facilitate modality alignment. Extensive experiments consistently demonstrate the effectiveness and superiority of the proposed CIB in terms of input robustness and accuracy.