Sampling lies at the heart of modern generative models in machine learning. Generative models aim to capture the underlying distribution of observed data so that you can generate new, realistic samples that resemble the original data. Sampling provides the mechanism to draw these new data points from complex, often high-dimensional distributions learned by the model. This connection is especially crucial in latent variable models, where you assume that each observed data point is generated from some unobserved, or latent, variables through a probabilistic process. By sampling from these latent variables and then mapping them through the model, you can synthesize new data points that share the characteristics of your training data.

Sampling also plays a central role in likelihood estimation and data generation. When training generative models, you often need to estimate how likely your data is under the model's distribution. This likelihood estimation is typically intractable for complex models, but sampling offers a practical workaround. By drawing samples from either the model or an approximate distribution, you can estimate expectations, gradients, or other quantities needed for training. This is especially important for models where direct calculation of probabilities is difficult or impossible, and sampling-based methods such as Monte Carlo estimation become essential tools for both training and evaluation.

There are several conceptual bridges connecting classical sampling techniques and modern generative approaches like Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs). In VAEs, you use sampling from a simple distribution (such as a Gaussian) in the latent space to generate new data points through a learned decoder network. The training process relies on sampling-based approximations to optimize the model parameters. GANs, on the other hand, use a generator network to transform random samples (typically from a uniform or normal distribution) into synthetic data, which are then evaluated by a discriminator network. Both VAEs and GANs are built on sampling principles that trace back to classical methods, such as importance sampling and Markov Chain Monte Carlo, but they adapt these ideas to work with deep neural networks and high-dimensional data. The evolution from classical sampling to modern generative models highlights how foundational concepts in probability and statistics continue to shape the most advanced tools in machine learning today.