Part 6a — From rate to counts: composition, library size, and the NB mean, worked through¶
A companion to Part 6 §2 for readers who want the sentence "a relative expression profile \(\rho\) that sums to one, then scaled by the library size \(\ell\)" turned into arithmetic. We run one small example — a handful of genes — all the way from the decoder's output to a sampled count vector, and use it to show why the rate-times-depth split is the right way to model counts.
Why this chapter exists. Part 6 §2 introduces the count decoder's core trick in one line: emit a rate \(\rho\) over genes (a softmax, so it sums to one), then multiply by the cell's library size \(\ell\) to get the negative-binomial mean \(\mu = \ell \rho\). That line packs in a real modeling decision — separate the biological composition from the technical sequencing depth — and it is far easier to feel with numbers than with symbols. This chapter assumes only Part 6 §2's vocabulary (the NB mean \(\mu\), dispersion \(\kappa\), library size \(\ell\), rate \(\rho\)) and works one example end to end.
The plan: first the two ingredients (\(\rho\) and \(\ell\)) on a six-gene toy cell; then assembling the mean \(\mu = \ell \rho\); then the two scenarios that justify the split — same biology at different depths, and a perturbation that reshapes the profile (where effect size lives); and finally the step from the mean to an actual noisy count draw. Real single-cell numbers are bigger but the mechanics are identical, and we scale up once at the end.
1. The two ingredients — a composition and a depth¶
Take a deliberately tiny cell with just six genes, call them \(A\) through \(F\). (Real data has thousands; six is enough to see everything.)
The rate \(\rho\) — what fraction of the cell's transcripts each gene accounts for. The decoder reads the latent \(z\) and emits one real number ("logit") per gene, then a softmax turns those logits into a probability vector: every entry positive, all entries summing to one. Concretely, if the decoder's logits for three genes were \((2.0, 1.0, 0.0)\), the softmax exponentiates and normalizes —
— and those three shares sum to one. For our six-gene cell, suppose the softmax produces
Read this as composition: gene \(A\) makes up 40% of this cell's transcripts, \(B\) a quarter, down to \(F\) at 3%. It is a relative statement — it says nothing yet about how many molecules were actually counted, only their proportions. This is the part the decoder predicts, and it is where the biology lives.
The library size \(\ell\) — how many transcripts this cell yielded in total. Sequencing does not read every molecule in a cell; it captures a sample of them, and that sample's total size varies a lot from cell to cell — driven by technical factors (capture efficiency, sequencing depth) far more than biology. We call that per-cell total the library size \(\ell\). For our example cell, say the sequencer captured
The crucial point Part 6 §2 makes: \(\ell\) is a given covariate, not a prediction. You read it straight off the observed cell (sum its counts); the decoder never has to guess it. The decoder's whole job is the shape \(\rho\); the scale \(\ell\) is handed in.
2. Assembling the mean — \(\mu = \ell \rho\)¶
Now the one-line assembly. Multiply each gene's share by the total depth to get its expected count:
For our cell, gene by gene:
| gene | rate \(\rho_g\) | \(\times\ \ell = 10{,}000\) | mean count \(\mu_g\) |
|---|---|---|---|
| \(A\) | 0.40 | 4000 | |
| \(B\) | 0.25 | 2500 | |
| \(C\) | 0.15 | 1500 | |
| \(D\) | 0.10 | 1000 | |
| \(E\) | 0.07 | 700 | |
| \(F\) | 0.03 | 300 | |
| sum | 1.00 | 10,000 |
So \(\mu = (4000, 2500, 1500, 1000, 700, 300)\) — the number of transcripts we expect to count for each gene in this cell. Notice the bottom row: because the rates sum to one, the expected counts sum to exactly the library size, \(\sum_g \mu_g = \ell \sum_g \rho_g = \ell \cdot 1 = \ell\). That is the whole elegance of the parameterization — \(\rho\) distributes the budget \(\ell\) across genes, and the budget is always spent exactly. (The mean is not the decoder's direct output; it is assembled from the predicted shape \(\rho\) and the given depth \(\ell\), exactly as Part 6 §2 stressed.)
3. Why split it this way — depth is a nuisance, composition is the signal¶
The decomposition earns its keep the moment you compare cells. Here are the two scenarios that make it click.
3.1 Same biology, different depth¶
Take two cells with identical biology — same cell type, same state, so the same composition \(\rho\) — but sequenced to different depths: cell 1 at \(\ell_1 = 10{,}000\), cell 2 at \(\ell_2 = 4{,}000\). The means scale with depth, term by term:
| gene | \(\rho_g\) | cell 1: \(\mu_g = 10{,}000 \cdot \rho_g\) | cell 2: \(\mu_g = 4{,}000 \cdot \rho_g\) |
|---|---|---|---|
| \(A\) | 0.40 | 4000 | 1600 |
| \(B\) | 0.25 | 2500 | 1000 |
| \(C\) | 0.15 | 1500 | 600 |
| \(D\) | 0.10 | 1000 | 400 |
| \(E\) | 0.07 | 700 | 280 |
| \(F\) | 0.03 | 300 | 120 |
The raw counts look completely different — cell 1's gene \(A\) reads ~4000, cell 2's ~1600 — yet nothing biological differs between them. The entire gap is sequencing depth, a technical artifact. If you fed a model the raw counts and asked it to "learn the difference," it would waste capacity modeling a difference that is pure nuisance.
The rate-times-depth split dissolves this. Both cells share the same \(\rho\); the model only ever has to predict that one depth-invariant profile, and the per-cell \(\ell\) — read off the data — absorbs the technical scale. Equivalently: divide any observed count vector by its total and you recover \(\rho\), which is why \(\rho\) is the normalized, depth-free description of the cell. The decoder predicts the biology (\(\rho\)); the library size carries the technical depth (\(\ell\)); they are kept apart on purpose.
3.2 A perturbation reshapes the profile — and that is effect size¶
Now the case the whole series is about. A perturbation does not change a cell's depth — it changes its composition. Suppose a drug strongly up-regulates gene \(C\) (and, because shares must still sum to one, pulls the others down to compensate). Hold the depth fixed at \(\ell = 10{,}000\) to isolate the biological change:
| gene | control \(\rho^{\text{ctrl}}_g\) | \(\mu^{\text{ctrl}}_g\) | perturbed \(\rho^{\text{pert}}_g\) | \(\mu^{\text{pert}}_g\) | change |
|---|---|---|---|---|---|
| \(A\) | 0.40 | 4000 | 0.32 | 3200 | −800 |
| \(B\) | 0.25 | 2500 | 0.20 | 2000 | −500 |
| \(C\) | 0.15 | 1500 | 0.30 | 3000 | +1500 (2×) |
| \(D\) | 0.10 | 1000 | 0.08 | 800 | −200 |
| \(E\) | 0.07 | 700 | 0.06 | 600 | −100 |
| \(F\) | 0.03 | 300 | 0.04 | 400 | +100 |
Gene \(C\)'s expected count doubles, \(1500 \to 3000\); the others dip slightly. That vector of changes — \((-800, -500, +1500, -200, -100, +100)\) — is the perturbation's effect, and the effect size from Part 5 §3 is how well a model reproduces it (the correlation between predicted and true change on the most-affected genes). A model that predicts the reshaped \(\rho^{\text{pert}}\) accurately recovers this; a model that stops at a latent and never decodes to counts has no place to express "+1500 on gene \(C\)" at all. This is the concrete mechanism behind Part 6 §2's claim that the count decoder is where effect size is recovered.
An honest wrinkle — composition is relative. Because \(\rho\) sums to one, a share cannot rise without others falling: the table's increases and decreases must balance. So the decoder models relative expression, and a perturbation that truly raised every gene's absolute transcript count (with depth held fixed) is not representable as a change in \(\rho\) alone — it would show up only through \(\ell\). This is the well-known compositional nature of sequencing counts, and it is one reason library-size handling and normalization are treated so carefully in practice. For most differential-expression questions ("which genes shift relative to the rest?") the relative profile is exactly what you want; just know the constraint is there.
4. From the mean to an actual count — the NB draw¶
The mean \(\mu_g\) is only the expected count; the observed count is noisy around it, and the dispersion \(\kappa\) sets how noisy. Recall from Part 6 §2 that the NB variance is \(\mu + \mu^2/\kappa\). Take gene \(A\) with \(\mu_A = 4000\) and see how the spread depends on \(\kappa\):
| dispersion \(\kappa_A\) | variance \(= \mu + \mu^2/\kappa\) | std. dev. | reading |
|---|---|---|---|
| \(2\) (high overdispersion) | \(4000 + 8{,}000{,}000 = 8{,}004{,}000\) | \(\approx 2829\) | counts swing wildly, ~1000–7000 |
| \(100\) (mild) | \(4000 + 160{,}000 = 164{,}000\) | \(\approx 405\) | counts cluster ~3600–4400 |
| \(\kappa \to \infty\) (Poisson) | \(4000\) | \(\approx 63\) | tight, ~3940–4060 |
So a single predicted state (\(\mu_A = 4000\)) can produce an observed count anywhere from ~1000 to ~7000 when the gene is heavily overdispersed (\(\kappa = 2\)). That is the measurement spread Part 6 §3 warns is not the same as outcome heterogeneity: even with the cell's true state pinned, the readout is noisy, and \(\kappa\) is what lets the NB model that noise honestly instead of pretending (as a Poisson or a fixed-variance Gaussian would) that counts cluster tightly around their mean.
Putting the pipeline together for our cell: the decoder emits \(\rho\) (and a \(\kappa_g\) per gene); we assemble \(\mu = \ell \rho\); then each gene's observed count is a draw \(x_g \sim \mathrm{NB}(\mu_g, \kappa_g)\). One plausible sampled count vector for the control cell (with, say, moderate \(\kappa\)) might read \((3870, 2560, 1410, 980, 720, 280)\) — close to \(\mu\) but jittered, and summing to roughly but not exactly \(\ell\). Draw again and you get a different vector: that is one cell's measurement noise, on top of whatever outcome variation the latent already carried.
At generation time, where does \(\ell\) come from? During training \(\ell\) is read off each real cell. When you generate a new cell, there is no observed total to read, so you supply a target depth — sample \(\ell\) from the empirical library-size distribution of the dataset, or fix it to a reference depth so generated cells are comparable. The decoder still only predicts the shape \(\rho\) and dispersion \(\kappa\); you choose the scale.
5. Scaling up — the same arithmetic at real size¶
Real single-cell data only changes the numbers, not the mechanics. A typical experiment has a few thousand genes (often ~2000 after selecting the highly variable ones) and per-cell library sizes with a median around 10,000–15,000 counts — so \(\rho\) is a ~2000-entry probability vector and \(\ell\) is a five-figure total. The same three steps run unchanged: softmax to a rate \(\rho\) that sums to one, scale by the cell's \(\ell\) to get \(\mu = \ell \rho\), draw counts from \(\mathrm{NB}(\mu, \kappa)\). The only practical difference is sparsity: with the budget \(\ell\) spread across thousands of genes, most \(\mu_g\) are small (well under 1), and many genes read exactly zero in any given cell — which is the dropout that Part 6 §2's ZINB refinement adds an explicit zero-inflation term for.
6. Recap — the rate-times-depth picture in one breath¶
- \(\rho\) is composition — a softmax over genes, summing to one, the depth-invariant biology the decoder predicts.
- \(\ell\) is depth — the cell's total captured counts, a given technical covariate read off the data, not predicted.
- \(\mu = \ell \rho\) assembles the mean — \(\rho\) distributes the budget \(\ell\) across genes, so \(\sum_g \mu_g = \ell\) exactly.
- The split is the point — same biology at different depths shares one \(\rho\) (depth is nuisance); a perturbation reshapes \(\rho\) (composition is signal), and that reshaping is the effect size the count decoder exists to recover.
- \(\kappa\) then sets the measurement noise of the actual count draw \(x_g \sim \mathrm{NB}(\mu_g, \kappa_g)\) around that mean.
With the assembly concrete, the Part 6 §2 line "a rate \(\rho\) that sums to one, scaled by the library size \(\ell\)" should now read as a small, sturdy piece of machinery rather than a notational flourish — and the count likelihood written out in the rest of §2 is just this same \(\mu = \ell\rho\) fed into the negative-binomial probability, gene by gene.
Companion to Part 6 — Route A: a decoder on the latent. New to single-cell counts? The data-modalities primer. Symbols: the notation reference.