## 2 Materials and Methods
### Tempered EM
#### 2.3.4 Examples of Models That Verify the Conditions  
Gaussian Mixture Model  
Despite being one of the most common models the EM is applied to, the GMM have many known irregularities and pathological behaviours, see [37, 38]. Although some recent works, such as [19, 39], tackled the theoretical analysis of EM for GMM, none of the convergence results for the traditional EM and its variants proposed by [2, 4, 5, 7] apply to the GMM. The hypothesis that the GMM fail to verify is the condition that the level lines have to be compact (\(M2\)\((d)\) in this paper). All is not lost however for the GMM, indeed, the model verifies all the other hypotheses of the general Theorem 1 as well as the tempering hypotheses of Theorem 3. Moreover, in this paper as in the others, the only function of the unverified hypothesis \(M2\)\((d)\) is to ensure in the proof that the EM sequence stays within a compact. The latter condition is the actual relevant property to guarantee convergence at this stage of the proof. This means that, in practice, if an tempered EM sequence applied to a GMM is observed to remain within a compact, then all the conditions for convergence are met, Theorem 3 applies, and the sequence is guaranteed to converge towards a critical point of the likelihood function. In the following, we provide more details on the GMM likelihoods and the theorem hypotheses they verify. First, note that the GMM belongs to the exponential family with the complete likelihood:  
\[ \begin{split} h(z;\theta)=\prod_{i=1}^{N}\sum_{k=1}^{K}exp\Bigg{(} \frac{1}{2}\Big{(}&-(x^{(i)}-\mu_{k})^{T}\Theta_{k}(x^{(i)}-\mu_{ k})\\ &+\ln(|\Theta_{k}|)+2\ln(\pi_{k})-p\ln(2\pi)\Big{)}\Bigg{)}\mathds{1}_{z^{(i)}=k}\,, \end{split}  \tag{8}\]  
and the observed likelihood:  
\[ \begin{split} g(\theta)=\prod_{i=1}^{N}\sum_{k=1}^{K}exp\Bigg{(} \frac{1}{2}\Big{(}&-(x^{(i)}-\mu_{k})^{T}\Theta_{k}(x^{(i)}-\mu_{ k})\\ &+\ln(|\Theta_{k}|)+2\ln(\pi_{k})-p\ln(2\pi)\Big{)}\Bigg{)}\,. \end{split}  \tag{9}\]  
This is an exponential model with parameter:  
\[\theta:=\Big{(}\{\pi_{k}\}_{k=1}^{K},\{\mu_{k}\}_{k=1}^{K},\{\Theta_{k}\}_{k= 1}^{K}\Big{)}\in\Big{\{}\{\pi_{k}\}_{k}\in[0,1]^{K}\bigg{|}\sum_{k}\pi_{k}=1 \Big{\}}\otimes\mathbb{R}^{p\times K}\otimes S_{p}^{++K}.\]  
where \(S_{p}^{++}\) is the cone of symmetric positive definite matrices of size \(p\). The verification of conditions \(M1\)-\(3\) for the GMM (except \(M2\)\((d)\) of course) is a classical exercise since these are the conditions our Theorem shares with any other EM convergence result on the exponential family. We focus here on the hypotheses specific to our Deterministic Approximate EM. _Condition on \(\int_{z}p_{\theta}^{\alpha}(z)dz\)._ Let \(\alpha\in\mathbb{R}_{+}^{*}\), in the finite mixture case, the integrals on \(z\) are finite sums:  
\[\int_{z}p_{\theta}^{\alpha}(z)dz=\sum_{k}p_{\theta}^{\alpha}(z=k)\,.\]  
Which is continuous in \(\theta\) since \(\theta\mapsto p_{\theta}(z=k)=h(z=k;\theta)/g(\theta)\) is continuous. Hence  
\[\forall\alpha\in\mathbb{R}_{+}^{*},\quad\sup_{\theta\in K}\ \int_{z}p_{\theta}^{ \alpha}(z)dz<\infty\,.\]  
_Condition on \(\int_{z}S_{u}^{2}(z)p_{\theta}^{\alpha}(z)dz\)._ The previous continuity argument is still valid. Poisson Count with Random Effect  
This model is discussed in [7], the authors prove, among other things, that this model verifies the conditions \(M1-3\). Here is a brief description of the model: the hidden variables \(\left\{z^{(i)}\right\}_{i=1}^{N}\) are distributed according to an autoregressive process of order 1: \(z^{(i)}:=az^{(i-1)}+\sigma\epsilon_{i}\), with \(|a|<1\), \(\sigma>0\) and the \(\left\{\epsilon^{(i)}\right\}_{i=1}^{N}\) are iid standard gaussian. Conditionally to \(\left\{z^{(i)}\right\}_{i=1}^{N}\), the observed variables \(x^{(i)}\) are independent Poisson variables with parameter \(\exp(\theta+z^{(i)})\).