How to read this symbol "$\varpropto$" in the following equation and what is its meaning?

Question

I stumbled upon the following equation in a research paper and I don't know how to read the symbol "$\varpropto$" in the context of the equation. Is it read "proportional to"? If so, what does it mean in the following equation?

$$p(e\mid a,f) = \frac{p(e,a,f)}{p(a,f)}\varpropto p(e\mid a)\cdot p(f\mid e,a)$$

I don't have any idea what "proportional to" would mean in the context of this equation if that is what the symbol "$\varpropto$" means here. Please clarify.

Answer 1

Yes, its means proportional to. Observe that: \begin{align*} \dfrac{p(e, a, f)}{p(a,f)} &= \dfrac{p(e,a)p(f \mid e,a)}{p(a,f)} \\ &= \dfrac{p(a)p(e \mid a)p(f \mid e,a)}{p(a,f)} \\ &= K \cdot p(e \mid a)p(f \mid e,a) \\ \end{align*} where $K = \dfrac{p(a)}{p(a,f)}$ is a constant.

Answer 2

Hint:

Check out Bayes' theorem:

$$P(A\mid B) \propto P(A) \cdot P(B\mid A) $$