## 4 Discussion and Conclusions  
We proposed the Deterministic Approximate EM class to bring together the many possible deterministic approximations of the E step. We proved a unified Theorem, with mild conditions on the approximation, which ensures the convergence of the algorithms in this class. Then, we showcased members of this class that solve the usual practical issues of the EM algorithm. For intractable E steps in low dimension, we introduced the Riemann approximation EM, a less parametric and deterministic alternative to the extensive family of MC-EM. We showed on an empirical intractable example how the Riemann approximation EM was able to increase the likelihood and recover every parameter in a satisfactory manner with its simplest design, and no hyper parameter optimisation.  
Second, we studied the tempered EM, introduced by [PERSON] and [PERSON] [31] to escape the attractive sub-optimal local extrema of non-convex objectives. We proved that tmp-EM is a specific case of the Deterministic Approximate EM, benefiting from the convergence property as long as the temperature profile converges towards 1. This mild condition justifies the use of many more temperature profiles than the ones tried in [31; 32]. To  
Figure 7: Results over many simulations of the Riemann EM and tmp-Riemann EM on the Beta-Gaussian model. The tempered Riemann EM reaches relative errors on the real parameters that are orders of magnitude below the Riemann EM with no temperature. The likelihood reached is also lower with the tempering.  
illustrate the interest of complex, non-monotonous, temperature profiles, we demonstrated on experiments with adversarial initial positions the superiority of an _oscillating_ profile over a simple _decreasing_ one.  
Finally, we added the Riemann approximation in order to apply the tempering in intractable cases. We were then able to show that the tmp-Riemann approximation massively improved the performances of the Riemann approximation, when the initialisation is ambiguous.  
Future works will improve both methods. The Riemann approximation will be generalised to be applicable even when the latent variable is not bounded, and an intelligent slicing of the integration space will improve the computational performances in high dimension. Regarding the tempered EM, since the theory allows the usage of any temperature profile, the natural next step is to look for efficient profiles with few hyper-parameters for fast tuning. Afterwards, implementing an adaptive tuning of the temperature parameters during the procedure will remove the necessity for preliminary grid search altogether.  
The following are available at [[https://www.mdpi.com/article/10.3390/a15030078/s1](https://www.mdpi.com/article/10.3390/a15030078/s1)]([https://www.mdpi.com/article/10.3390/a15030078/s1](https://www.mdpi.com/article/10.3390/a15030078/s1)).  
Conceptualization, T.L. and S.A.; methodology, T.L. and S.A.; software, T.L.; validation, T.L. and S.A.; formal analysis, T.L. and S.A.; investigation, T.L.; resources, S.D. and S.A.; data curation, T.L.; writing--original draft preparation, T.L.; writing--review and editing, T.L., S.D. and S.A.; visualization, T.L.; supervision, S.D and S.A.; project administration, S.A.; funding acquisition, S.D and S.A. All authors have read and agreed to the published version of the manuscript.  
The research leading to these results has received funding from the European Research Council (ERC) under grant agreement No 678304, European Union's Horizon 2020 research and innovation program under grant agreement No 666992 (EuroPOND) and No 826421 (TVB-Cloud), and the French government under management of Agence Nationale de la Recherche as part of the \"Investissements d'avenir\" program, reference ANR-19-P3 IA-0001 (PRAIRIE 3 IA Institute) and reference ANR-10-IAIHU-06 (IHU-A-ICM).  
Not applicable.  
The authors declare no conflict of interest.