Algorithm 1.

Elliptical slice sampler for posterior distribution $\pi (\cdot \mid \mathbf{Y},\mathbf{\theta })$

1	Input: Latent variable from the previous iteration $\mathbf{X}\in {\mathbb{R}}^d$. Observed data $\mathbf{Y}$, previous updated parameter $\mathbf{X}$..
2	Output Updated latent variable value ${\mathbf{X}}^{\prime }$
3	Sample ellipse $\mathbf{Z}\sim 𝒩(0,\mathbf{\Sigma }(\mathbf{\theta }\left)\right)$
4	Compute log-likelihood threshold: sample $U\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,1)$ and let
	$\tau \leftarrow \log L(\mathbf{Y}\mid \mathbf{X},\mathbf{\theta })+\log (U)$
5	Sample angle parameter $\alpha \sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}[0,2\pi ]$ and $\left[{\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]\leftarrow [\alpha -2\pi ,\alpha ]$
6	${\mathbf{X}}^{\prime }\leftarrow \mathbf{X}\cdot \mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathbf{Z}\cdot \mathrm{s}\mathrm{i}\mathrm{n}\alpha$
7	while $\mathrm{l}\mathrm{o}\mathrm{g}\left(L\left(\mathbf{Y}\mid {\mathbf{X}}^{\prime },\mathbf{\theta }\right)\right)<\tau$ do
8	if $\alpha <0$ then
9	${\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}}\leftarrow \alpha$
10	else
11	${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\leftarrow \alpha$
12	Sample $\alpha \sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\left({\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)$.
13	Make new proposal
	${\mathbf{X}}^{\prime }\leftarrow \mathbf{X}\cdot \cos \alpha +\mathbf{Z}\cdot \sin \alpha$
14	Return ${\mathbf{X}}^{\prime }$.