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```{math}

\newcommand\pos{\boldsymbol{x}}
\newcommand\mom{\boldsymbol{p}}
\newcommand\mass{\mathcal{M}}
```

(sec:slice-sampling)=
# Slice sampling

The idea of slice sampling is well documented in
two arXiv references for the ensemble slice sampler `zeus` (which is a particular implementation):
1. [*Ensemble Slice Sampling: Parallel, black-box and gradient-free inference for correlated & multimodal distributions*](https://arxiv.org/abs/2002.06212) by Minas Karamanis and Florian Beutler.
2. [*zeus: A Python implementation of Ensemble Slice Sampling for efficient Bayesian parameter inference*](https://arxiv.org/abs/2105.03468) by Minas Karamanis, Florian Beutler, and John A. Peacock.

::::{grid}
:gutter: 2

:::{grid-item}
```{figure} ../assets/figure_1_from_arXiv2002.06212_ensemble_slice_sampling.png
:width: 100%
:name: zeus_fig_1

Figure 1 from [Ref. 1](https://arxiv.org/abs/2002.06212) above.
```
:::

:::{grid-item}
```{figure} ../assets/figure_2_from_arXiv2105.03468_zeus_ensemble_slice_sampling.png
:width: 100%
:name: zeus_fig_2

Figure 2 from [Ref. 2](https://arxiv.org/abs/2105.03468) above.
```
:::

::::

Slice sampling in action for a one-dimensional distribution is shown in {numref}`zeus_fig_1`. 
The idea is that sampling from a distribution $p(x)$ is the same as uniform sampling from the *area* under the plot of $f(x) \propto p(x)$. E.g., the highest probability $p(x)$ is where the area is largest, and this is a simple proportionality. The height $y$ is introduced with $0 < y < f(x)$ as an auxiliary variable and one samples the uniform joint probability $p(x,y)$. Then $p(x)$ is obtained by marginalizing over $y$ (meaning just dropping $y$ from the sample).

The blue star comes from uniformly sampling in $y$ at the initial point $x_0$ to identify $y_0$. The "slice" is the set of $x$ such that $y_0 < f(x)$. In the figure the interval initially from $L$ to $R$ is expanded to $L'$ to $R'$, which includes the full slice. When the final interval is found, a new $x_1$ is drawn uniformly from the intersection of the interval and the slice.
The green star is an accepted new point $x_1$ (inside the slice) while the red star is a rejected point (outside the slice). 
There is one parameter to tune: the width $\mu$ of the interval from $L$ to $R$ that sets the original interval around $x_0$ and then the step size for expanding the interval to be from $L'$ to $R'$. In `zeus` the value of $\mu$ is tuned by stochastic optimization.

The "ensemble" aspect of `zeus` is that there are multiple walkers, as in the ensemble sampler `emcee`. 
{numref}`zeus_fig_2` illustrates how the ensemble of walkers is used to do what is called a "differential move". There are several choices for moves, including a global one that is effective for multi-modal distributions.

<!--* An example from the `zeus` documentation of sampling from a multimodal distribution is given in {ref}`project:zeus-multimodal`.-->
* A text case for parallel tempering (using `ptemcee`) vs. slice sampling (using `zeus`) in a very simple multi-modal example is in {ref}`demo:multimodal-distributions-with-two-samplers`.
* The basic call for `zeus` compared to `emcee` and `PyMC` is illustrated in {ref}`demo:comparing-samplers-for-a-simple-problem`.