The Monte Carlo Markov Chain method requires sampling from a multivariate distribution. This is because the Markov Chain process requires dependent draws. See 1:55 at https://www.youtube.com/watch?v=7LB1VHp4tLE. It is claimed that it is difficult to simultaneously draw from a multivariate distribution. But modern computers and software can sample from a multivariate distribution. So my first question is in which sense is it difficult? Because that is why a solution is proposed which is to sample from a conditional distribution. There is no need to watch the entire video because my question is only about sampling from a multivariate distribution. Why is it difficult to sample from a multivariate distribution as compared to a univariate distribution?

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    $\begingroup$ It's not--this is a false generalization. Indeed, what do you really mean by "difficult"--in what sense? $\endgroup$
    – whuber
    Aug 1, 2022 at 22:15
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    $\begingroup$ Difficulty is curse of dimensionally. sampling from n-D $\endgroup$ Aug 2, 2022 at 1:34
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    $\begingroup$ Most people will not want to spend the time to watch a video to understand what your question might be. Please include an explanation in the question itself. $\endgroup$
    – whuber
    Aug 2, 2022 at 3:28
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    $\begingroup$ There are three ongoing problems with the question. The first is that "difficult" has no definite meaning. The second is that "multivariate distribution" covers a lot of ground, allowing for any general statements to be true or false depending on the details. The third is that you are leaping, without justification, from a vague statement to one that is false (namely, that it is at problematic in any way to sample from bivariate Normal distributions or even multivariate Normal distributions generally). $\endgroup$
    – whuber
    Aug 4, 2022 at 16:11
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    $\begingroup$ Thank you for making those improvements. I still question your final question about bivariate Normal distributions, because they provide examples of dependent random variables where it is easy--conceptually and computationally--to draw values all at once without going through the sequence of conditional distributions. $\endgroup$
    – whuber
    Aug 5, 2022 at 11:53