How to derive Probabilities from likelihood ratios when size of set is greater than 2?

Question

I am defining a set $\mathbf{Z} = [p,q,r,s]$ such that $Pr(p)+Pr(q)+Pr(r)+Pr(s)=1$.

I have access to the likelihoods that are defined as follows

\begin{align} \lambda_p&=\log \frac{Pr(x=p)}{Pr(x=s)}, \hspace{2mm} \lambda_q=\log \frac{Pr(x=q)}{Pr(x=s)}\\[15pt] \lambda_r&=\log \frac{Pr(x=r)}{Pr(x=s)}, \hspace{2mm} \lambda_s=\log \frac{Pr(x=s)}{Pr(x=s)}=0 \end{align}

How can I compute $Pr(x=p)$ from this information.

Note

For $\mathbf{Z'} = [m,n]$, we have

\begin{align} \lambda&=\log \frac{Pr(x=m)}{Pr(x=n)}=\log \frac{Pr(x=m)}{1-Pr(x=m)}\\ \implies Pr(x=m) &= \frac{1}{1+e^{-\lambda}} \end{align}

I am trying to derive similar expression when size of set is greater than 2.

Answer 1

We have

$$ \mathrm e^{\lambda_p}+\mathrm e^{\lambda_q}+\mathrm e^{\lambda_r}+\mathrm e^{\lambda_s}=\frac1{\operatorname{Pr}(x=s)} $$

and

$$ \mathrm e^{\lambda_p}=\frac{\operatorname{Pr}(x=p)}{\operatorname{Pr}(x=s)}\;, $$

so

$$ \operatorname{Pr}(x=p)=\frac{\mathrm e^{\lambda_p}}{\mathrm e^{\lambda_p}+\mathrm e^{\lambda_q}+\mathrm e^{\lambda_r}+\mathrm e^{\lambda_s}}\;. $$