## 2 Materials and Methods  
Consider the unknown temporal \(n\)-vector \(\mathbf{r}_{t}\) for \(t\in\mathcal{T}_{r}:\{0,1,\ldots,T,T+1\}\). Let \(\mathbf{r}=\{\mathbf{r}_{0},\mathbf{r}_{1},\ldots,\mathbf{r}_{T},\mathbf{r}_{T+1}\}\) denote the variable of interest and let \(\mathbf{r}_{ij}\) denote \(\{\mathbf{r}_{i},\mathbf{r}_{i+1},\ldots,\mathbf{r}_{j}\},\forall(i,j)\in\mathcal{T}_{r}^{ 2},i\leq j\)Assume that the temporal \(m\)-vectors of observations \(\mathbf{d}_{t}\) for \(t\in\mathcal{T}_{d}:\{0,1,\ldots,T\}\) are available, and define \(\mathbf{d}=\{\mathbf{d}_{0},\mathbf{d}_{1},\ldots,\mathbf{d}_{T}\}\) and \(\mathbf{d}_{i:j}=(\mathbf{d}_{i},\ldots,\mathbf{d}_{j})\) accordingly. The model specified hereafter defines a hidden Markov (HM) model [23] as displayed in Figure 1.  
Prior modelThe prior model on \(\mathbf{r}\) consists of an initial and a forward model,  
\[f(\mathbf{r})=f(\mathbf{r}_{0})f(\mathbf{r}_{1:T+1}|\mathbf{r}_{0}), \tag{1}\]  
where \(f(\mathbf{r}_{0})\) is the pdf of the initial state and \(f(\mathbf{r}_{1:T+1}|\mathbf{r}_{0})\) defines the forward model. (a) _Initial distribution:_ The distribution for the initial state \(f(\mathbf{r}_{0})\) is assumed to be in the class of selection-Gaussian distributions [20; 21]. Consider a Gaussian \((n+n)\)-vector \([\mathbf{\tilde{r}},\mathbf{\
u}]\),  
\[ \begin{bmatrix}\mathbf{\tilde{r}}\\ \mathbf{\
u}\end{bmatrix} \sim\varphi_{2n}\left( \begin{bmatrix}\mathbf{\tilde{r}}\\ \mathbf{\
u}\end{bmatrix} ; \begin{bmatrix}\mathbf{\mu}_{r}\\ \mathbf{\mu}_{\
u}\end{bmatrix} , \begin{bmatrix}\mathbf{\Sigma}_{\mathbf{\tilde{r}}}&\mathbf{ \Sigma}_{\mathbf{\tilde{r}}}\mathbf{\Gamma}_{v|\mathbf{\tilde{r}}}^{T}\\ \mathbf{\Gamma}_{v|\mathbf{\Sigma}_{\mathbf{\tilde{r}}}}&\mathbf{\Sigma}_{v}\end{bmatrix}  \right), \tag{2}\]  
with \(n\)-vectors \(\mathbf{\mu}_{r}\) and \(\mathbf{\mu}_{\
u}\), \((n\times n)\)-matrix \(\mathbf{\Gamma}_{v|\mathbf{\tilde{r}}}\), and where \(\mathbf{\Sigma}_{\mathbf{\tilde{r}}}\), \(\mathbf{\Sigma}_{v}\), and \(\mathbf{\Sigma}_{v|\mathbf{\tilde{r}}}\) are all three covariance \((n\times n)\)-matrices with \(\mathbf{\Sigma}_{\mathbf{\tilde{r}}}=\mathbf{\Gamma}_{v|\mathbf{\Sigma}_{\mathbf{\tilde{r}}}} \mathbf{\Gamma}_{v|\mathbf{\tilde{r}}}^{T}+\mathbf{\Sigma}_{v|\mathbf{\tilde{r}}}\). Define a selection set \(A\subset\mathbb{R}^{n}\) of dimension \(n\) and let \(\mathbf{r}_{0}=[\mathbf{\tilde{r}}|\mathbf{\
u}\in A]\); then, \(\mathbf{r}_{0}\) is in the class of selection-Gaussian distribution and its pdf is,  
\[f(\mathbf{r}_{0}) =[\Phi_{n}(A;\mathbf{\mu}_{\
u},\mathbf{\Sigma}_{\
u})]^{-1} \tag{3}\] \[\times\Phi_{n}(A;\mathbf{\mu}_{\
u}+\mathbf{\Gamma}_{v|\mathbf{\tilde{r}}}( \mathbf{r}_{0}-\mathbf{\mu}_{\mathbf{\tilde{r}}}),\mathbf{\Sigma}_{v|\mathbf{\tilde{r}}})\times \varphi_{n}(\mathbf{r}_{0};\mathbf{\mu}_{\mathbf{\tilde{r}}},\mathbf{\Sigma}_{\mathbf{\tilde{r}}}).\]  
Note that the class of Gaussian distributions constitutes a subset of the class of selection-Gaussian distributions with \(\mathbf{\Gamma}_{v|\mathbf{\tilde{r}}}=0\times\mathbf{I}_{n}\). The dependence in \([\mathbf{\tilde{r}},\mathbf{\
u}]\) represented by \(\mathbf{\Gamma}_{v|\mathbf{\tilde{r}}}\) and the selection subset \(A\) are crucial user-defined parameters with the latter being temporally constant. The selection-Gaussian model may represent multimodal, skewed, and/or peaked marginal distributions, see [21].