On-Line Expectation–Maximization Algorithm for latent Data Models

Abstract

We propose a generic on-line (also sometimes called adaptive or recursive) version of the expectation-maximization (EM) algorithm applicable to latent variable models of independent observations. Compared with the algorithm of Titterington, this approach is more directly connected to the usual EM algorithm and does not rely on integration with respect to the complete-data distribution. The resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the Kullback-Leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate, i.e. that of the maximum likelihood estimator. In addition, the approach proposed is also suitable for conditional (or regression) models, as illustrated in the case of the mixture of linear regressions model. Copyright (c) 2009 Royal Statistical Society.

Online model estimation of ultra-wideband TDOA measurements for mobile robot localization

“…Since our online approach builds upon the EM formalism presented by Cappé et al [4], our notation in the following derivations is closely related to that of the latter. If we write our parameter vector θ as…”

“…Figure 5 shows an example of online parameter estimation using our EM framework, on a data set comprising 5000 samples. Cappé et al [4] resolve issues due to the dependency on the initialization by updating onlyŝ for a number of first iterations. Thus, we use the first 100 observations to build up an estimate ofŝ (100) , before calculating the first parameter estimateθ (101) .…”

“…Variations of γ (k) = 1/k have shown to produce good convergence speed [4] -typically, the choice γ defines the trade-off between adaptability and stability of the estimate. In practice, one should bound γ from below to allow the estimation algorithm to continuously adapt to a changing environment.…”

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“…The online EM was first introduced in [25]. It was subsequently generalized in [26], where the authors present a proof convergence and a proposition of asymptotic equivalency to natural gradient ascent. Moreover, the algorithm performs simple and efficient updates, making good use of the sufficient statistics of the exponential family, so as to avoid redundant calculations.…”

Prospects for Bandit Solutions in Sensor Management

“…For instance, if the coefficients of the reward functions evolve at each time-step according to a linear model, then its parameters can be estimated through the online expectation maximization algorithm [20]. In the dynamic MABC problem, an accurate model of the dynamics can improve not only the estimation accuracy of the reward function parameters, but also provide valuable information for the action selection process.…”