Are the terms in the diffusion model equation random variables or probability density functions?

Question

Are all terms in the first line(71) of the equation random variables or probability density functions? If they are probability density functions, is there a possibility of obtaining a value that is not equal to 1 when integrating the right-hand side of the equation after all calculations have been completed?
Based on the answer given in line 72, it seems that all terms are considered as probability density functions. If so, is it possible to transform them into the probability density function of a Gaussian distribution?
in q(x_{t-1}|x_t,x_0), x_t, x_0, x_{t-1} are EVENTS? or Distribution?

I feel like I'm lacking some basic concepts in statistics. Can you help me, please?

That's truly awful notation ;-) but doesn't the last line $(73)$ answer your questions? — whuber, Commented Apr 2, 2023 at 14:30
Yes, It seems to be simply the product of p.d.f.s, but I'm not sure if there's any issue with that. — statishard, Commented Apr 3, 2023 at 4:31
In that case it sounds like you are questioning the meaning of (71). That's an application of basic rules and formulas for conditional probabilities, translated into densities. A search, then, for information about conditional density might set you in the right direction. — whuber, Commented Apr 3, 2023 at 15:12
Thank you. I will search for the Bayes rule keyword related to distribution(density func), not the concept of Bayes rule for a specific event. — statishard, Commented Apr 4, 2023 at 0:24

erenovic · Accepted Answer · 2023-05-27 18:29:27Z

They are all pdfs, you are basically applying Bayes' Rule. Since the first line has an exact equality, if you integrate just over the full domain of $x_{t-1}$ for fixed $x_t$ and $x_0$, you will get 1 as you said. But the integration is intractable in such a high dimensional space (i.e. even an image as small as $32\times 32$).
The main assumption of Diffusion models is that the noise is Gaussian. So , yes, you can think that the distributions of $x_i$ are Gaussian.
Those are random variables. Every $x_i$ is an image that is processed through the Markov chain. So, you are basically saying that the images are Gaussian, given the initial image and/or the previous image in the Markov chain.

Hope this helps.

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