[FEATURE] Dirichlet-multinomial likelihood for composition data

[ChristineStawitz-NOAA]

Is your feature request related to a problem? Please describe.
A Dirichlet multinomial distribution for composition data can help capture error more accurately

Describe the solution you'd like
A clear and concise description of what you want to happen.

Describe alternatives you've considered
Plain ol multinomial or multinomial robust (with small constant added)

Describe a reference describing the statistical validity of this approach
Thorson et al. 2017. Model-based estimates of effective sample size in stock assessment models using the Dirichlet-multinomial distribution.

Describe if this is needed for a management application
???

Additional context
@iantaylor-NOAA noted in issue 136 that measures of uncertainty specified in the input data might change based on what distribution is being used. For the Dirichlet-multinomial there still needs to be an input sample size but it is unclear if we should be using the same input sample size for the Dirichlet-multinomial as we do the multinomial. Something to think about when coding and advocating for a non-standard distribution. [Moved from #136 to this issue by @kellijohnson-NOAA].

[k-doering-NOAA]

related to this is the Tweedie distribution option: nmfs-ost/ss3-source-code#266 although this hasn't seen operational use yet (to my knowledge, maybe I am incorrect). It sounds like @timjmiller would be the most informed on this.

[timjmiller]

Most of these multivariate composition likelihoods are not available in TMB. Although many are pretty easy to code to return the appropriate value, we will want to allow osa residual calculation for as many of them as possible, which complicates the coding. Chris and I are helping DTU Aqua folks with a paper on this topic and I have applied the code from that work to WHAM age comp likelihood functions <https://github.com/timjmiller/wham/blob/master/src/age_comp_osa.hpp> which might be a good starting point for FIMS.

[k-doering-NOAA]

Another paper regarding the statistical validity for using the dirichlet multinomial: Fisch et al. 2022. Looks like the take home message was that in the 2 stock assessments, the linear formation of the Dirichlet multinomial performed best (although there are probably some nuances I'm not capturing with the statement). Looks like @KyleShertzer-NOAA is a coauthor on this one!