Neutral models which assume ecological equivalence between species provide null models for community assembly. In Hubbell's unified neutral theory of biodiversity (UNTB), many local communities are connected to a single metacommunity through differing immigration rates. Our ability to fit the full multisite UNTB has hitherto been limited by the lack of a computationally tractable and accurate algorithm. We show that a large class of neutral models with this mainland-island structure but differing local community dynamics converge in the large population limit to the hierarchical Dirichlet process. Using this approximation we developed an efficient Bayesian fitting strategy for the multisite UNTB. We can also use this approach to distinguish between neutral local community assembly given a nonneutral metacommunity distribution and the full UNTB where the metacommunity too assembles neutrally. We applied this fitting strategy to both tropical trees and a data set comprising 570$\,$851 sequences from 278 human gut microbiomes. The tropical tree data set was consistent with the UNTB but for the human gut neutrality was rejected at the whole community level. However, when we applied the algorithm to gut microbial species within the same taxon at different levels of taxonomic resolution, we found that species abundances within some genera were almost consistent with local community assembly. This was not true at higher taxonomic ranks. This suggests that the gut microbiota is more strongly niche constrained than macroscopic organisms, with different groups adopting different functional roles, but within those groups diversity may at least partially be maintained by neutrality. We also observed a negative correlation between body mass index and immigration rates within the family Ruminococcaceae. This provides a novel interpretation of the impact of obesity on the human microbiome as a relative increase in the importance of local growth versus external immigration within this key group of carbohydrate degrading organisms.