# Why is Algorithm 8 of Neal (2000) a valid sampler?

I have been having difficulty understanding why Algorithm 8 of Neal (2000) is a valid sampler.

I am looking for lecture notes that include a nice explanation of the proof. Does anyone know of any such reference?

• Please provide the background so that a reader does not need reading Neal (200) first. Commented Oct 30, 2019 at 8:02
• Thank you for the edit, I will be sure to do this in the future. Commented Oct 30, 2019 at 16:03
• @Xi'an It's a good reading though :D Commented Nov 2, 2022 at 8:51

$$P(c_i = c\mid \ldots) \propto \begin{cases} n_{\neg i, c}f(Y_i\mid\theta_c) &\text{ for }1 \leq c \leq k^{\neg i}\\ \alpha\int_{\theta} f(y_i\mid\theta)dG_0(\theta) &\text{ for }c = k^{\neg i} + 1 \end{cases}$$
Doing that integral analytically is difficult, if $$G_0$$ is not conjugate to $$F$$.
Some of Neal's algorithms offer various perturbations of this model trying to bypass the need for conjugacy, by doing that integral implicitly. By sampling $$\theta$$ from $$G_0$$, then weighting by $$f(y_i\mid\theta)$$, you effectively get samples from $$f(y_i\mid\theta)dG_0(\theta)$$. In fact, I think algorithm 7 does just this (with a single sample per iteration). Algorithm 8 recognizes that a single sample doesn't represent that integral very well, so we generate $$m$$ samples from $$G_0$$, and weight them accordingly.