incorporate within-sample information

[rmcminds]

currently, the model conflates effects on alpha diversity with effects on generalism/specialism. This is because, for a given factor or host node, a general increase in the probability of finding things increases all symbiont types in all samples, which is probably easiest to interpret as a change in ADiv, but also means less-likely symbiont types are more likely to be found in some samples, influencing generalism. what might be ideal is recognizing that an increase in alpha diversity would increase the probability for all types (the current model), whereas an increase in generalism would actually keep the overall probability of finding something (the sum of the various type-specific probabilities?) the same while essentially only increasing the probability of finding the rarer types (essentially evening out or smoothing the probabilities). to do this, the various probabilities within each sample would have to be 'aware' of each other somehow.

one thought is to explicitly model the number of types found in a sample using, say, a poisson distribution, and then pass the current model output (slightly modified so it doesn't have a grand mean parameter) to a softmax function and a Wallenius' noncentral multivariate hypergeometric distribution (https://en.wikipedia.org/wiki/Wallenius%27_noncentral_hypergeometric_distribution) with all k=1 to determine which particular types are represented. since the existing parameters would then be normalized on a per-sample basis, they would then only represent generalism-specialism, not alpha diversity.

[rmcminds]

above proposal wouldn't work because softmax is invariant to universal addition. Thus, current 'main effect' parameters that are meant to represent generalism would in fact have zero influence on the posterior (if all symbiont types get the same added parameter in a given sample, nothing changes). the concept of generalism would really require some kind of smoothing parameter (e.g. a pre-softmax multiplier rather than an addition term, or adding a dirichlet distribution to the softmax output with a parameter to control the smoothness)

[rmcminds]

could model be reformulated so that only positive occurrences are considered? e.g. in a given sample, there are two microbe types: what are the chances that their identities are what they are? this would assume that a given microbe's ability to infect a host is independent from its ability to compete with other microbes?

[rmcminds]

actually, the concept of generalism-specialism as some kind of biological mechanism is maybe worth rethinking. Basically, if a coral became 'more generalist' without simultaneously increasing its alpha diversity, wouldn't that mean it simultaneously became easier to infect by all symbionts /except/ the one previously most likely? more likely is a scenario where the baseline ability of all other symbionts doesn't change, but the previous winner is suddenly less successful. fundamental to this is the idea that there is some kind of competition within samples, though - the alpha diversity question really is a separate mechanism