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$$ q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_{0}\right)=\prod_{t=1}^{T} q\left(\mathbf{x}_{t} \mid \mathbf{x}_{t-1}\right) $$

What dose the $\mathbf{x}_{1: T}$ mean? This is from https://drive.google.com/file/d/1noQ2d4-nzSh3Yp3-mOOYN8YGJoSa4pmz/view page 17.

LNseyJt2X-24UJ4r7KTL
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    It denotes the range of x... From $X_1$ to $X_T$ – Nandeesh Bhatrai Jul 21 '22 at 04:13
  • Thx for such an immediate reply. So the $T$ can be replaced with a value in the range from 1 to $T$? – LNseyJt2X-24UJ4r7KTL Jul 21 '22 at 04:19
  • No. $x_{1:T}$ represents the sequence of images $x_1, \cdots, x_T$. It's a path through image space. The right-hand side tells you that your process is independent at each step (to get to $x_1$, you only use information from $x_0$ is "$q(x_1 \mid x_0)$" and so on. So this is basically a statement that the process under discussion only uses the previous image to get the next one -- that is, the increments are independent. – Eric Towers Jul 21 '22 at 04:58
  • Thx guys, really needed it. – LNseyJt2X-24UJ4r7KTL Jul 21 '22 at 09:37

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