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understanding Understanding the E step of EM for GMM

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I'm reading this chapter about EM (9.3.1) of the book "Pattern Recognition and Machine Learning".

I understand the basic EM algorithm for GMM, but I'm having some problems understanding the probabilistic interpretation of the E step.

Most of the following formulas make sense to me except 9.39. Can someone please explain it for me? Thanks a lot.

It seems that it is in the form of $$E[z_{nk}]=\frac{\sum_{z_{nk}}z_{nk}f(n,k)}{normalizer}$$ where the numerator is a summation over two possible states of $z_{nk}$ (0 and 1), so it becomes $\pi_kN(x_n|\mu_k,\Sigma_k)$ in the second step, am I right so far?

If so, what are we summing over here in the denominator? 0s and 1s of all possible $j$s (but the result doesn't quite match)? it seems to make more sense if the denominator is the normalized zeroth momentWhy (in the same form$f(n,k)=[\pi_kN(x_n|\mu_k,\Sigma_k)]^{z_{nk}}$ is used as the denominator in the second step).an unnormalized probability?

It would make more sense if the denominator is the unnormalized zeroth moment and the numerator is the unnormalized first moment, did I understand it correctly?

enter image description here

enter image description here

I'm reading this chapter about EM (9.3.1) of the book "Pattern Recognition and Machine Learning".

I understand the basic EM algorithm for GMM, but I'm having some problems understanding the probabilistic interpretation of the E step.

Most of the following formulas make sense to me except 9.39. Can someone please explain it for me? Thanks a lot.

It seems that it is in the form of $$E[z_{nk}]=\frac{\sum_{z_{nk}}z_{nk}f(n,k)}{normalizer}$$ where the numerator is a summation over two possible states of $z_{nk}$ (0 and 1), so it becomes $\pi_kN(x_n|\mu_k,\Sigma_k)$ in the second step, am I right so far?

If so, what are we summing over here in the denominator? 0s and 1s of all possible $j$s (but the result doesn't quite match)? it seems to make more sense if the denominator is the normalized zeroth moment (in the same form as the denominator in the second step).

enter image description here

enter image description here

I'm reading this chapter about EM (9.3.1) of the book "Pattern Recognition and Machine Learning".

I understand the basic EM algorithm for GMM, but I'm having some problems understanding the probabilistic interpretation of the E step.

Most of the following formulas make sense to me except 9.39. Can someone please explain it for me? Thanks a lot.

It seems that it is in the form of $$E[z_{nk}]=\frac{\sum_{z_{nk}}z_{nk}f(n,k)}{normalizer}$$ where the numerator is a summation over two possible states of $z_{nk}$ (0 and 1), so it becomes $\pi_kN(x_n|\mu_k,\Sigma_k)$ in the second step, am I right so far?

If so, what are we summing over here in the denominator? 0s and 1s of all possible $j$s (but the result doesn't quite match)? Why $f(n,k)=[\pi_kN(x_n|\mu_k,\Sigma_k)]^{z_{nk}}$ is used as an unnormalized probability?

It would make more sense if the denominator is the unnormalized zeroth moment and the numerator is the unnormalized first moment, did I understand it correctly?

enter image description here

enter image description here

6 added 137 characters in body
source | link

I'm reading this chapter about EM (9.3.1) of the book "Pattern Recognition and Machine Learning".

I understand the basic EM algorithm for GMM, but I'm having some problems understanding the probabilistic interpretation of the E step.

Most of the following formulas make sense to me except 9.39. Can someone please explain it for me? Thanks a lot.

It seems that it is in the form of $$E[z_{nk}]=\frac{\sum_{z_{nk}}z_{nk}f(n,k)}{normalizer}$$ where the numerator is a summation over two possible states of $z_{nk}$ (0 and 1), so it becomes $\pi_kN(x_n|\mu_k,\Sigma_k)$ in the second step, am I right so far?

If so, what are we summing over here in the denominator? 0s and 1s of all possible $j$s (but the result doesn't quite match)? it seems to make more sense if the denominator is the normalized zeroth moment (in the same form as the denominator in the second step).

enter image description here

enter image description here

I'm reading this chapter about EM (9.3.1) of the book "Pattern Recognition and Machine Learning".

I understand the basic EM algorithm for GMM, but I'm having some problems understanding the probabilistic interpretation of the E step.

Most of the following formulas make sense to me except 9.39. Can someone please explain it for me? Thanks a lot.

It seems that it is in the form of $$E[z_{nk}]=\frac{\sum_{z_{nk}}z_{nk}f(n,k)}{normalizer}$$ where the numerator is a summation over two possible states of $z_{nk}$ (0 and 1), so it becomes $\pi_kN(x_n|\mu_k,\Sigma_k)$ in the second step, am I right so far?

If so, what are we summing over here in the denominator? 0s and 1s of all possible $j$s (but the result doesn't quite match)?

enter image description here

enter image description here

I'm reading this chapter about EM (9.3.1) of the book "Pattern Recognition and Machine Learning".

I understand the basic EM algorithm for GMM, but I'm having some problems understanding the probabilistic interpretation of the E step.

Most of the following formulas make sense to me except 9.39. Can someone please explain it for me? Thanks a lot.

It seems that it is in the form of $$E[z_{nk}]=\frac{\sum_{z_{nk}}z_{nk}f(n,k)}{normalizer}$$ where the numerator is a summation over two possible states of $z_{nk}$ (0 and 1), so it becomes $\pi_kN(x_n|\mu_k,\Sigma_k)$ in the second step, am I right so far?

If so, what are we summing over here in the denominator? 0s and 1s of all possible $j$s (but the result doesn't quite match)? it seems to make more sense if the denominator is the normalized zeroth moment (in the same form as the denominator in the second step).

enter image description here

enter image description here

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