# Gibbs sampling for drawing samples and estimating parameters

I'm learning Bayesian inference by myself and having a difficulty for understanding Gibbs sampling.

From what I understood, Gibbs sampling is to draw samples from a given probability distribution $$p(x_1, x_2 | \theta)$$ where $$\theta$$ is parameters for the distribution as $$x_1 \sim p(x_1|x_2,\theta)$$ and $$x_2 \sim p(x_2 | x_1, \theta)$$ repeatedly so that we can see the distribution.

But at the same time I see (as far as I read) it's used to estimate parameters of a probability distribution and am not sure I understand it correctly so asking this question here.

Let's say $$X = \{x_1, x_2, ..., x_n\}$$ are observed data from a distribution and we assume the data were generated by a probability distribution $$p(x | \theta_1, \theta_2)$$ and also assume we can calculate the likelihood $$L = p(\theta_1, \theta_2 | X)$$.

In this case is it a valid use case of Gibbs sampling to estimate $$\theta_1$$ and $$\theta_2$$ as drawing samples as $$\theta_1 \sim p(\theta_1 | \theta_2, X)$$ and $$\theta_2 \sim p(\theta_2 | \theta_1, X)$$?

• Yes, assuming you can sample directly from the full conditionals $p(\theta_1 | \theta_2, X)$.... From a Bayesian perspective, data are fixed and parameters are random, so estimating the parameters of a probability distribution is done by drawing samples from a probability distribution -- the posterior distribution of the parameter values given your prior and your observed data. – duckmayr May 16 at 17:34