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I'm reading about Monte Carlo methods. Suppose that $X_1,...,X_n$ are i.i.d $p(x_i|\theta)$, where $\theta$ is an unknown parameter of interest. My textbook states: Suppose we could sample some number $S$ of independent, random $\theta$-values from the posterior $p(\theta|x)$. Then the empirical distribution of the sample $(\theta_1,...,\theta_S)$ would approximate the posterior $p(\theta|x)$.

What does it mean to sample from a probability distribution that is unknown? How is this possible? I would greatly appreciate a simple example.

DavidSilverberg
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    It is slightly misleading to say that the probability distribution here is `unknown' - using Bayes' theorem, one can write down a formula for $p ( \theta | x )$, up to an unknown constant, and so it is at least partially known, in that sense. – πr8 Jan 12 '20 at 12:56

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In MCMC methods, you typically sample from partially known (as also pointed out in the comments section) distributions. Knowing its form up to a scalar (i.e. constant wrt $\theta$) is enough. Therefore, MCMC can also be used to approximate integrals by implicitly calculating the normalization constant. For example, you probably won't be able to calculate the normalization constant for $e^{-\theta^8}$, but will be able to sample from it using MCMC.

gunes
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