How can I use ratios to set priors on multinomial probabilities? I have a vector, $k$, that determines allocation to five pools.
I'd like to set priors on these probabilities, and I can provide informative priors on a few of the ratios, e.g.:
$$ \frac{k1}{k2} \sim \text{beta}(6,1)$$
$$ \frac{k1}{k3} \sim \text{beta}(2,12)$$
$$ \frac{k1+k2+k3}{k4+k5} \sim \text{beta}(50,1)$$
$$ \frac{k4}{k5} \sim \text{beta}(1,1)$$
The fact that these values are all correlated makes it unclear how I can start with these priors and then sample $k$ from a multinomial distribution.
Are there any good papers or common methods for doing so?
 A: Check out the Dirichlet distribution: https://en.wikipedia.org/wiki/Dirichlet_distribution
It's a generalization of the Beta distribution and is ideal for modeling beliefs about multinomial probabilities.
A: If you just want to sample $k$, then this is easy from the distributions given (plus a distribution for $k1$).  The procedure is:


*

*Draw $k1$.

*Draw $b2 \sim {\rm beta}(6,1)$ and set $k2 = \frac{k1}{b2}$.

*Draw $b3 \sim {\rm beta}(2,12)$ and set $k3 = \frac{k1}{b3}$.

*Draw $b4 \sim {\rm beta}(50,1)$ and $b5 \sim {\rm beta}(1,1)$ then solve the system
$$
k4+k5 = \frac{k1+k2+k3}{b4}
\\
k5 = \frac{k4}{b5}
$$
whose solution is $k5 = \frac{k1+k2+k3}{b4(1+b5)}$, $k4 = b5 k5$.

*Finally, if any of the $k$'s are larger than 1, reject the sample and start over.\


You could avoid the rejection step by noting that all of the later $k$'s scale linearly with $k1$.  So once you've got $(b2, ..., b5)$, find the range of $k1$ values that ensure everyone is less than 1 and draw $k1$ within this range.
