What a generative model actually is

There is a density behind the data, and the data do not contain it

Suppose the world can produce an image, a sentence, or a song. Whatever the medium, there is a set of possible outputs and some notion of how likely each one is — a distribution $p(x)$ over everything the process could make. A photograph of a specific cat on a specific sofa is not stored anywhere; it is one point that a generator, if it were the right one, would assign some density to. In the probability volume the distribution is handed to you. In generative modelling it is the thing you are trying to reconstruct from examples, and you never see it directly.

A dataset is a finite set of samples from $p(x)$. It is a handful of points, not the distribution. This is the whole difficulty in one sentence: a dataset is sparse, and a distribution is everywhere. Two datasets of a hundred images can differ completely while coming from the same distribution, and a model that merely stores the hundred images has learned almost nothing about the process that made them.

The goal of a generative model is therefore not to compress the dataset. It is to fit a distribution $p_\theta(x)$ close enough to the true $p(x)$ that when you draw new points from $p_\theta$, they are plausible in the same way that fresh data would be — sometimes close to a training point, sometimes in the gaps between them, but always in places where the true process could plausibly have put a point.

Hold onto one image before the demo: data are dots, the distribution is the terrain the dots sit on, and modelling means recovering the terrain rather than the dots.

Click on the plane to place points

Below is a two-dimensional plane, which is the smallest space in which a distribution is still something you can look at. Every click adds a data point at that location. Two buttons drop in a preset shape — two clusters or a ring — and Undo and Clear let you sculpt a dataset by hand. The grey cloud starts as two clusters because that is the easiest shape to reason about; try the ring afterwards, because a ring has a hole in the middle, and the hole is where the interesting question lives.

Once you have a dataset, ask yourself a question that has no obvious answer: where is the distribution? Not the points — the process that would produce points like these. If I asked you to draw three more points that belong, where would you put them? That judgment is a model, and it is the judgment we are about to hand to an algorithm.

Two properties of a dataset matter everywhere in this guide. The first is its spacing: how far apart neighbouring points are. Spacing is a proxy for how much of the plane the data actually cover, and it sets the natural scale at which a model should invent new points. The second is its shape: a cluster, a ring, a curve, several separate modes. A model that captures the shape but not the spacing overfits; one that captures the spacing but not the shape is too smooth. The demo in the next step uses both.

New points, not the ones you drew

Here is the simplest honest model of the dataset you just made: put a small bump of probability at every data point, then draw new points from the union of those bumps. This is a kernel density estimate — the histogram's smoother cousin — and it is a real generative model, fitted to your points, from which we can sample. Each sample picks one data point at random, then wanders a short distance from it. Press Draw 240 samples and the blue points appear over the grey data.

The key readout is novelty: the typical distance from a model sample to the nearest data point, measured in units of the dataset's own spacing. If that number is zero, the model has learned to copy. If it is comfortably above zero, the model is putting points where the data are not but easily could have been.

Notice what sampling buys you. The model has a compact description — the dataset plus one bandwidth — and from that it can emit as many points as you like. The samples are not memorised outputs; they are draws from a distribution that was fitted to the data. That is the entire contract of a generative model, and the rest of this guide is about doing it in spaces where "put a bump at every point" is hopeless: a million-dimensional image, where a bump per training example is not a model, it is a lookup table.

The jitter slider is the difference, and it is one number

Drag the jitter slider to zero and press the button again. Every sample lands exactly on a data point. The model has become a copy machine: it can only ever return one of the inputs, so it cannot produce anything the dataset did not already contain. Now drag the slider up. The samples spread out, following the shape of the data — the clusters stay clustered, the ring stays a ring — and the novelty readout climbs. Some samples fall in the gaps between points. That is generalisation, made visible.

Push it too far and the model breaks in the opposite direction: the bumps swell until clusters merge, the hole in the ring fills in, and the samples stop looking like the data at all. So there is a sweet spot, and finding it is what every training algorithm in this guide does — not by one slider, but by fitting millions of parameters so that the implied distribution matches the data without collapsing onto it. In diffusion the equivalent knob is the noise schedule and the amount of training; too little and the model memorises, too much and it over-smooths.

The same tension is the reason this guide spends four more parts on noise. Adding noise to an image is a way of moving it off the data manifold into that between-space, and training a network to remove the noise is a way of forcing it to model the between-space rather than the points. The forward process is next.

The same picture, in higher dimensions

In the next part the points stop being two-dimensional dots and become pixels. A 64×64 image is a point in 12,288 dimensions, which is far too many to plot but exactly the same idea — a dataset of points, a distribution behind them, and a model that must learn the terrain rather than the dots. The machinery that makes it work is a procedure that destroys a data point with noise, one small step at a time, until nothing but the distribution remains.

Further reading

The references below are the canonical treatments of generative models as density estimation, in increasing order of machinery. If you take one idea away, take the separation between a dataset and the distribution behind it; almost every confusion later in this volume comes from collapsing those two.

Bishop's chapter on density estimation is the cleanest introduction to kernels and mixtures; Goodfellow's tutorial and Kingma and Welling's VAE paper are where the modern framing of "sample from a learned distribution" begins; and the DDPM paper is the point at which this guide's next four parts start. 3Blue1Brown's videos are the visual reference for the style of the demos here.

• Christopher Bishop, Pattern Recognition and Machine Learning, chapter 2 — binary and multinomial distributions, the Gaussian, and kernel density estimation.
• Ian Goodfellow, "NIPS 2016 Tutorial: Generative Adversarial Networks" — the taxonomy of generative models as density estimation, explicit and implicit.
• Diederik Kingma and Max Welling, "Auto-Encoding Variational Bayes", 2013 — the latent-variable view of a generative model and the reparameterisation trick.
• Jonathan Ho, Ajay Jain and Pieter Abbeel, "Denoising Diffusion Probabilistic Models", 2020 — the model the next part begins to take apart.
• 3Blue1Brown, "But what is a diffusion model?" — the visual intuition for adding and removing noise.

Cheat sheet