Since its original formulation, Jensen's inequality has played a fundamental role across mathematics, statistics, and machine learning, with its probabilistic version highlighting the nonnegativity of the so-called Jensen's gap, i.e., the difference between the expectation of a convex function and the function at the expectation. Of particular importance is the case when the function is logarithmic, as this setting underpins many applications in variational inference, where the term variational gap is often used interchangeably. Recent research has focused on estimating the size of Jensen's gap and establishing tight lower and upper bounds under various assumptions on the underlying function and distribution, driven by practical challenges such as the intractability of log-likelihood in graphical models like variational autoencoders (VAEs). In this paper, we propose new, general bounds for Jensen's gap that accommodate a broad range of assumptions on both the function and the random variable, with special attention to exponential and logarithmic cases. We provide both analytical and empirical evidence for the performance of our method. Furthermore, we relate our bounds to the PAC-Bayes framework, providing new insights into generalization performance in probabilistic models.

Training 3D generative models often faces bottle-necks due to dependencies on precise camera pose estimation, particularly when using Neural Radiance Fields (N eRFs) for photorealistic novel-view synthesis. We introduce HyperNeR-FGAN, a generative framework that eliminates camera pose requirements by integrating a hypernetwork with a Generative Adversarial Network (GAN). This architecture maps Gaussian noise directly to the weights of a NeRF model, bypassing viewing direction inputs during training. Our experiments demonstrate that HyperNeRFGAN achieves state-of-the-art performance on datasets where camera position estimation is impractical - notably in medical imaging scenarios with limited or ambiguous viewpoint metadata. Despite its architectural simplicity compared to existing methods, the model produces high-fidelity 3D reconstructions across diverse modalities, including MRI and X-ray-derived 2D scans. The framework's efficiency and robustness suggest broad applicability in domains requiring 3D generation from unstructured or poorly annotated 2D data. Key revisions emphasize the camera-pose independence, clinical relevance, and architectural efficiency while maintaining technical nrecision.