Denoised Expression Versus Observed Counts

Three Expression Quantities, Three Different Questions

A count matrix contains the values actually recorded for each cell and gene. A count VAE also supplies model-based quantities that can look like expression values but answer different questions. Confusing these quantities can turn a useful noise-reduction estimate into an unsupported claim that cells have similar expression.

For cell \(n\) and gene \(g\), let \(x_{ng}\) denote the observed count. In a count VAE, the decoder specifies a distribution for this count conditional on latent variables and other model inputs. Write its conditional mean as \(\mu_{ng}(z_n,l_n)\), where \(z_n\) is the cell's latent state and \(l_n\) is a library-size quantity. For example, a model may set \(\mu_{ng}(z_n,l_n)=l_n\rho_{ng}(z_n)\), with \(\rho_{ng}\) a gene proportion from the decoder. The exact parameterization depends on the model; the scale of the output must therefore be checked before it is interpreted.

If the encoder approximates the posterior distribution of the latent state by \(q_\phi(z_n\mid x_n)\), a count-scale posterior denoised estimate can be defined as \(\hat{\mu}_{ng}=\mathbb{E}_{q_\phi(z_n\mid x_n)}[\mu_{ng}(z_n,l_n)]\). In computation, this expectation is commonly approximated by averaging decoder outputs at posterior draws. If the reported target is normalized expression rather than expected counts, average the corresponding normalized decoder output instead. These are different targets when the library-size factor varies across cells.

Decoder Means Are Not Posterior Predictive Counts

A decoder expectation is conditional on its inputs. If those inputs are uncertain, averaging decoder means gives a posterior estimate of the conditional mean. A posterior predictive count instead represents a possible new count under the fitted model, including count-sampling variation as well as uncertainty about the latent state.

For a negative binomial observation model parameterized by mean \(\mu\) and dispersion \(\theta\), the conditional variance is \(\mu+\mu^2/\theta\). The law of total variance gives the posterior predictive variance as the average conditional variance plus the variance of the conditional means. In this expression, \(Z\) is drawn from the approximate posterior for the cell.

The first term averages negative binomial sampling variance; the second captures variation in the decoder mean across posterior draws. The posterior denoised estimate is the average mean, not a draw from this predictive distribution. Thus, it can be fractional and typically has less variability than posterior predictive counts.

Why Denoising Can Hide Cell-Level Differences

Posterior averaging can reduce noise when the model captures the relevant structure and the count observations are sparse. But the same averaging can also pull estimates toward patterns shared across cells. If a model treats genuinely different cells as if they had similar latent expression, denoised values can understate real heterogeneity. A low observed count alone does not tell us which case applies: it may reflect sampling from a modest mean, or it may be one piece of evidence for a truly low-expressing cell.

The following calculation illustrates shrinkage with a simple Poisson-Gamma model, not a VAE. It makes the mechanism explicit. Assume a cell has count \(X\mid\lambda\sim\operatorname{Poisson}(\lambda)\), with exposure fixed at one, and assign its rate the prior \(\lambda\sim\operatorname{Gamma}(a=2,b=2)\), using the shape-rate convention. The prior mean rate is \(a/b=1\). The posterior mean is pulled toward that prior mean, with the strength of the pull depending on the count.

Choosing a Quantity and Checking Its Behavior

There is no single expression quantity that is best for every task. Use observed counts when the analysis requires the recorded data or when modeling the sampling distribution directly. Use decoder expectations when the question concerns the model-implied mean at specified inputs. Use posterior denoised estimates when a model-based, uncertainty-averaged expression summary is appropriate. Use posterior predictive draws when the task requires plausible replicated counts or a check of count-level variation.

Before comparing or plotting values, establish whether they are on the count scale, a normalized scale, or another scale. For example, \(\rho_{ng}\) may represent a relative expression proportion, whereas \(l_n\rho_{ng}\) is an expected count. An expectation on one scale cannot be described as though it were on another. Also record whether the posterior estimate averages over latent draws, whether library size is fixed or uncertain, and what covariates or conditions are held constant.

Useful diagnostics include comparing observed and posterior-predictive distributions across cells, checking whether zeros and high counts are reproduced, and inspecting denoised values alongside the raw counts for selected genes. Look particularly at genes with biological reasons to vary across cell states and at small groups of cells, where pooling can dominate. If posterior estimates are nearly identical across cells while observed counts, known covariates, or independent measurements suggest structured variation, investigate model fit and the representation of that structure rather than treating smoothness as proof of accuracy.

A common mistake is to say, “The denoised count is 2, so the cell had two transcripts.” The correction is that 2 may be a posterior average of expected counts; the observed count is still the recorded integer, and the estimate carries model assumptions and uncertainty. Similarly, a denoised value closer to a group average is not by itself evidence that a cell lacks a distinct expression state.

These distinctions matter whenever expression summaries are used to describe cells, compare groups, or select genes for further analysis. Denoising can make patterns easier to see, but it changes the object being examined: a model-based estimate replaces the raw observation. Keeping both quantities available, checking the scale, and examining predictive fit help prevent smoothing from being mistaken for biological uniformity.