I'm watching a video on topic modeling by David Blei (from 27:00 of Part 2) and I don't understand how setting the $\alpha$ hyperparameter to a value close to zero basically results in a mixture model (and not mixed-membership model).
Setting $\alpha$ to a low value will result in samples of $\theta$ drawn from $Dir(\alpha)$ to be sparse, with majority of its mass concentrated in only few of its dimensions and the rest close to zero. But wouldn't the likelihood $p(D | \theta)$ dominate this effect and result in a posterior $p(\theta | D)$ where this is no longer the case?
In the video, David Blei says that, in such case, we can set $\alpha$ to be even smaller and make the prior dictate what the posterior is going to look like, but where in the formulation of the Dirichlet-multinomial (wiki) is this apparent? Without looking at too much detail, it seems that the posterior distribution, where the $\alpha$ and $counts$ are added (for each dimension), will be dictated by the $counts$ if $\alpha$ is very small.