The equations for DDPM, the original and most popular class of Diffusion models, has some pretty gnarly math behind it. But, it turns out it's not too bad. Let's derive it!

It turns out, you can build an understanding of how to make diffusion models work by working in 2 steps:

• First, derive the equations for forwards and backwards for a single sample, assuming a perfect model i.e. you know exactly what noise was added
• Second, substitute in the model that estimates the noise from a sample. In theory, requires no change in the forward or backwards code, just build a U-Net model and put its fowrard cal in place of the function that knows the perfect noise

If that doesn't make much sense yet, that's fine. We'll go through more of it as we go.