Clarification on the random variable W in the decoder

[jcara514]

Hello,

So I’ve been doing a deep dive into the technical aspects of this software for my dissertation project. Something that I noticed while going through the decoder algorithm was the following: w is defined as Gamma(f_w(z,s), theta). However, in the supplemental section, it is mentioned that w is defined to be a gamma variable with shape parameter r and scale p/1-p. Additionally in that section, when the ZINB model is shown, following statement is made:

However, wouldn’t the mean of w be f_w(z)theta (or even yet, rp/1-p)? I’ve been trying to figure out why the mean of w is just the decoder output, but I haven’t seen any clear answers online on why this is so. I hope this can be clarified.

[ori-kron-wis]

See if this helps:

The mean of the the scVI ZINB model distribution (related to supplementary note 3)

To bring more explanation in the supplement the definition is the logits total_counts convention for a negative binomial (r, p and lambda instead of rho, theta and l_n). See our code on how this changes computation: scvi-tools/src/scvi/distributions/_negative_binomial.py at main · scverse/scvi-tools · GitHub. To make it very short, theta is the mean variance relationship and the mean is independent of theta.

In short, there is some confusion of the notations of the supplemental section

[jcara514]

So w is distributed as a gamma variable with shape theta and rate theta/mu? I guess now I’m confused on why w~Gamma(shape = theta, theta/mu) and w~Gamma(shape= rho, theta) are equivalent.