Expectation maximisation

A particular optimisation method for statistics for that gets you a maximum likelihood estimate despite various annoyances such as missing data.

Vague description of the algorithm:

We have an experimental process that generates a random vector \(B\cup Y\) according to parameter \(\theta\). We wish to estimate the parameter of interest \(\theta\) by maximum likelihood. However, we only observe i.i.d. samples \(b_i\) drawn from \(B\). The likelihood function of the incomplete data \(L(\theta, b)\) is tedious or intractable to maximise. But the “complete” joint likelihood of both the observed and unobserved components, \(L(\theta, \{b_i\}, y)\), is easier to maximise. Then we are potentially in a situation where expectation maximisation can help.

Call \(\theta^{(k)}\) the estimate of of \(\theta\) at step \(k\). Write \(\ell(\theta, \{b_i\}, y)\equiv\log L(\theta, \{b_i\}, y)\) because we virtually always work in log likelihoods and especially here.

The following form of the algorithm works when the log-likelihood \(\ell(\theta, b, y)\) is linear in \(b\). (Which is equivalent to it being in a exponential family I believe, but should check.)

At time \(k=0\) we start with an estimate of \(\theta^{(0)}\) chosen arbitrarily or by our favourite approximate method.

We attempt to improve our estimate of the parameter of interest by the following iterative algorithm:

  1. “Expectation”: Under the completed data model with joint distribution \(F(b,y,\theta^{(k)})\) we estimate \(y\) as

    \[ y^{(k)}=E_{\theta^{(k)}}[Y|b] \]

  2. “Maximisation”: Solve a (hopefully easier) maximisation problem:

    \[ \theta^{(k+1)}=\operatorname{argmax}_\theta \ell(\theta, b, y^{(k)}) \]

In the case that this log likelihood is not linear in \(b\), you are supposed to instead take

\[ \theta^{(k+1)}=\operatorname{argmax}_\theta E_{\theta^{(k)}}[\ell(\theta, b, Y)|b] \]

In practice this nicety is often ignored.

Even if you do the right thing, EM may not converge especially well, or to the global maximum, but it can be easy and robust to get started with, and at least it doesn’t make things worse.

Literature note – apparently the proofs in Dempster and Laird (1977) are dicey; See the Wu (1983) paper for improved (i.e. correct) versions.

  • A Transparent Interpretation of the EM Algorithm by James Coughlan has a short point

    We write data \(z\), latent variable \(y\), parameter of interest \(\theta\). Then…

    maximizing Neal and Hinton’s joint function of \(\theta\) and a distribution on \(y\) is equivalent to maximum likelihood estimation.

    The key point is to note that maximizing \(\log P(z|\theta)\) over \(\theta\) is equivalent to maximizing

    \[ \log P (z|\theta)-D(\tilde{P}(y)\|P(y|z,\theta)) \]

    jointly over \(\theta\) and \(\tilde{P}(y)\).

    […We rewrite this cost function]

    \[ H(\tilde{P}) + \sum_y\tilde{P}(y)\log \{P(y|z,\theta)P(z|\theta)\}, \]

    where \(H(\tilde{P})=-\sum_y\tilde{P}(y)\log\tilde{P}(y)\) is the entropy of \(\tilde{P}\). This expression is in turn equivalent to

    \[ H(\tilde{P}) +\sum_y\tilde{P}(y)\log P(y,z|\theta), \]

    which is the same as the function \(F(\tilde{P},\theta)\) given in Neal and Hinton. This function is maximized $1 iteratively, where each iteration consists of two separate maximizations, one over \(\theta\) and another \(\tilde{P}\)

  • Dan Piponi, Expectation-Maximization with Less Arbitrariness

My goal is to fill in the details of one key step in the derivation of the EM algorithm in a way that makes it inevitable rather than arbitrary.

Bilmes, Jeff A. 1998. “A Gentle Tutorial of the EM Algorithm and Its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models.” International Computer Science Institute 4 (510): 126. http://lasa.epfl.ch/teaching/lectures/ML_Phd/Notes/GP-GMM.pdf.

Celeux, Gilles, Didier Chauveau, and Jean Diebolt. 1995. “On Stochastic Versions of the EM Algorithm.” Report. https://hal.inria.fr/inria-00074164/document.

Celeux, Gilles, Stephane Chretien, Florence Forbes, and Abdallah Mkhadri. 2001. “A Component-Wise EM Algorithm for Mixtures.” Journal of Computational and Graphical Statistics 10 (4): 697–712. https://doi.org/10.1198/106186001317243403.

Celeux, Gilles, Florence Forbes, and Nathalie Peyrard. 2003. “EM Procedures Using Mean Field-Like Approximations for Markov Model-Based Image Segmentation.” Pattern Recognition 36 (1): 131–44. https://doi.org/10.1016/S0031-3203(02)00027-4.

Delyon, Bernard, Marc Lavielle, and Eric Moulines. 1999. “Convergence of a Stochastic Approximation Version of the EM Algorithm.” The Annals of Statistics 27 (1): 94–128. https://doi.org/10.1214/aos/1018031103.

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B (Methodological) 39 (1): 1–38. http://www.jstor.org/stable/2984875.

Kuhn, Estelle, and Marc Lavielle. 2004. “Coupling a Stochastic Approximation Version of EM with an MCMC Procedure.” ESAIM: Probability and Statistics 8 (September): 115–31. https://doi.org/10.1051/ps:2004007.

Lee, Gyemin, and Clayton Scott. 2012. “EM Algorithms for Multivariate Gaussian Mixture Models with Truncated and Censored Data.” Computational Statistics & Data Analysis 56 (9): 2816–29. https://doi.org/10.1016/j.csda.2012.03.003.

McLachlan, Geoffrey J, and T Krishnan. 2008. The EM Algorithm and Extensions. Hoboken, N.J.: Wiley-Interscience. http://site.ebrary.com/id/10296227.

McLachlan, Geoffrey J., Thriyambakam Krishnan, and See Ket Ng. 2004. “The EM Algorithm.” 2004,24. Humboldt-Universität Berlin, Center for Applied Statistics and Economics (CASE). http://www.econstor.eu/handle/10419/22198.

Miyahara, Hideyuki, Koji Tsumura, and Yuki Sughiyama. 2016. “Relaxation of the EM Algorithm via Quantum Annealing for Gaussian Mixture Models.” In, 4674–9. https://doi.org/10.1109/CDC.2016.7798981.

Navidi, William. 1997. “A Graphical Illustration of the EM Algorithm.” The American Statistician 51 (1): 29–31. https://doi.org/10.1080/00031305.1997.10473582.

Neal, Radford M., and Geoffrey E. Hinton. 1998. “A View of the EM Algorithm That Justifies Incremental, Sparse, and Other Variants.” In Learning in Graphical Models, edited by Michael I. Jordan, 355–68. NATO ASI Series 89. Springer Netherlands. http://machinelearning.wustl.edu/uploads/Main/EM_algorithm.pdf.

Prescher, Detlef. 2004. “A Tutorial on the Expectation-Maximization Algorithm Including Maximum-Likelihood Estimation and EM Training of Probabilistic Context-Free Grammars,” December. http://arxiv.org/abs/cs/0412015.

Roche, Alexis. 2011. “EM Algorithm and Variants: An Informal Tutorial,” May. http://arxiv.org/abs/1105.1476.

Wei, Greg C. G., and Martin A. Tanner. 1990. “A Monte Carlo Implementation of the EM Algorithm and the Poor Man’s Data Augmentation Algorithms.” Journal of the American Statistical Association 85 (411): 699–704. https://doi.org/10.1080/01621459.1990.10474930.

Wu, C. F. Jeff. 1983. “On the Convergence Properties of the EM Algorithm.” The Annals of Statistics 11 (1): 95–103. https://doi.org/10.1214/aos/1176346060.