SCDE inference methods: traditional Bayesian statistics or something else?

http://hms-dbmi.github.io/scde/

Specifically, it is about the third equation on page 5 of the SCDE paper, which reads

$$p_S(x) = E\left[\prod_{c\in B} p(x|r_c, \Omega_c)\right]$$.

SCDE deals with single-cell RNA sequencing data, and $\Omega_c$ is a cell-specific error model while $r_c$ is the observed expression of a gene in that cell. A central issue is dropout: observed expression near zero due to technical effects. The parameter $x$ is the mean expression within a cell subpopulation, while $B$ is a bootstrap sample from that cell subpopulation. I want to know where this formula comes from and what it is saying. Here is what I have done so far to dig into this question.

The expectation

What does it mean to take an expected value of a posterior probability? If the expectation is taken w.r.t. the posterior density itself, then it is just the $L_2$ norm of the posterior density. I doubt that's what Kharchenko et al. meant. Perhaps the expectation is taken as an average over many bootstrap samples.

Likelihood versus posterior

The paper never specifies a flat prior distribution. Nevertheless, it seems as if SCDE uses likelihoods throughout and presents them as posterior distributions. I assume that SCDE uses a generative model where expression levels are independent across cells conditioned on the subpopulation average, so that the likelihood factorizes as $$\prod_{c\in S} p(r_c | x, \Omega_c)$$ where the multiplicands are probabilities of observed expression values given the average $x$ and the dropout-tolerant error model. This lines up nicely with the posterior they indicate, especially when we stop to ask...

... how do Kharchenko et al define $p(x|r_c, \Omega_c)$? The paragraph below the equation of interest says it equals $$p_d(x)Poisson(x) + (1 - p_d(x))NB(x|r_c)$$, which refers to a mixture model with a Poisson component that tends to kick in more often at low values of $x$ and a negative binomial component that dominates when $x$ is big. It's referred to as a posterior probability, but it looks like a mixture model likelihood. $NB(x|r)$ typically means a negative binomial density with parameter $r$ and random variable $x$, but the text defines it as the probability of $r_c$ given $x$ (and no dropout). The text also defines $Poisson(x)$ to depend on $r_c$: "the probabilities of observing expression magnitude of $r_c$ in case of a dropout".

Related

There's a related question here, but it asks for an overview whereas I want to know specifics.

Overview of concepts of the methods that this Single-Cell Differential Expression (SCDE) tool uses?

Tags

Someone please tag this appropriately. I could only find "genetics", but this is more like genomics or transcriptomics.