###### Abstract
Note that with such a design, if the distance between the true conditional and the chosen family is non-zero, then the approximation can never converge towards the true conditional. In addition to intractable E steps, EM procedures also commonly face a second issue: their convergence, when guaranteed, can be towards any maximum. This theoretical remark has crucial numerical implications. Indeed, most of the time, convergence is reached towards a sub-optimal local maximum, usually very dependent on the initialisation. To tackle this issue and improve the solutions of the algorithm, other types of, usually deterministic, approximations of the E step have been proposed. One notable example is the tempering (or \"annealing\") of the conditional probability function used in the E step. Instead of replacing an intractable problem by a tractable one, the tempering approximation is used to find better local maxima of the likelihood profile during the optimisation process, in the spirit of the Simulated Annealing of [PERSON] et al. [28] and the Parallel Tempering (or Annealing MCMC) of [PERSON] and [PERSON] [29], [PERSON] and [PERSON] [30]. The deterministic annealing EM was introduced by [PERSON] and [PERSON] [31] with a decreasing temperature profile; another temperature profile was proposed by [PERSON] and [PERSON] [32]. Contrary to most of the studies on stochastic approximations, these two works do not provide theoretical convergence guarantees for the proposed tempered methods, which, as a consequence, does not provide insight on the choice of the temperature scheme. Moreover, the tempered methods do not allow the use of the EM in case of an intractable E step. In their tempered SAEM algorithm, [PERSON] and [PERSON] [10] combine the stochastic and tempering approximations, which allows the SAEM to run, even with an intractable E step, while benefiting from the improved optimisation properties brought by the tempering. In addition, theoretical convergence guarantees are provided. However, this method is once again stochastic and parametric. Overall, most of the literature on approximated E steps focuses on stochastic approximations that estimate intractable conditional probability functions. The few purely deterministic approximations proposed, such as the tempered/annealed EM, are used for other purposes, improving the optimisation procedure, and lack convergence guarantees. In this paper, we prove that, with mild model assumptions and regularity conditions on the approximation, any Deterministic Approximate EM benefits from the state of the art theoretical convergence guarantees of [2,4,5]. This theorem covers several already existing methods, such as the tempered EM, and paves the way for the exploration of new ideas. To illustrate this, we study a small selection of methods with practical applications that verify our theorem's hypotheses. First, for E steps without closed form, we propose to use Riemann sums to estimate the intractable normalising factor. This \"Riemann approximation EM\" is a deterministic, less parametric, alternative to the MC-EM and its variants. It is suited to the small dimension cases, where the analytic estimation of the intractable integral is manageable, and MC estimation unnecessary. Second, we prove that the deterministic annealed EM (or \"tempered EM\") of [PERSON] and Nakano [31] is a member of our general deterministic class as well. We prove that the convergence guarantees are achieved with almost no condition of the temperature scheme, justifying the use of a wider range of temperature profile than those proposed by [31] and [32]. Finally, since the Riemann and tempered approximations are two separate methods that fulfil very different practical purposes, we also propose to associate the two approximations in the \"tempered Riemann approximation EM\" when both their benefits are desired. In Section 2.1, we introduce our general class of deterministic approximated versions of the EM algorithm and prove their convergence guarantees, for models of the exponential family.