Shared Groups Do Not Require Identical Profiles
A joint clustering model can assign the same samples to the same groups across omics blocks without requiring those groups to look identical in every block. For example, one group of samples may have elevated expression of a gene set but only a modest shift in a corresponding protein measurement. If the model forces one common profile across both blocks, it may conceal that difference or compromise the fit to both.
The distinction is between membership and profile. Let \(z_i\) denote the cluster assignment of sample \(i\), shared across modalities. Let \(\theta_{mkj}\) denote the parameter for feature \(j\), modality \(m\), and cluster \(k\). Shared assignments mean that the same \(z_i\) is used in every block; they do not mean that \(\theta_{mkj}\) must equal the corresponding parameter in another block. Each block can have its own profile, suited to its measurement scale and likelihood.
Likelihood and Profile Structure
Suppose there are \(N\) samples, \(M\) modalities, and \(K\) clusters. In modality \(m\), sample \(i\) has measurements \(Y_{mij}\) for features \(j\). A standard conditional-independence formulation is
Here \(f_{mj}\) is an appropriate sampling model, \(\theta_{mkj}\) is the cluster profile parameter, and \(\phi_{mj}\) represents any additional likelihood parameter, such as a variance or dispersion. Conditional on the assignments and model parameters, the likelihood factors over the observations as
This factorization does not require the likelihoods in different modalities to have the same form. A continuous block can use a Gaussian likelihood, a sequencing block can use a count likelihood, and a binary block can use a Bernoulli likelihood. The shared assignments connect the blocks at the level of which samples belong together; the block-specific parameters describe what those groups look like in each block.
One can give each \(\theta_{mkj}\) its own prior. When profiles are comparable across modalities—for instance, standardized measurements of the same biological quantity—it can be useful to add partial pooling. Write \(t_{mkj}\) for the profile parameter on a comparable scale and \(q_{kj}\) for a shared reference profile. A simple hierarchical specification is
The deviation \(t_{mkj}-q_{kj}\) allows modality \(m\) to depart from the reference profile for cluster \(k\). Small \(\tau_{mj}^{2}\) implies stronger pooling toward that reference; large \(\tau_{mj}^{2}\) permits more modality-specific variation. This hierarchy is not appropriate merely because two measurements come from different blocks: the parameters must have a meaningful common scale or a justified mapping to one. Otherwise, use modality-specific priors without interpreting their differences as deviations from a shared numerical profile.
The main assumptions should be explicit. The chosen likelihood must represent each block’s measurement distribution adequately; the shared assignment must be scientifically plausible; and conditional independence must be a reasonable approximation after conditioning on assignments and parameters. A hierarchical prior additionally assumes that comparable modality-specific profiles are exchangeable around the specified reference, with the chosen deviation scale representing plausible differences. These are modeling assumptions, not consequences of sharing \(z_i\).
Why Partial Pooling Produces a Compromise
A short Gaussian calculation shows how the hierarchy allows different profiles while stabilizing estimates from small clusters. Consider one feature in one modality, with the cluster assignments treated as given. Assume observations in cluster \(k\) are conditionally independent with \(Y_i\mid t\sim N(t,\sigma^2)\), where \(\sigma^2\) is known. Put the prior \(t\mid q\sim N(q,\tau^2)\), with \(q\) and \(\tau^2\) fixed. For \(n_k\) observations with sample mean \(\bar{x}\), the likelihood as a function of \(t\) is proportional to
Multiplying by the normal prior adds its quadratic term, \((t-q)^2/\tau^2\), to the exponent. Completing the square gives a normal posterior with precision equal to the sum of the prior and data precisions:
Thus the posterior mean is a precision-weighted average of the reference \(q\) and the observed cluster mean \(\bar{x}\). With fewer observations, the prior contributes relatively more; with more observations, the data contribute relatively more. This is partial pooling, not a constraint that the modality-specific profile equal \(q\). The derivation conditions on the assignments and hyperparameters; it does not describe how cluster labels are sampled.
Worked Example: Shrinking Two Continuous Profiles Separately
Consider one standardized feature measured in two modalities for the same four samples assigned to cluster \(k\). In modality A, suppose the values are \(0.4, 0.8, 1.2,\) and \(2.0\); in modality B, they are \(-0.8, -0.3, 0.1,\) and \(0.2\). For each modality, assume independent Gaussian observations with known variance \(\sigma^2=1\), and use the same reference \(q=0.5\) and deviation variance \(\tau^2=0.25\). These are hypothetical standardized measurements.
The sample means are \(\bar{x}_A=(0.4+0.8+1.2+2.0)/4=1.1\) and \(\bar{x}_B=(-0.8-0.3+0.1+0.2)/4=-0.2\). For each block the posterior variance is \(1/(1/0.25+4/1)=1/8=0.125\). The posterior mean for modality A is \((0.5/0.25+4(1.1)/1)/(1/0.25+4/1)=(2+4.4)/8=0.8\).
For modality B, the posterior mean is \((0.5/0.25+4(-0.2)/1)/(1/0.25+4/1)=(2-0.8)/8=0.15\). The conditional posterior means for the two modality-specific profiles are therefore \(0.8\) and \(0.15\), not one forced common value. Relative to \(q\), their estimated deviations are \(0.3\) and \(-0.35\) standardized units. The shared cluster assignment is unchanged; the two blocks show distinct estimated profiles within that group.
Different Blocks, Different Likelihood Parameters
The profile parameter need not be a mean. In count data it may be a rate, and in binary data it may be a success probability. Each block’s likelihood and prior should match its measurement type; sharing assignments does not justify borrowing a parameterization from one block for another.
Worked Example: Distinct Count Rates in One Shared Cluster
Suppose six samples belong to the same cluster in two sequencing modalities. For a hypothetical feature, modality A has 18 total reads across the six samples and total exposure 6 library-size units; modality B has 6 total reads and the same total exposure. Assume counts are conditionally independent Poisson observations with a common within-modality cluster rate \(\lambda_m\) and known sample exposures \(e_i\), so \(Y_i\mid\lambda_m\sim\operatorname{Poisson}(e_i\lambda_m)\). Use independent \(\operatorname{Gamma}(2,2)\) priors for the rates, with the second argument denoting the rate.
For total count \(y_+=\sum_iY_i\) and total exposure \(e_+=\sum_i e_i\), the likelihood for \(\lambda_m\) is proportional to \(\lambda_m^{y_+}\exp(-e_+\lambda_m)\). Multiplying by a \(\operatorname{Gamma}(a,b)\) prior gives a \(\operatorname{Gamma}(a+y_+,b+e_+)\) posterior. Thus modality A has posterior \(\operatorname{Gamma}(2+18,2+6)=\operatorname{Gamma}(20,8)\), with posterior mean rate \(20/8=2.5\) reads per library-size unit. Modality B has posterior \(\operatorname{Gamma}(2+6,2+6)=\operatorname{Gamma}(8,8)\), with posterior mean rate \(8/8=1.0\) read per library-size unit.
The cluster labels are shared, but the rate parameters and their posterior means differ. The calculation assumes the Poisson rate model and the stated exposures are appropriate; it does not imply that every sequencing feature has a constant rate across samples.
Worked Example: Modality-Specific Binary Probabilities
Suppose a binary feature is recorded in two modalities for the ten samples in a shared cluster. Modality A has 7 positive observations and modality B has 2. Assume independent Bernoulli observations within each modality and independent \(\operatorname{Beta}(1,1)\) priors on the probabilities \(p_A\) and \(p_B\). The conjugate update is \(\operatorname{Beta}(a+s,b+n-s)\) after \(s\) positives among \(n\) observations.
For modality A, the posterior is \(\operatorname{Beta}(1+7,1+10-7)=\operatorname{Beta}(8,4)\), whose mean is \(8/(8+4)=0.667\). For modality B, it is \(\operatorname{Beta}(1+2,1+10-2)=\operatorname{Beta}(3,9)\), whose mean is \(3/(3+9)=0.250\). These are posterior means of the positive-observation probabilities in their respective blocks. The calculation uses separate probabilities; it does not assume that a positive call has identical meaning or measurement error in both modalities.
Model Choices, Diagnostics, and Limits
There are three common structural choices. A fully shared profile constrains corresponding modality parameters to be equal, which is restrictive and only meaningful for genuinely comparable parameters. Independent modality-specific profiles permit differences but provide no cross-modality pooling. A hierarchical deviation model lies between these: it allows differences while shrinking comparable profiles toward a reference. Shared partitions with modality-specific parameters are also distinct from modality-specific partitions, where the membership itself can differ by block.
| Model choice | What is shared? | Main consequence |
|---|---|---|
| Fully shared profile | Assignments and corresponding profile parameters | Strong constraint; can miss real block differences |
| Independent block profiles | Assignments only | Flexible, but no pooling of profile information |
| Hierarchical deviations | Assignments and a profile reference | Partial pooling; deviations depend on the hierarchy and its scale |
| Modality-specific partitions | Not necessarily the assignments | Groups may differ across modalities |
Diagnostics should target the assumptions that make these choices useful. Compare observed and fitted feature distributions within each block and cluster; persistent residual structure may indicate an inadequate likelihood, missing covariates, or dependence among features. Check whether estimated deviations are consistently large or one-sided across a block, which may indicate that the reference profile is poorly chosen. Assess sensitivity to plausible deviation scales: if conclusions change sharply as \(\tau_{mj}\) changes, the amount of pooling is driving the result. Sparse clusters deserve particular attention because their profiles can be strongly influenced by the prior.
A common mistake is to reason, “The samples share a cluster, so the profile parameter must also be shared.” That confuses common membership with equal measurements. The correction is to use the same \(z_i\) across blocks but define \(\theta_{mkj}\) separately for each block, adding a hierarchical link only when the parameters are meaningfully comparable. A second mistake is to interpret a small estimated deviation as proof that the modalities are biologically identical; strong shrinkage or little data can also produce small deviations.
These models remain approximations. Conditional independence can fail when features are correlated or when unmodeled batch effects remain. A shared partition can be unsuitable if modalities genuinely divide samples differently. Hierarchical pooling can transfer information in an unhelpful way when scales, feature definitions, or measurement processes are incompatible. Finally, a cluster-specific association in an omics profile is descriptive under this model; it is not evidence that one modality causally changes another.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a shared-partition model, what does sharing \(z_i\) imply, and what does it not imply?
- A Gaussian cluster profile has \(q=0\), \(\tau^2=1\), known \(\sigma^2=1\), \(n_k=3\), and \(\bar{x}=2\). Calculate the posterior mean and variance.
- Why is a common numerical profile reference not automatically appropriate for an RNA count rate and a binary probability?
- Name one diagnostic sign that the hierarchical deviation model may be poorly specified, and explain why it matters.
Solutions. (1) The same sample assignment is used in every modality, so samples are grouped jointly. It does not require equal modality-specific means, rates, probabilities, or other profile parameters. (2) The posterior mean is \((0/1+3(2)/1)/(1/1+3/1)=6/4=1.5\), and the variance is \(1/(1+3)=0.25\). (3) A rate and a probability have different meanings, supports, and likelihoods; a common reference is defensible only after specifying a meaningful scale or mapping. (4) For example, deviations that remain large and systematically signed across a modality suggest that the reference profile or exchangeability assumption is inadequate, so shrinkage toward that reference may distort the block’s profile estimates.