How can we convert values proportional to probabilities to Bernoulli probabilities?

Question

According to Wikipedia, the parameter in a Bernoulli distribution should be $0<p<1$.

I am reading this famous paper proposing Hierarchical Dirichlet Process, and on page 1580, A.6 and the sentence right after it, they state

$$q(s_j|\alpha_0)\propto\Bigg(\frac{n_{j..}}{\alpha_0}\Bigg)^{s_j}$$

but $n_{j..}$ is an integer which can take any large value, while $\alpha_0$ is a real value which almost always is way below $n_{j..}$. Also $s_j$ is a binary value either 0, or 1.

How can I convert it to an actual Bernoulli?

Answer 1

Since $p(1)=p$ and $p(0)=1-p$ are both proportional to a known expression* (the unscaled probabilities, $u(i)=c.p(i)$, with the same unknown constant of proportionality, $c$) and you know the $p(i)$ values must add to $1$, then $u(0)+u(1)=c$.

Which is to say $p(i) = \frac{u(i)}{u(0)+u(1)},\: i=0,1$.

(This notion is widely used in Bayesian statistics with many kinds of discrete variables.)

Note that $u(0)=1$ (always, since the power is $s_i=0$), so $p(0) = 1/(1+u(1))$ and $p(1)=u(1)/(1+u(1))$

*(these $p$'s are $q(s_i|\alpha_0)$ in the paper, for $s_i=0$ and $1$ respectively)