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Following is a normal distribution equation:

$$q(\mathbf x_t\mid\mathbf x_{t-1})=\mathcal N(\mathbf x_t;\sqrt{1-\beta_t}\mathbf x_{t-1},\beta_t\mathbf I).$$

I understand there should be mean and variance in normal distribution. But what is the meaning of this? It contains three variables.

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    $\begingroup$ This is conventional (IMHO awful) notation in which the first argument on the right hand side names the variable, the second specifies the mean, and the third specifies some function of the variance (the variance itself, the SD, or perhaps the precision). The appearance of "$x_t$" on the right hand side is therefore superfluous --which is why it's not great notation. $\endgroup$
    – whuber
    Commented Dec 19, 2022 at 22:00
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    $\begingroup$ @whuber Thanks! $\endgroup$
    – Jing Gu
    Commented Dec 19, 2022 at 22:06

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The semicolon separates off the parameters of the distribution. This function represents the probability density of $x_t$ given those two parameters.

This should be interpreted as the probability density $q$ is defined by this normal distribution.

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    $\begingroup$ Another problem with the notation is that it isn't at all evident whether $q$ refers to a distribution, a density, or perhaps even something else. $\endgroup$
    – whuber
    Commented Dec 19, 2022 at 22:03

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