\begin{algorithm}
	\label{fig: step_shrink}
	A hybrid slice sampling transition of the stepping-out and shrinkage procedure from $x$ to $y$, i.e. input $x$ and output $y$.
	The stepping-out procedure has input $x$ (current state), $t$ (chosen level),
	$w>0$ (step size parameter from $\mathcal{R}_w$) and outputs an interval $[L,R]$.
	The shrinkage procedure has input $[L,R]$ and output $y$:
	\begin{enumerate}
		\item Choose a level $t \sim \mathcal{U}(0,\rho(x))$;
		\item Stepping-out with input $x,t,w$ outputs an interval $[L,R]$:
		\begin{enumerate}
			\item Choose $u \sim \mathcal{U}[0,1]$. Set $L=x-u w$ and $R=L+w$;
			\item Repeat until $t \geq \rho(L)$, i.e. $L \not \in K(t)$: Set $L=L-w$;
			\item Repeat until $t \geq \rho(R)$, i.e. $R \not \in K(t)$: Set $R=R+w$;
		\end{enumerate}
		\item Shrinkage procedure with input $[L,R]$ outputs $y$:
		\begin{enumerate}
			\item Set $\bar{L}=L$ and $\bar{R}=R$;
			\item Repeat:
			\begin{enumerate}
				\item   Choose $v\sim \mathcal{U}[0,1]$ and set $y=\bar{L}+ v (\bar{R}-\bar{L})$;
				\item   If $y  \in K(t)$ then return $y$ and exit the loop;
				\item   If $y<x$ then set $\bar{L}=y$, else $\bar{R}=y$.
			\end{enumerate}
		\end{enumerate}
	\end{enumerate}
\end{algorithm}

\begin{algorithm}
A hybrid slice sampling transition of the stepping-out and shrinkage procedure from $x$ to $y$, i.e. input $x$ and output $y$.
The stepping-out procedure has input $x$ (current state), $t$ (chosen level),
$w>0$ (step size parameter from $\mathcal{R}_w$) and outputs an interval $[L,R]$.
The shrinkage procedure has input $[L,R]$ and output $y$:
\begin{enumerate}
\item Choose a level $t \sim \mathcal{U}(0,\rho(x))$;
\item Stepping-out with input $x,t,w$ outputs an interval $[L,R]$:
\begin{enumerate}
\item Choose $u \sim \mathcal{U}[0,1]$. Set $L=x-u w$ and $R=L+w$;
\item Repeat until $t \geq \rho(L)$, i.e. $L \not \in K(t)$: Set $L=L-w$;
\item Repeat until $t \geq \rho(R)$, i.e. $R \not \in K(t)$: Set $R=R+w$;
\end{enumerate}
\item Shrinkage procedure with input $[L,R]$ outputs $y$:
\begin{enumerate}
\item Set $\bar{L}=L$ and $\bar{R}=R$;
\item Repeat:
\begin{enumerate}
\item Choose $v\sim \mathcal{U}[0,1]$ and set $y=\bar{L}+ v (\bar{R}-\bar{L})$;
\item If $y \in K(t)$ then return $y$ and exit the loop;
\item If $y<x$ then set $\bar{L}=y$, else $\bar{R}=y$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{algorithm}
{
    "title": "convergence of hybrid slice sampling via spectral gap",
    "id": "1409.2709",
    "abstract": "it is known that the simple slice sampler has robust convergence properties, however the class of problems where it can be implemented is limited. in contrast, we consider hybrid slice samplers which are easily implementable and where another markov chain approximately samples the uniform distribution on each slice. under appropriate assumptions on the markov chain on the slice we show a lower bound and an upper bound of the spectral gap of the hybrid slice sampler in terms of the spectral gap of the simple slice sampler. an immediate consequence of this is that spectral gap and geometric ergodicity of the hybrid slice sampler can be concluded from spectral gap and geometric ergodicity of its simple version which is very well understood. these results indicate that robustness properties of the simple slice sampler are inherited by (appropriately designed) easily implementable hybrid versions. we apply the developed theory and analyse a number of specific algorithms such as the stepping-out shrinkage slice sampling, hit-and-run slice sampling on a class of multivariate targets and an easily implementable combination of both procedures on multidimensional bimodal densities.",
    "categories": "stat.me",
    "doi": "",
    "created": "2014-09-09",
    "updated": "2024-02-09",
    "authors": [
        "krzysztof \u0142atuszy\u0144ski",
        "daniel rudolf"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/1409.2709"
}
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