Generative models: factorization and denoising

The probability chain rule expresses a sequence's joint probability as a product of next-element conditional probabilities. Choosing an ordering is a modeling decision; the identity itself does not assume independence. A neural network parameterizes each conditional using an allowed prefix. Training on observed prefixes can evaluate many token losses in parallel with a causal mask, while ordinary left-to-right sampling must choose earlier tokens before conditioning on them to generate later ones.

Negative log likelihood sums the negative logarithms of conditional probabilities. This teaches the model to assign probability to observed sequences, not to verify the truth of every sentence. At generation time, a model may condition on its own earlier mistakes rather than the clean prefixes seen during training. Sampling temperature or truncation changes the generation policy and its output distribution. A fluent sample can still be false, unsafe for a particular use, or unrepresentative of rare training modes.

A standard Gaussian DDPM defines a forward process that gradually adds noise to continuous data. A useful property is that a noisy sample at any chosen timestep can be generated directly from a clean example and Gaussian noise, without simulating every earlier step. The cumulative noise schedule determines how much original signal remains. A network receives the noisy sample and timestep and can be trained to predict the added noise under a commonly used simplified squared-error objective.

Generation begins from an approximate terminal noise distribution and applies a learned reverse procedure. The reverse conditionals depend on the data distribution; simply negating the forward noise does not recover an unknown original. Finite-step DDPMs use a parameterized reverse model, and alternative samplers can trade step count against error and sample behavior. Do not generalize the continuous Gaussian formula to discrete tokens without changing the corruption model and associated derivation. Conditioning, such as text guidance, also changes the learned or sampling distribution.

A denoising objective tells the network which noise component to estimate at each corruption level. A small training loss alone does not establish high-quality generation or complete coverage of the data distribution. A sampler can generate visually appealing common cases while missing uncommon modes. Likelihood-based scores, perceptual measures, diversity tests, and human or task-specific evaluations probe different properties and may disagree. State which failure would matter in the intended application before choosing a metric.

Inspect training-data provenance and evaluation contamination. A generator may reproduce a memorized item rather than create a meaningfully new one. Compare near-duplicates and evaluate on cases separated from training sources where possible. Generated data can support simulation or prototyping, but a plausible sample is not an observation of the world. For a scientific or operational claim, require external evidence or a validated generative model with clearly stated assumptions, uncertainty, and a defined domain of use.