## 2 Materials and Methods
### Tempered Riemann Approximation EM  
Context, Theorem and Proof  
The Riemann approximation of Section 2.2 makes the EM computations possible in hard cases, when the conditional distribution has no analytical form for instance. It is an alternative to the many stochastic approximation methods (SAEM, MCMC-SAEM, etc.) that are commonly used in those cases. The tempering approximation of Section 2.3 is used to escape the initialisation by allowing the procedure to explore more the likelihood profile before committing to convergence. We showed that both these approximation are particular cases of the wider class of Deterministic Approximate EM, introduced in Section 2.1. However, since they fulfil different purposes, it is natural to use them in coordination and not as alternatives of one another. In this section, we introduce another instance of the Approximate EM: a combination of the tempered and Riemann sum approximations. This \"tempered Riemann approximation EM\" (tmp-Riemann approximation) can compute EM steps when there is no closed form thanks to the Riemann sums as well as escape the initialisation thanks to the tempering. For a bounded latent variable \(z\in[0,1]\), we define the approximation as: \(\hat{p}_{n,\theta}(z):=h(\lfloor nz\rfloor/n;\theta)^{\frac{1}{n_{1}}}/\int_{ z^{\prime}}h(\lfloor nz^{\prime}\rfloor/n;\theta)^{\frac{1}{n_{n}}}dz^{\prime}\), for a sequence \(\{T_{n}\}_{n}\in(\mathbb{R}_{+}^{*})^{N}\), \(T_{n}\underset{n\rightarrow\infty}{\longrightarrow}1\). In the following Theorem, we prove that the tempered Riemann approximation EM verifies the applicability conditions of Theorem 1 with no additional hypothesis from the regular Riemann approximation EM covered by Theorem 2. Theorem 4: _Under conditions \(M1-3\) of Theorem 1, and when \(z\) is bounded, the (Stable) Approximate EM with \(\hat{p}_{n,\theta}(z):=\frac{h(\lfloor nz\rfloor/n;\theta)^{\frac{1}{n_{1}}}}{ \int_{z^{\prime}}h(\lfloor nz^{\prime}\rfloor/n;\theta)^{\frac{1}{n_{2}}}dz^{ \prime}}\), which we call \"tempered Riemann approximation EM\", verifies the remaining conditions of applicability of Theorem 1 as long as \(z\mapsto S(z)\) is continuous and \(\{T_{n}\}_{n}\in(\mathbb{R}_{+}^{*})^{N}\), \(T_{n}\underset{n\rightarrow\infty}{\longrightarrow}1\)._  
Proof. : This proof of Theorem 4 is very similar to the proof of Theorem 2 for the regular Riemann approximation EM. The first common element is that for the tempered Riemannapproximation EM, the only remaining applicability condition of the general Theorem 1 to prove is also:  
\[\forall\text{compact}\ K\subseteq\Theta,\ \sup_{\theta\in K}\ \int_{z}(\bar{p}_{ \theta,n}(z)-p_{\theta}(z))^{2}dz\underset{n\to\infty}{\longrightarrow}0\,.\]  
In the proof of Theorem 2, we proved that having the uniform convergence of the approximated complete likelihood \(\{\bar{h}_{n}\}_{n}\) towards the real \(h\) - with both \(\bar{h}_{n}(z;\theta)\) and \(h(z;\theta)\) uniformly bounded - was sufficient to fulfil this condition. Hence, we prove In this section, that these sufficient properties still hold, even with the tempered Riemann approximation, where \(\bar{h}_{n}(z;\theta):=h(\lfloor nz\rfloor/n;\theta)^{\frac{1}{1n}}\). We recall that \(h(z;\theta)\) is uniformly continuous on the compact set \([0,1]\times K\), and verifies:  
\[0<m\leq h(z;\theta)\leq M<\infty\,.\]  
where \(m\) and \(M\) are constants independent of \(z\) and \(\theta\). Since \(T_{n}>0,\ T_{n}\underset{n\to\infty}{\longrightarrow}1\), then the sequence \(\{1/T_{n}\}_{n}\) is bounded. Since \(\bar{h}_{n}(z;\theta)=h(\lfloor nz\rfloor/n;\theta)^{\frac{1}{1n}}\), with \(0<m\leq h(\lfloor nz\rfloor/n;\theta)\leq M<\infty\) for any \(z,\theta\) and \(n\), then we also have:  
\[0<m^{\prime}\leq\bar{h}_{n}(z;\theta)\leq M^{\prime}<\infty\,,\]  
with \(m^{\prime}\) and \(M^{\prime}\) constants independent of \(z,\theta\) and \(n\).