Generative Models
Generative models are machine learning models that learn a probability distribution from data and can draw new samples from it. Two broad families are common. One represents the distribution directly, for example as a mixture of Gaussians that can be sampled and conditioned analytically. The other learns a transformation from a simple reference distribution (such as Gaussian noise) to the data distribution, as in diffusion models, which learn to remove noise added to the data, and flow-based models such as continuous normalizing flows and flow matching, which learn a vector field that carries one distribution to the other. Conditional generation, where some variables are fixed and the rest are generated, is what makes these models useful beyond producing samples, including for imputing missing values in tabular data.
Conditional generation for inverse problems (2025)
Kitchin showed that generative models trained on tabular input and output data from the Claude-Light instrument can solve an inverse problem by conditioning on desired outputs and generating the inputs (Kitchin 2025). A single-component Gaussian mixture model (gmr package) and a ForestDiffusion model (gradient-boosted trees with flow matching) both reproduced the solution found by conventional forward and backward models, and repeated conditional sampling gave standard deviations near 0.002 in the inputs. The Gaussian mixture can be conditioned on any subset of variables, so one model gives forward or inverse predictions, while the ForestDiffusion implementation requires choosing the conditioning variables at training time. Neither model is reliable outside the data: the mixture extrapolates linearly from its nearest Gaussian and the tree-based model returns constant leaf values, in both cases with misleadingly small spread. Summary: kitchin2025a.