I've been trying to get my head around Expectation maximization algorithms, and I thought I'd start simple. I found this 3-coin example here: http://cs.dartmouth.edu/~cs104/CS104_11.04.22.pdf I understand the calculation of all of the probabilities but I don't understand how the recalculation of lambda, p1 and p2 is actually done. (Page 18)

I understand how the maximization of a log-likelihood/likelihood function by differentiation works, but can't figure out the recalculation method here.

Can anyone explain why the recalculation of lambda, p1 and p2 take the form they do?

  • 5
    $\begingroup$ Please make this question a little bit more self-contained. $\endgroup$
    – frafl
    Commented Jun 21, 2013 at 11:49
  • $\begingroup$ See this question. It covers that exact document. $\endgroup$ Commented Jul 1, 2013 at 21:59
  • $\begingroup$ @AndrásSalamon As far as I can tell, Nicholas' answer should take care of this question here? $\endgroup$
    – Raphael
    Commented Jul 16, 2013 at 10:27

1 Answer 1


From mcdowella's answer to IcySnow's question Expectation Maximization coin toss examples, Michael Collins' EM Algorithm (pdf) paper presets the "Three Coins Example" in section 3.1 (pdf page 8-9) and explains the steps for recalculating the parameters $\lambda, p_1, p_2$ in accordance with the EM Algorithm.

The derivation is too lengthy to type up here, but to summarize, you'll formulate the conditional expectation in terms of the previous iteration of parameter values and then differentiate the expectation with respect to each parameter from the previous iteration then perform algebra to obtain the maximum after setting the derivative to zero.


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