2
$\begingroup$

I am studying gaussian mixture models. The first step defines the following equation.

enter image description here

They then proceed to marginalize $z_n$ out

enter image description here

My question is, how did they arrive at that equation ? Where did the product over $K$ go to ? Marginalizing over $z_n$ means to sum over $z_n$. But there was a multiplication over $n$ in the original equation. What happened to it ?

$\endgroup$
1
$\begingroup$

The first equation, $p(\textbf{X, z} \mid \bf{\theta})$ refers to the joint likelihood function of all observed data, $\textbf{X} = x_1, x_2, \ldots, x_N$ and the latent variables, $\textbf{z} = z_1, z_2, \ldots, z_N$, given the model parameters, $\bf{\theta} \equiv \{\bf{\mu, \Sigma, \pi}\}$ hence the first equation has a product over $N$ and $K$.

The second equation refers to the likelihood of a single observation, $p(x_n \mid \bf{\theta})$. It comes from the following intuition,

Given the latent variable assignment, $z_n = k$, the given observation $x_n$ is drawn from the $k^{th}$ Gaussian component of the mixture model.

$$ p(x_n \mid z_n = k, \theta) = \mathcal{N}(\mu_k, \Sigma_k) $$

Now, for a given observation, if you marginalize $z_n$, you get

$$ \begin{align} p(x_n \mid \theta) &= \sum_{k=1}^{K} p(z_n = k) \times p(x_n \mid z_n = k, \bf{\theta}) \\ &= \sum_{k=1}^{K} \pi_k \times p(x_n \mid z_n = k, \bf{\theta}) \end{align} $$

Hope that helps!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.