Confidence intervals for multinomial proportions

Question

I know that there are a lot of methods to find the confidence intervals for binomial proportions using methods like Agresti-Coull, but is there any important papers mentioning to find confidence intervals for multinomial case?

Answer 1

Not sure about important papers, but there's the obvious traditional conjugate Bayesian updating option (using a Dirichlet prior for the multinomial parameters).

In the binomial case there's of course a direct match between using different Beta priors (a $\text{Beta}(1,0)$ and $\text{Beta}(0,1)$ for the upper and lower limits to get exact binomial confidence intervals, but I'm not sure whether there's a directly corresponding approach for the multinomial. However, a Jeffreys $\text{Beta}(0.5, 0.5)$ prior often performs pretty similar to that, too. The Jeffreys prior for the Multinomial is a $\text{Dirichlet}(0.5, \ldots, 0.5)$ distribution. After observing $y_c$ items falling into categories $c=1,\ldots,C$, the posterior distribution is then given by $\text{Dirichlet}(0.5+y_1, 0.5+y_2, \ldots, 0.5+y_C)$.

If you are then interested in any particular multinomial proport, the marginal distribution for it is just a Beta distribution. E.g. if you are interested in proportion $c$, then it's marginal distribution is a $\text{Beta}(0.5+y_c, \sum_{i\neq c} (0.5+y_i))$ distribution. If you need any more unusual transformations of the proportions, you can always sample from the Dirichlet posterior, do the transformations to the samples, et voilà you've got posterior samples for the transformations of these proportions.

If you need things like covariate adjustments, you can use Bayesian nominal / multinomial regression (e.g. in R the brms package covers this nicely) with similar nice properties for the posterior samples.