This paper explores the interplay between statistics and generative artificial intelligence. Generative statistics, an integral part of the latter, aims to construct models that can {\it generate} efficiently and meaningfully new data across the whole of the (usually high dimensional) sample space, e.g. a new photo. Within it, the gradient-based approach is a current favourite that exploits effectively, for the above purpose, the information contained in the observed sample, e.g. an old photo. However, often there are missing data in the observed sample, e.g. missing bits in the old photo. To handle this situation, we have proposed a gradient-based algorithm for generative modelling. More importantly, our paper underpins rigorously this powerful approach by introducing a new F-entropy that is related to Fisher's divergence. (The F-entropy is also of independent interest.) The underpinning has enabled the gradient-based approach to expand its scope. For example, it can now provide a tool for Possible future projects include discrete data and Bayesian variational inference.

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