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I recently read this primer that does a great job of explaining the principles of the Expectation-Maximization (EM) "algorithm."

However, I'm confused about how they calculated the probability distributions of each possible "completion" (coin) that we see in this figure after the "E-step":

enter image description here

My attempt:

I decided to try and use a simpler example. Let's say the coin biases are $\theta_A = 0.8$ and $\theta_B = 0.2$ Also, we only flipped one coin twice and observed the following: $HH$

Now the probability for the $HH$ permutation for coin A is $\frac{4}{5}*\frac{4}{5} = \frac{16}{25}$

And the probability for the $HH$ permutation for coin B is $\frac{1}{5}*\frac{1}{5} = \frac{1}{25}$

So in other words, for every one observation of the $HH$ permutation from coin B, we would expect 16 observations of $HH$ permutation from coin A.

If we assume that these are the only two possible outcomes, the probability of $HH$ permutation from coin A is $\frac{16}{16 + 1}$ and the probability of $HH$ permutation from coin B is $\frac{1}{16 + 1}$


If my approach is correct, is this the general form of the "expectation" step of the EM algorithm:

  • Find the probability of each observation occurring using t = 0 parameters.
  • Find the "relative" (not sure if this is the right term) probabilities to get the distribution
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