Negative binomial modeling of RNA-Seq data A common way to model RNA-seq data is using a negative binomial distribution, where each sample-gene pair is modeled by a different negative binomial distribution with mean $\mu_{ij}$ where $i$ and $j$ are indices for genes and samples, respectively (see article). My question is, given we only have a single observation ($X_{ij}$) from each sample-gene pair, how come we can learn the $\mu_{ij}$ from that single observation? We need multiple observations to estimate a mean, right? Maybe I am missing something critical.
The $\mu_{ij}$ in that section of the DESeq manuscript is the expected value of sample $j$ in gene $i$ given its group association (with expected value $\mu_{ij}$). This is why $K_{ij}$ (this is presumably what you meant by $X_{ij}$) is a function of $\mu_{ij}$, rather than $\mu_{ij}$ itself. I think you're just swapping $K_{ij}$ and $\mu_{ij}$ in your head.