Why Use Poisson Factorization?
A cell-by-gene count matrix contains nonnegative integers, but a topic model also needs a mechanism connecting those counts to a small set of shared gene programs. Poisson factorization provides one such mechanism: each cell has a nonnegative weight for each topic, each topic has a nonnegative rate for each gene, and the cell-gene count is generated from the sum of the topic-specific rates.
The key distinction from Dirichlet-based latent Dirichlet allocation (LDA) is what the cell weights mean. In LDA, a cell’s topic weights are proportions that sum to one, and the total number of observed counts in the cell is typically conditioned on. In Poisson factorization, cell-topic weights are nonnegative intensities; they are not inherently proportions, and the model can generate the total count as well as its distribution across genes.
The word conditional matters: the Poisson distribution describes the count after the latent weights and rates are specified. The basic model assumes that the conditional mean and conditional variance of \(Y_{ig}\) are both \(\lambda_{ig}\). In a common Bayesian version, the nonnegative factors have independent Gamma priors. We use the shape-rate convention: if \(U\sim\operatorname{Gamma}(a,b)\), then \(E[U]=a/b\) and \(\operatorname{Var}(U)=a/b^2\). These priors are one modeling choice, not a requirement of the factorization itself.
Expected Counts and Topic Contributions
For fixed factors, the expected count in a cell-gene pair is the sum of its topic-specific contributions. This gives a nonnegative matrix product: the matrix of expected counts is \(\Theta\Beta\), where \(\Theta\) has cell-topic entries \(\theta_{ik}\) and \(\Beta\) has topic-gene entries \(\beta_{kg}\). A particular contribution, \(\theta_{ik}\beta_{kg}\), is an expected count contribution under the model; it is not automatically a probability or a directly observed allocation of reads to a topic.
Summing expected counts over genes gives the expected total count for cell \(i\). Write \(B_k=\sum_{g=1}^{G}\beta_{kg}\) for the total rate across genes of topic \(k\). Then the expected cell total is \(\sum_k\theta_{ik}B_k\), not generally \(\sum_k\theta_{ik}\). The latter equality holds if each topic’s gene rates have been normalized so that \(B_k=1\).
Worked Example: Calculating the Expected Count Matrix
Suppose there are two cells, three genes, and two topics. Let the cell-topic weights be \(\theta_1=(2,1)\) and \(\theta_2=(1,3)\). Let the topic-gene rates be \(\beta_1=(0.5,1,0.5)\) and \(\beta_2=(1,0.5,2)\). These are hypothetical factor values for illustration.
For cell 1, the expected count for gene 1 is \(2(0.5)+1(1)=2\). For genes 2 and 3, the corresponding calculations are \(2(1)+1(0.5)=2.5\) and \(2(0.5)+1(2)=3\).
For cell 2, the expected counts are \(1(0.5)+3(1)=3.5\), \(1(1)+3(0.5)=2.5\), and \(1(0.5)+3(2)=6.5\). Thus
The expected total counts are \(2+2.5+3=7.5\) for cell 1 and \(3.5+2.5+6.5=12.5\) for cell 2. These are model expectations, not claims that the observed counts must equal those values.
What the Poisson Assumption Implies
The model’s conditional independence and Poisson variance give concrete implications that can be checked. Given the factors, \(Y_{ig}\) has variance \(\lambda_{ig}\), and counts for different genes in the same cell have covariance zero. If uncertainty in the factors is included, the marginal distribution is generally more variable than a Poisson distribution with a fixed rate. Thus marginal overdispersion does not contradict conditional Poisson sampling; it can arise from variation in the latent factors.
For example, under independent Gamma factors for a single topic, the product of the random cell weight and gene rate is itself variable. The count’s marginal variance then includes both the Poisson sampling variance and variation in its random mean.
Worked Example: Marginal Mean and Variance Under Gamma Factors
Consider one cell, one gene, and one topic, with independent factors \(\theta\sim\operatorname{Gamma}(2,1)\) and \(\beta\sim\operatorname{Gamma}(3,2)\), using shape-rate parameters. Conditional on them, \(Y\sim\operatorname{Poisson}(\theta\beta)\). Their means are \(E[\theta]=2\) and \(E[\beta]=1.5\), so independence gives \(E[Y]=E[\theta\beta]=2(1.5)=3\).
The factor variances are \(\operatorname{Var}(\theta)=2\) and \(\operatorname{Var}(\beta)=3/4=0.75\). Therefore \(E[\theta^2]=2+2^2=6\), and \(E[\beta^2]=0.75+1.5^2=3\). Independence gives \(\operatorname{Var}(\theta\beta)=E[\theta^2]E[\beta^2]-E[\theta\beta]^2=6(3)-3^2=9\).
By the law of total variance, \(\operatorname{Var}(Y)=E[\operatorname{Var}(Y\mid\theta,\beta)]+\operatorname{Var}(E[Y\mid\theta,\beta])=E[\theta\beta]+\operatorname{Var}(\theta\beta)=3+9=12\). The marginal variance is 12, greater than the marginal mean of 3, even though the conditional count distribution is Poisson.
Connection to LDA and Fixed Cell Totals
In standard LDA for cell gene counts, a cell has topic proportions that sum to one, and each topic has a probability distribution over genes that also sums to one. Conditional on the cell’s total count \(N_i=\sum_gY_{ig}\), the gene counts follow a multinomial distribution with probabilities given by the topic-proportion-weighted mixture of topic gene probabilities. The total \(N_i\) is treated as fixed in that conditional model.
Poisson factorization instead specifies a rate for each gene and lets the gene counts arise as independent Poisson variables, conditional on the factors. But the two forms are connected: conditional on their sum, independent Poisson counts have a multinomial distribution. For the factorization above, define \(\lambda_{i+}=\sum_g\lambda_{ig}\). Conditional on the total count, the gene probabilities are \(\lambda_{ig}/\lambda_{i+}\), provided \(\lambda_{i+}>0\). The topic weights in that conditional mixture are proportional to \(\theta_{ik}B_k\); they need not be the raw \(\theta_{ik}\).
This conditioning relationship clarifies both the similarity and the difference. If the topic-gene rates are normalized so \(B_k=1\), and the cell weights are normalized to sum to one, the Poisson factorization’s gene probabilities conditional on the total have the LDA mixture form. Without those normalizations, the cell factors describe intensities and the topic rates affect both gene composition and total expected counts. A fixed-total LDA analysis does not model variation in cell totals in the same way.
Worked Example: Conditioning Poisson Counts on Their Total
For one hypothetical cell, suppose the two topic contributions produce gene rates \(\lambda_{i1}=7\) and \(\lambda_{i2}=4\). The independent Poisson model gives a total rate of \(\lambda_{i+}=11\). Suppose the observed counts are 3 and 1, for a total of 4.
Before conditioning, their joint probability is \(P(Y_{i1}=3,Y_{i2}=1)=e^{-7}7^3/3!\times e^{-4}4^1/1! =e^{-11}(343/6)(4)\).
Conditional on a total of 4, the gene probabilities are \(7/11\) and \(4/11\). The multinomial probability of these same counts is \(\frac{4!}{3!1!}(7/11)^3(4/11)=4(343/1331)(4/11)=5488/14641\), approximately \(0.3748\).
The joint probability is also the probability that the total is 4 times this conditional probability. The total has distribution \(\operatorname{Poisson}(11)\), so the joint probability is \(\left(e^{-11}11^4/4!\right)(5488/14641)=e^{-11}(343/6)(4)\), matching the independent-Poisson calculation. The Poisson model accounts for the probability of the total as well as the conditional allocation across genes; the multinomial calculation alone does not.
Assumptions, Diagnostics, and Model Choice
The basic model is useful when a low-rank, nonnegative rate matrix is a plausible description of the data, but its assumptions are substantive. Conditional on the factors, counts are independent and Poisson; expected counts are additive across topics; and the factors are nonnegative. An extension may include a known cell-specific exposure \(s_i\), such as a library-size scaling factor, by setting \(\lambda_{ig}=s_i\sum_k\theta_{ik}\beta_{kg}\). That exposure changes the expected totals and must be defined consistently; it does not make the cell weights proportions.
A practical diagnostic is to compare observed cell totals and gene-wise variability with posterior predictive data generated from the fitted model. Persistent excess variability relative to the model, systematic residual patterns across genes or cells, or cell totals poorly reproduced by generated data can indicate a mismatch. These signs do not uniquely identify the cause: omitted covariates, inadequate topics, and count noise assumptions may all contribute. A good fit to average counts alone is not enough, because it can hide dependence or variability the model does not capture.
| Feature | Poisson factorization | Dirichlet-based LDA |
|---|---|---|
| Cell-topic quantities | Nonnegative intensities; not necessarily normalized | Topic proportions constrained to sum to one |
| Cell total | Generated through the Poisson rates, unless conditioned on | Usually conditioned on as the observed total |
| Count structure | Conditionally independent Poisson counts given factors | Multinomial counts given the total and mixture probabilities |
| Topic-gene quantities | Nonnegative rates; row sums may vary | Gene probabilities within each topic, summing to one |
There are also ambiguities intrinsic to the rate factorization. For any positive constant \(c\), replacing \(\theta_{ik}\) by \(c\theta_{ik}\) and \(\beta_{kg}\) by \(\beta_{kg}/c\) leaves every \(\lambda_{ig}\) unchanged. Relabeling topics also leaves the likelihood unchanged. Priors and constraints can influence the resulting representation, but the data likelihood alone does not select a unique scale or topic ordering. Consequently, factor magnitudes should not be compared as though they had a uniquely determined scale without stating the model’s normalization and prior conventions.
Poisson factorization can be attractive when modeling cell totals is useful or when a rate-based representation matches the analysis goal. LDA is a natural alternative when conditioning on each cell’s total and interpreting topic mixtures as proportions is central. Neither formulation is automatically superior: the choice depends on whether the total count should be part of the modeled outcome and whether the corresponding assumptions suit the data.
Common Mistakes and Limitations
- Calling \(\theta_{ik}\) a topic proportion by default. In the rate model it is an intensity. A mixture proportion requires a suitable normalization, and conditional topic weights are proportional to \(\theta_{ik}B_k\).
- Assuming Poisson factorization fixes the scale. Reciprocal rescaling of a topic’s cell weights and gene rates leaves the expected matrix unchanged.
- Concluding that marginal overdispersion disproves conditional Poisson sampling. Random latent factors can make the marginal variance exceed the mean even when counts are Poisson conditional on those factors.
- Treating the expected count matrix as the observed matrix. The product \(\Theta\Beta\) gives fitted conditional means; observed counts fluctuate around them.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For one cell, let the two topic weights be \((1,2)\), and let the topic rates for one gene be \((3,0.5)\). What is the expected count for that gene?
- State the conditional variance of \(Y_{ig}\) under the basic model and explain why the marginal variance can be larger.
- For a cell with gene rates \((2,6)\), what are the gene probabilities conditional on the cell total?
- Give one important modeling distinction between this Poisson factorization and fixed-total LDA.
Solutions. (1) The expected count is \(1(3)+2(0.5)=4\). (2) Conditional on the factors, \(\operatorname{Var}(Y_{ig}\mid\theta,\beta)=\lambda_{ig}\). When factors vary, the law of total variance adds \(\operatorname{Var}(\lambda_{ig})\) to \(E[\lambda_{ig}]\), so the marginal variance can exceed the marginal mean. (3) The total rate is \(2+6=8\), so the conditional probabilities are \(2/8=0.25\) and \(6/8=0.75\). (4) The Poisson model generates cell totals through its rates unless conditioned on; fixed-total LDA conditions on the cell total and uses simplex-valued topic proportions.