Asking for the content of Introduction to EM: Gaussian Mixture Models

[ngvananh2508]

I am sorry. I do not report any issues, but I have one question. I tried to find the answer, but I could not get it.

I understand this formula, but I don't comprehend the product sign in the formal part. Why is the sum sign replaced by the product sign?

Thank you very much for helping me.

[stephens999]

it's a bit tricky to explain in words. But let me say the two expressions are not the same. The likelihood is P(X | mu, sigma, pi) whereas the second expression is P(X, Z | mu, sigma, pi) = P(Z| mu, sigma,pi) p(X|mu, sigma, pi, Z) = prod_i p(Z_i |pi) p(X_i | mu, sigma, Z_i)

So to understand the second expression you need to understand the above, and then you need to understand why p(Z_i |pi) p(X_i | mu, sigma, Z_i) = prod_k (...) where (...) is the expression after the product sign above.

[ngvananh2508]

Finally, I can understand this formula. Does this expression p(Z_i |pi) p(X_i | mu, sigma, Z_i) = prod_k (...) contain only 1 term pi_k.N(x_k|mu_k, sigma_k) and the others are 1s?

Thank you very much.

[stephens999]

Yes, you got it!