Some information is factual (e.g., "Paris is in France"), whereas other information is probabilistic (e.g., "the coin flip will be a [Heads/T ails]."). We believe that good Language Models (LMs) should understand and reflect this nuance. Our work investigates this by testing if LMs' output probabilities are calibrated to their textual contexts. We define model "calibration" as the degree to which the output probabilities of candidate tokens are aligned with the relative likelihood that should be inferred from the given context. For example, if the context concerns two equally likely options (e.g., heads or tails for a fair coin), the output probabilities should reflect this. Likewise, context that concerns non-uniformly likely events (e.g., rolling a six with a die) should also be appropriately captured with proportionate output probabilities. We find that even in simple settings the best LMs (1) are poorly calibrated, and (2) have systematic biases (e.g., preferred colors and sensitivities to word orderings). For example, gpt-4o-mini often picks the first of two options presented in the prompt regardless of the options' implied likelihood, whereas Llama-3.1-8B Our other consistent finding is mode-collapse: Instruction-tuned models often over-allocate probability mass on a single option. These systematic biases introduce non-intuitive model behavior, making models harder for users to understand. We investigate the extent to which language model (LM) output probabilities are calibrated to the numeric content of their contexts. Figure 1: Models produce un-calibrated results. Inputting Examples 1 and 2 to gpt-4o different, uncalibrated behaviors arise in the model probabilities.

Prior knowledge about the imaging physics provides a mechanistic forward operator that plays an important role in image reconstruction, although myriad sources of possible errors in the operator could negatively impact the reconstruction solutions. In this work, we propose to embed the traditional mechanistic forward operator inside a neural function, and focus on modeling and correcting its unknown errors in an interpretable manner. This is achieved by a conditional generative model that transforms a given mechanistic operator with unknown errors, arising from a latent space of self-organizing clusters of potential sources of error generation. Once learned, the generative model can be used in place of a fixed forward operator in any traditional optimization-based reconstruction process where, together with the inverse solution, the error in prior mechanistic forward operator can be minimized and the potential source of error uncovered. We apply the presented method to the reconstruction of heart electrical potential from body surface potential. In controlled simulation experiments and in-vivo real data experiments, we demonstrate that the presented method allowed reduction of errors in the physics-based forward operator and thereby delivered inverse reconstruction of heart-surface potential with increased accuracy.