According to this lecture note, Eq. 25 gives the coordinate ascent update for latent variable $z_k$ as follows $$q^*(z_k)\propto\exp(E_{-k}[\log{p(z_k,Z_{-k},x)}])$$ and I understand the derivation for this formula. But in the following Bayesian mixtures of Gaussians example section, try to find the specific update formula for latent variable $z_i$ as the following picture: enter image description here

Eq. 36 is the same as the above Eq. 25, what confuses me is Eq. 38, why omit the terms which don't rely on $z_i$?

I think if we substitute Eq. 37 back into 36, we will have more terms than Eq. 38, like this $$q^*(z_i)\propto\exp(\log\pi_{z_i}+E[\log p(x_i|\mu_{z_i})]+E[x])$$

where $E[x]$ is a short-hand notation for those more terms, I just don't quite understand why we could omit them? Just because it could be re-written as follows, which is like a constant multiplying the $\exp$? $$q^*(z_i)\propto \exp(E[x])\cdot\exp(\log\pi_{z_i}+E[\log p(x_i|\mu_{z_i})])=\alpha\cdot \exp(\log\pi_{z_i}+E[\log p(x_i|\mu_{z_i})])$$


1 Answer 1


Your last equation has just about arrived at the answer. Take that last equation:

$$q^*(z_i)\propto \alpha\cdot \exp(\log\pi_{z_i}+E[\log p(x_i|\mu_{z_i})])$$

and explicitly write the $\propto$ relation like so

$$q^*(z_i) = \frac{\alpha\cdot \exp(\log\pi_{z_i}+E[\log p(x_i|\mu_{z_i})])}{\sum_i \alpha\cdot \exp(\log\pi_{z_i}+E[\log p(x_i|\mu_{z_i})])}$$

then we realize that because it's constant with respect to $z$, $\alpha$ cancels out.

In a more general, if hand-wavy, way: When developing a gradient method, we take the gradient of the objective, and any terms that are constant with respect to the variable of interest have a derivative of zero.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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