We present an information-theoretic framework for discrete diffusion models that yields principled estimators of log-likelihood using score-matching losses. Inspired by the I-MMSE identity for the Gaussian setup, we derive analogous results for the discrete setting. Specifically, we introduce the Information-Minimum Denoising Score Entropy (I-MDSE) relation, which links mutual information between data and its diffused version to the minimum denoising score entropy (DSE) loss. We extend this theory to masked diffusion and establish the Information-Minimum Denoising Cross-Entropy (I-MDCE) relation, connecting cross-entropy losses to mutual information in discrete masked processes. These results provide a timeintegral decomposition of the log-likelihood of the data in terms of optimal scorebased losses, showing that commonly used losses such as DSE and DCE are not merely variational bounds but tight and principled estimators of log-likelihood. The I-MDCE decomposition further enables practical extensions, including time-free formula, conditional likelihood estimation in prompt-response tasks, and coupled Monte Carlo estimation of likelihood ratios. Experiments on synthetic and realworld data confirm the accuracy, variance stability, and utility of our estimators.

The majority of language model training builds on imitation learning.

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Object pose estimation plays a vital role in embodied AI and computer vision, enabling intelligent agents to comprehend and interact with their surroundings. Despite the practicality of category-level pose estimation, current approaches encounter challenges with partially observed point clouds, known as the multihypothesis issue. In this study, we propose a novel solution by reframing categorylevel object pose estimation as conditional generative modeling, departing from traditional point-to-point regression. Leveraging score-based diffusion models, we estimate object poses by sampling candidates from the diffusion model and aggregating them through a two-step process: filtering out outliers via likelihood estimation and subsequently mean-pooling the remaining candidates. To avoid the costly integration process when estimating the likelihood, we introduce an alternative method that distils an energy-based model from the original score-based model, enabling end-to-end likelihood estimation. Our approach achieves state-of-the-art performance on the REAL275 dataset, surpassing 50% and 60% on strict 5 2cm and 5 5cm metrics, respectively. Furthermore, our method demonstrates strong generalization to novel categories without the need for fine-tuning and can readily adapt to object pose tracking tasks, yielding comparable results to the current state-of-the-art baselines. Our checkpoints and demonstrations can be found at https://sites.google.com/view/genpose.

Diffusion models are recent state-of-the-art methods for image generation and likelihood estimation. In this work, we generalize continuous-time diffusion models to arbitrary Riemannian manifolds and derive a variational framework for likelihood estimation. Computationally, we propose new methods for computing the Riemannian divergence which is needed for likelihood estimation. Moreover, in generalizing the Euclidean case, we prove that maximizing this variational lower-bound is equivalent to Riemannian score matching. Empirically, we demonstrate the expressive power of Riemannian diffusion models on a wide spectrum of smooth manifolds, such as spheres, tori, hyperboloids, and orthogonal groups. Our proposed method achieves new state-of-the-art likelihoods on all benchmarks.

In recent years, differentiable penalized likelihood methods have gained popularity, optimizing the causal structure by maximizing its likelihood with respect to the data. However, recent research has shown that errors in likelihood estimation, even on relatively large sample sizes, disallow the discovery of proper structures. We propose a new approach to amortized causal discovery that addresses the limitations of likelihood estimator accuracy. Our method leverages Prior-Fitted Networks (PFNs) to amortize data-dependent likelihood estimation, yielding more reliable scores for structure learning. Experiments on synthetic, simulated, and real-world datasets show significant gains in structure recovery compared to standard baselines. Furthermore, we demonstrate directly that PFNs provide more accurate likelihood estimates than conventional neural network-based approaches.

We present an information-theoretic framework for discrete diffusion models that yields principled estimators of log-likelihood using score-matching losses. Inspired by the I-MMSE identity for the Gaussian setup, we derive analogous results for the discrete setting. Specifically, we introduce the Information-Minimum Denoising Score Entropy (I-MDSE) relation, which links mutual information between data and its diffused version to the minimum denoising score entropy (DSE) loss. We extend this theory to masked diffusion and establish the Information-Minimum Denoising Cross-Entropy (I-MDCE) relation, connecting cross-entropy losses to mutual information in discrete masked processes. These results provide a time-integral decomposition of the log-likelihood of the data in terms of optimal score-based losses, showing that commonly used losses such as DSE and DCE are not merely variational bounds but tight and principled estimators of log-likelihood. The I-MDCE decomposition further enables practical extensions, including time-free formula, conditional likelihood estimation in prompt-response tasks, and coupled Monte Carlo estimation of likelihood ratios. Experiments on synthetic and real-world data confirm the accuracy, variance stability, and utility of our estimators. The code is publicly available at https://github.com/Dongjae0324/infodis.

Generative models that maximize model likelihood have gained traction in many practical settings. Among them, perturbation based approaches underpin many strong likelihood estimation models, yet they often face slow convergence and limited theoretical understanding. In this paper, we derive a tighter likelihood bound for noise driven models to improve both the accuracy and efficiency of maximum likelihood learning. Our key insight extends the classical KL divergence Fisher information relationship to arbitrary noise perturbations, going beyond the Gaussian assumption and enabling structured noise distributions. This formulation allows flexible use of randomized noise distributions that naturally account for sensor artifacts, quantization effects, and data distribution smoothing, while remaining compatible with standard diffusion training. Treating the diffusion process as a Gaussian channel, we further express the mismatched entropy between data and model, showing that the proposed objective upper bounds the negative log-likelihood (NLL). In experiments, our models achieve competitive NLL on CIFAR-10 and SOTA results on ImageNet across multiple resolutions, all without data augmentation, and the framework extends naturally to discrete data.

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In practice, this integral is approximated with averaging over a finite number of Monte Carlo samples.