# What is a $\propto$ update in mean-field approximation?

I am trying to understand some notation used in papers about Bayesian variational inference. In some papers that use mean-field approximation to fit a probabilistic model, they describe coordinate ascent algorithms which have updates with the symbol $\propto$ for the closed-form update of some parameters.

From what I've seen, it seems these $\propto$ always tend to be $\propto \: exp( . )$

For example these: (Gopalan, Prem K., Laurent Charlin, and David Blei. "Content-based recommendations with poisson factorization." Advances in Neural Information Processing Systems. 2014., page 5)

(Cemgil, Ali Taylan. "Bayesian inference for nonnegative matrix factorisation models." Computational intelligence and neuroscience 2009 (2009)., page 5)

And I'm really wondering what would that $\propto$ translate to in an algorithmic procedure.

• $\propto$ means "proportional to", usually used when a constant factor term has been dropped from one or both sides. Commented May 3, 2018 at 21:50
• Thanks, but I meant to ask, how does that work in terms of an algorithm? Let's say you were to design a program that follows an iterative update a := exp(b) and then something else - that means you assign the right hand side to a at each iteration. But since $\propto$ is proportional and not equal, how would that work in an algorithm or computer software? Commented May 4, 2018 at 7:34

As a side note, some authors prefer to write log-update equations e.g., $$\log q(Z) = \mathbb{E}[\log p(Z,W)] + \text{const},$$ where $\text{const}$ is an additive constant that can be interpreted the same way.
• Thanks for answering! That's also my first thoughy when I see $\propto$, but it turns out it means something different here: calculate the expression in the right, then divide it by the sum of all the elements (calculated like that) in that row so that rows sum to 1. Commented Sep 15, 2018 at 21:52