While this setup has enjoyed a lot of attention in the applied community, there hasn't be theoretical work that even formalizes the desired guarantees.

Let k, d, r2 N, r k, d, 2 (0,1/2). Let d, k, r2 N, r d, k, >0.

Large vision-language models (L VLMs) achieve strong performance on multi-modal tasks, yet they often default to their language prior (LP)--memorized textual patterns from pre-training while under-utilizing visual evidence. Prior analyses of LP mostly rely on input-output probing, which fails to reveal the internal mechanisms governing when and how vision influences model behavior. To address this gap, we present the first systematic analysis of language prior through the lens of chain-of-embedding, which examines the layer-wise representation dynamics within L VLMs. Our analysis reveals a universal phenomenon: each model exhibits a Visual Integration Point (VIP), a critical layer at which visual information begins to meaningfully reshape hidden representations and influence decoding. Building on this observation, we introduce the T otal Visual Integration (TVI) estimator, which aggregates representation distance beyond the VIP to quantify how strongly visual query influences response generation. Across 54 model-dataset combinations spanning 9 contemporary L VLMs and 6 benchmarks, we demonstrate that VIP consistently emerges, and that TVI reliably predicts the strength of language prior. Modern large vision-language models (L VLMs) (OpenAI, 2025; Comanici et al., 2025; Bai et al., 2025; Zhu et al., 2025) have extended the boundaries of AI applications at an unprecedented rate. Their remarkable capability in solving highly complex vision-language tasks originated from the internalized rich unimodal knowledge during the pre-training (Radford et al., 2021; Oquab et al., 2024; Brown et al., 2020) and also from the strong multimodal alignment (Liu et al., 2023; Dai et al., 2023; Zhu et al., 2024). Despite their successes, a central challenge remains: L VLMs are prone to over-relying on their language prior (LP)--the statistical patterns memorized during large-scale language model pretraining--while under-utilizing the actual visual evidence (Fu et al., 2024; Lee et al., 2025; Luo et al., 2025). This imbalance often results in hallucinations, shortcut reasoning, and brittle generalization when tasks truly demand visual grounding.

While MDPs are powerful tools for modeling sequential decision making problems under uncertainty, they are sensitive to the accuracy of their parameters. MDPs with uncertainty in their parameters are called Uncertain MDPs. In this paper, we introduce a general framework that allows off-the-shelf MDP algorithms to solve Uncertain MDPs by planning based on currently available information and replan if and when the problem changes. We demonstrate the generality of this approach by showing that it can use the VI, TVI, ILAO*, LRTDP, and UCT algorithms to solve Uncertain MDPs. We experimentally show that our approach is typically faster than replanning from scratch and we also provide a way to estimate the amount of speedup based on the amount of information being reused.