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In an image processing class, I dont really get behind the idea how to 'sample from a posterior' with Gibbs sampling. We have a posterior distribution:

$f(z_1, .. ,z_n \mid x_1,.. ,x_n) := f(z \mid x)$

From Bayes theorem we express this as:

$f(z \mid x)$ ∝ $f(x \mid z)f(z)$ ,

where $f(x \mid z)$ is the likelihood and $f(z)$ the prior, which we are modeling in different fashions.

In a first task, we are asked to 'sample from the prior' with Gibbs sampling. To that end, we sample from $f(z)$ by sampling from its conditional distribution by updating the random variables elementwise from $f(z_1 \mid z_2 , .. z_n)$.

Now, we have to 'sample from the complete posterior' and this is where my intuition fails. Iam confused as this is already a conditional distribution. Could anyone explain the general approach of using Gibbs sampling to sample from a posterior distribution?

Thanks

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  • $\begingroup$ Can you clarify what a "complete posterior" distribution is? $\endgroup$ – AdamO Nov 8 '17 at 20:35
  • $\begingroup$ We regard the posterior as f(z∣x). Our task is to use Gibbs sampling to draw samples from that distribution. The word complete might be misleading here. $\endgroup$ – kirtap Nov 8 '17 at 21:09
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Typically and in simplest form, Gibbs sampling would be sampling from each of the full conditional distributions

$$f(z_i|\mathbf{z}_{j\neq i},\mathbf{x})$$

using some suitable sampling order (e.g. random scan).

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