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I came across a problem for Gibbs sampler.

Suppose I want to draw samples from $f(x,y,z)$, Can I use the following scheme to draw samples?

Step 1, draw $f(x^{2t-1},y^t|z^{t-1})$.

Step 2, draw $ f(x^{2t},z^t|y^{t-1})$.

Then, after $T$ steps, we have $2T$ samples for $x$, and $T$ samples for $y$ and $z$.

Is the above procedure appropriate? Could you provide me some references?

Thanks!

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  • $\begingroup$ I made a mistake, in step 2, it should be conditional on $y^t$ $\endgroup$ – Ryan Feb 14 '17 at 5:59
  • $\begingroup$ The parameter of interest is $x$. $\endgroup$ – Ryan Feb 14 '17 at 6:00
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The method is correct by the following reasoning:

Start with $$(x,y,z)\sim f(x,y,z)$$ and apply Step 1. $$ (x',y') \sim f(x,y|z)$$ then $$(x',y',z)\sim f(x,y,z)$$ Now apply Step 2 $$ (x'',z') \sim f(x,z|y')$$ then $$(x'',y',z')\sim f(x,y,z)$$ This means $f$ is stationary for this two-step Gibbs algorithm.

Note: the title should be corrected as "fixing one parameter" is confusing. You could say Gibbs sampling/sampler with two block conditionals.

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  • $\begingroup$ @Ryan: can you please change the title? $\endgroup$ – Xi'an Feb 17 '17 at 8:25

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