Shared Partitions Versus Modality-Specific Partitions

Why the Partition Assumption Matters

A multi-omics study measures the same samples in several data blocks, such as RNA abundance and protein abundance. A clustering model must decide not only what profiles define its clusters, but also whether the same samples belong together in every block. A shared-partition model makes one assignment per sample and uses it for all modalities. A modality-specific-partition model makes a separate assignment for each block.

These are different assumptions about how cross-omics structure arises. Shared assignments express the view that one underlying grouping organizes the samples across the measured modalities, even if the profiles distinguishing groups differ by modality. Separate assignments allow, for example, two samples to be similar in one modality and dissimilar in another. Neither construction is automatically preferable: the choice depends on the scientific target and on whether the data support the constraint.

Two Generative Models

Let \(Y_i^{(q)}\) denote the observed data for sample \(i\) in modality \(q\), where \(i=1,\ldots,n\) and \(q=1,\ldots,M\). A block may contain many features. Assume initially that the same \(n\) samples are aligned across all \(M\) blocks, and that each block has a likelihood appropriate to its data type.

Definition: A shared partition assigns each sample one latent cluster label \(z_i\in\{1,\ldots,K\}\), used in every modality. A modality-specific partition assigns each sample a separate label \(z_i^{(q)}\in\{1,\ldots,K_q\}\) for each modality \(q\). A partition is the grouping itself; changing the numerical names of its clusters does not change the grouping.

For the shared model, let \(\pi=(\pi_1,\ldots,\pi_K)\) be cluster probabilities and let \(\theta_{qk}\) denote the profile parameters for cluster \(k\) in modality \(q\). A standard finite-mixture specification is

z_i\mid\pi\sim\operatorname{Categorical}(\pi_1,\ldots,\pi_K),\qquad Y_i^{(q)}\mid z_i=k,\theta_{qk}\sim f_q(\,\cdot\mid\theta_{qk}).

The key conditional-independence assumption is that, given the shared label and the relevant parameters, the modality blocks are independent:

p(Y\mid z,\theta)=\prod_{i=1}^{n}\prod_{q=1}^{M} f_q\!\left(Y_i^{(q)}\mid\theta_{q,z_i}\right).

Thus the shared model does not require the same feature distribution or profile in every modality. It requires the same sample grouping. The profile parameters can differ across blocks.

In the modality-specific model, each block has its own allocation probabilities \(\pi_q\), labels, and cluster profiles:

z_i^{(q)}\mid\pi_q\sim\operatorname{Categorical}(\pi_{q1},\ldots,\pi_{qK_q}),\qquad Y_i^{(q)}\mid z_i^{(q)}=k,\theta_{qk}\sim f_q(\,\cdot\mid\theta_{qk}).

A common baseline gives the allocation vectors and profile parameters independent proper priors across modalities. Under that specification, the data likelihood conditional on all latent variables factors by block. Dependence can be added through shared hyperparameters, but then the blocks need not be marginally independent even though their sample partitions are separate. The phrase “modality-specific partitions” alone does not determine those prior choices.

How the Constraint Changes the Likelihood

For fixed parameters, the shared model multiplies the evidence from each modality for the same candidate label. By Bayes’ rule, the conditional probability of assigning sample \(i\) to cluster \(k\), given the other labels and the parameters, is proportional to its allocation probability times its likelihood in every block:

P(z_i=k\mid Y_i^{(1)},\ldots,Y_i^{(M)},\pi,\theta) \propto \pi_k\prod_{q=1}^{M} f_q\!\left(Y_i^{(q)}\mid\theta_{qk}\right).

This follows because the factors for other samples do not depend on \(z_i\), while the factors for sample \(i\) are exactly the \(M\) displayed likelihood terms and the allocation probability \(\pi_k\). With uncertain parameters, the corresponding terms are posterior predictive probabilities integrated over parameter uncertainty. In either case, one block can reinforce or counteract another’s support for a shared assignment.

In the modality-specific model, the assignment in block \(q\) is informed by that block’s likelihood and allocation prior. It need not agree with assignments inferred from other blocks. This flexibility can improve block-wise fit, but it does not directly provide one unified clustering of the samples.

For model comparison, let \(\mathcal M_S\) denote the shared model and \(\mathcal M_D\) the separate-partition model. Their marginal likelihoods integrate over parameters and sum over possible allocations. For example, the shared-model marginal likelihood has the form

p(Y\mid\mathcal M_S) = \sum_{z}\int \left\{\prod_{i=1}^{n}\prod_{q=1}^{M} f_q(Y_i^{(q)}\mid\theta_{q,z_i})\right\} p(z,\pi,\theta\mid\mathcal M_S)\,d\pi\,d\theta.

The separate-partition model sums over a different label vector \(z^{(q)}\) for each block. When its priors are independent across modalities, its marginal likelihood factors into the product of the modality-specific marginal likelihoods. These marginal likelihoods account for both fit and the prior-supported range of possible partitions; comparing maximized likelihoods alone omits that integration.

Worked Example: One Label Must Reconcile Two Blocks

Suppose two clusters have equal allocation probabilities, and the profile parameters are treated as fixed for this illustration. For sample 1, the RNA likelihoods under clusters 1 and 2 are 0.80 and 0.20. The corresponding protein likelihoods are 0.30 and 0.70. Assume the blocks are conditionally independent given the shared label.

Under the shared model, the unnormalized support for cluster 1 is \(0.5(0.80)(0.30)=0.120\). For cluster 2 it is \(0.5(0.20)(0.70)=0.0700\). Therefore,

P(z_1=1\mid Y_1^{(1)},Y_1^{(2)},\pi,\theta) =\frac{0.120}{0.120+0.0700} =0.6316.

The shared model favors cluster 1, but the protein block favors cluster 2 on its own. The joint result is a compromise based on both likelihoods, not a claim that both modalities independently favor the same cluster.

With separate partitions, the RNA allocation probability for cluster 1 is \(0.80/(0.80+0.20)=0.800\), while the protein allocation probability for cluster 1 is \(0.30/(0.30+0.70)=0.300\). The separate model can therefore assign this sample to cluster 1 for RNA and cluster 2 for protein. This calculation uses fixed parameters and equal allocation probabilities; integrating uncertain profiles would replace the likelihood values with appropriate posterior predictive terms.

Comparing Models Through Evidence

A formal comparison can use posterior model odds. Let \(P(\mathcal M_S)\) and \(P(\mathcal M_D)\) be prior probabilities for the two model structures. Bayes’ rule gives

\frac{P(\mathcal M_S\mid Y)}{P(\mathcal M_D\mid Y)} = \frac{p(Y\mid\mathcal M_S)}{p(Y\mid\mathcal M_D)} \frac{P(\mathcal M_S)}{P(\mathcal M_D)}.

The marginal likelihood ratio is the Bayes factor for the shared model against the separate model. This comparison is meaningful only when the likelihoods, cluster-count assumptions, and proper prior distributions are specified for both models. In particular, priors on the number of clusters and on profile parameters can materially affect the evidence.

Worked Example: Converting Marginal Likelihoods to Posterior Model Probabilities

Consider a hypothetical two-modality analysis with marginal likelihoods \(p(Y\mid\mathcal M_S)=0.0120\) and \(p(Y\mid\mathcal M_D)=0.0300\). Suppose the two model structures have equal prior probabilities, \(P(\mathcal M_S)=P(\mathcal M_D)=0.5\). The Bayes factor for shared versus separate partitions is

BF_{S,D}=\frac{0.0120}{0.0300}=0.400.

The posterior odds are \(0.400\times(0.5/0.5)=0.400\). Converting these odds to a posterior probability gives

P(\mathcal M_S\mid Y) =\frac{0.400}{1+0.400} =0.2857,\qquad P(\mathcal M_D\mid Y)=1-0.2857=0.7143.

Under the stated prior probabilities and model assumptions, the posterior probability of the separate-partition model is 0.7143. These numbers are hypothetical integrated likelihoods, not likelihoods for one selected assignment. The conclusion is conditional on the specified candidate models and priors; it does not show that the modalities have no biological relationship.

Choosing a Structure and Diagnosing Tension

The shared model is appropriate when the scientific goal is a common sample taxonomy and there is a defensible reason to expect that grouping to be expressed across blocks. It can borrow evidence across modalities and yields one directly interpretable partition. Its central risk is forcing a compromise: strong structure in one block may overwhelm weaker or differently organized structure in another.

Separate partitions are useful when different biological processes plausibly organize different modalities, or when the aim is to describe within-block groupings rather than produce one consensus. Their additional flexibility can fit heterogeneous blocks, but comparisons of cluster labels across modalities then have no automatic meaning. A cluster numbered 1 in RNA is not thereby the same group as cluster 1 in protein.

Useful diagnostics include checking whether conclusions about the shared grouping are sensitive to removing or down-weighting one block, examining block-wise fit under the shared allocation, and checking whether a separate-partition fit reveals substantial disagreement. Poor fit in one modality, unstable assignments under modest modeling changes, or a shared partition that tracks only the strongest block are warnings that the shared constraint may be too restrictive. Conversely, many small, weakly supported groups in separate fits may reflect excess flexibility rather than genuine modality-specific organization. Such checks diagnose tension; they do not by themselves prove which structure is correct.

A middle ground is a partially shared or hierarchical allocation structure, in which some grouping information is common but modality-specific deviations are permitted. This can represent more nuanced relationships, but it introduces further parameters and assumptions that require their own justification. It should not be treated as a cost-free compromise between the two basic models.

Common pitfall: “The shared model uses one cluster profile for all modalities.” That is incorrect: it uses one sample label across modalities, while each modality can have its own cluster-specific profile parameters. A second mistake is to choose the separate model solely because it has a larger maximized likelihood. Its flexibility generally makes that unsurprising; compare properly specified marginal likelihoods or another criterion suited to the analysis goal.

Assumptions and Limitations

Both constructions rely on correctly specified block likelihoods and on a meaningful mapping of samples across modalities. The shared model additionally assumes that one partition is adequate for all blocks. Conditional independence of blocks given allocations and profile parameters is a modeling assumption, not a guaranteed property of multi-omics data. Residual cross-block dependence can make evidence appear stronger than it should under the model.

Missing measurements, batch effects, outliers, and feature dependence can also distort the comparison if the likelihood does not represent them. A model with a single shared label need not capture every cross-modal relationship, and separate partitions do not explain why assignments differ. Model evidence is sensitive to prior choices, especially in mixture models with uncertain cluster counts and many possible partitions. Therefore, report the model structure, the relevant priors, and the basis for choosing it; do not present a selected clustering as a uniquely established biological truth.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In a shared-partition model, must the RNA and protein cluster profiles be identical?
If the shared-model likelihood for one sample and cluster \(k\) is 0.40 in block 1 and 0.25 in block 2, what is their joint likelihood contribution under conditional independence?
A modality-specific model gives different groupings in two blocks. Does that mean cluster 1 in one block corresponds to cluster 2 in the other?
With prior odds of 2 to 1 in favor of the shared model and a Bayes factor of 0.30 for shared versus separate, what are the posterior odds and the posterior probability of the shared model?
Name one diagnostic warning that could indicate the shared-partition assumption is too restrictive.

Solutions. (1) No. The sample labels are shared, but profiles may be modality-specific. (2) \(0.40\times0.25=0.100\). (3) No. The labels are arbitrary within each modality, and separate partitions do not establish cross-block cluster correspondence. (4) Posterior odds are \(0.30\times2=0.600\) to 1; hence the shared-model probability is \(0.600/(1+0.600)=0.375\). (5) For example, poor block-wise fit or substantial assignment disagreement when the blocks are analyzed separately can indicate that one common grouping is forcing a compromise.