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Theoretically, the slice sampling has equilibrium distribution as the target distribution. If we can sample exactly as follows,

$y' = U(0, p^*(x))$

$x' = U\{x: p^*(x) > y' \}$

However, in the implementation of sampling $x'$, people usually use a sequence of intervals, such as the description here. I think this will violate the uniform description of course, because $x'$ stays close to $x$. I want to know

  1. What's the transition probability in this case.

  2. Is this valid? why?

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Are you objecting to the scheme described on this picture from the Wikipedia page:

enter image description here

which finds a large enough interval containing the entire slice then possibly shrinks it when simulating points outside the slice. As described by the picture the scheme is wrong since it misses one part of the slice, the rhs blue interval. The difficulty in applying the scheme in realistic situations is to be certain that the overall interval (in black) indeed contains the entire slice, which can be made of several intervals.

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  • $\begingroup$ Thanks Xi'an. It seems the algorithm has no guarantee to include all pieces of $\{x: p^*(x') > y' \}$. For example, if the distribution has two modes, and they are very far from each other. In this case, I guess the algorithm does not converge to the target distribution, right? Even the slice covers all pieces, but when you sample a point in between two segments, like the two blue lines in the picture, the slice will shrink to the sample point, which will drop out a piece. Again, in this case, you can't evenly sample over the whole area. $\endgroup$ Jan 19, 2019 at 23:31
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    $\begingroup$ If the collection of black intervals covers the collection of blue intervals, the algorithm is correct. $\endgroup$
    – Xi'an
    Jan 20, 2019 at 7:51

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