How to write a proof with the use of $\propto$

Question

I have an issue of style, for example with Bayesian stats:

\begin{align} h(\theta|X) &\propto f(X|\theta) \pi(\theta)\\ & = \theta(1-\theta) . 2\theta & \text{(should I use '$=$' or '$\propto$' here if this is just a substitution?)}\\ & \propto \theta^2(1-\theta)&\\ & etc~etc \end{align}

Is the equals sign stating that the continued line is equal to the LHS of the original equation, or equal to the preceding RHS, in the following line?

If the former, one would strictly keep to using the $\propto$ symbol each line downwards; for the latter, one could see the $\propto$ only used in steps where one was eliminating constant scaling terms.

Answer 1

There is no ambiguity in

$$a=b\propto c=d=e\propto f\propto g\cdots$$

and the net meaning is $a\propto g$.

This is (a little) more informative than would be $$a\propto b\propto c\propto d\propto e\propto f\propto g\cdots$$

The situation is similar to a development like

$$a=b< c=d=e\le f< g\cdots$$

Answer 2

In my copy of Robert's The Bayesian Choice, for example on p. 25, he uses the proportionality sign all the way so that for example, $\pi(\theta|x) \propto f(x|\theta)\pi(\theta) \propto \exp(-\frac{-(x-\theta)^2}{2} - \theta^2/20)$.

So this is at least one data point that the proportion sign is used throughout.