Interpreting Cross-Omics Cluster Profiles

What a Cluster Profile Summary Describes

A fitted multi-omics clustering model can produce a posterior distribution for both sample assignments and modality-specific profile parameters. A useful profile summary must say which quantity is being summarized: a cluster’s mean expression for a feature, its rate of a binary event, or the posterior probability that a model-defined feature indicator is active. These are different quantities, and none is interchangeable with the probability that a particular sample belongs to the cluster.

Let \(i=1,\ldots,n\) index samples, \(m=1,\ldots,M\) modalities, \(j\) features within modality \(m\), and \(k=1,\ldots,K\) clusters. Write \(Z_i\) for sample \(i\)'s cluster label and \(\theta_{mkj}\) for the parameter describing feature \(j\) in modality \(m\), cluster \(k\). Posterior draws, indexed by \(s=1,\ldots,S\), provide joint values \(Z_i^{(s)}\) and \(\theta_{mkj}^{(s)}\). In practice, the draws should come from a converged sampling procedure, and cluster labels must have a consistent interpretation across draws. The earlier material on Label Switching and Identifiability in Mixture Posteriors addresses that label-alignment requirement.

Summarizing Means, Rates, and Feature Probabilities

For a cluster-specific mean, the posterior mean is the average of the sampled values \(\mu_{mkj}^{(s)}\). A posterior median can be useful when the draws are skewed, and posterior quantiles provide a credible interval. For example, a 95% equal-tail credible interval is bounded by the 2.5th and 97.5th percentiles of the posterior draws. Under the model and prior, this interval contains 95% of the posterior probability for the parameter. It is not a statement that the fixed parameter has a 95% probability of falling in this particular interval under repeated-sampling reasoning.

For a binary outcome, \(\pi_{mkj}\) is the probability of the event specified by the model, such as detecting a marker in a sample from cluster \(k\). Its posterior mean and quantiles summarize uncertainty about that cluster-specific rate. In contrast, if \(\gamma_{mkj}\) is a binary feature indicator, averaging its draws estimates the posterior probability that the feature is active or included. A posterior probability such as \(P(\gamma_{mkj}=1\mid Y)=0.8\) is a statement about that model-defined indicator, not an 80% positive detection rate.

All summaries should retain their context and units. A mean on a log-transformed expression scale is in log-scale units, not in the original measurement units. When several modalities are presented together, label the modality, feature, cluster, scale, and summary type. A table entry of “0.7” is not interpretable without knowing whether it is a rate, a feature probability, or something else.

Assignment Uncertainty and Parameter Uncertainty

A cluster profile can be uncertain for at least two connected reasons: the profile parameter may be uncertain even for a fixed assignment configuration, and uncertainty about the assignments can change the posterior distribution of that parameter. The distinction is clearest when all cluster labels are aligned and the same profile quantity is defined in every draw.

Let \(Z\) denote the vector of all sample assignments, and let \(\mu\) be a selected cluster-specific mean. The law of total variance gives an exact decomposition of its posterior variance:

The first term averages the remaining parameter variance within assignment configurations. The second term measures how the conditional posterior mean varies across assignment configurations. This is a decomposition of posterior uncertainty, not a claim that the two sources are independent or that assignment uncertainty causes the parameter uncertainty. Their interaction is reflected in how the conditional distribution of \(\mu\) changes with \(Z\).

Rates and Feature Probabilities Across Modalities

The same posterior-draw approach applies across modalities, but the likelihood and parameter must match the data type. For a binary feature, a model may specify \(X_{imj}\mid Z_i=k,\pi_{mkj}\sim\operatorname{Bernoulli}(\pi_{mkj})\). Here \(X_{imj}=1\) denotes the event defined for that feature, and \(\pi_{mkj}\) is its cluster-specific probability. This interpretation assumes the Bernoulli model is appropriate for the recorded outcome and that the cluster-specific rate is the quantity of interest. It does not justify treating a continuous measurement as binary without a defined threshold and modeling rationale.

For an indicator \(\gamma_{mkj}\), a posterior feature probability is computed by counting its active draws. A sample assignment probability is computed differently: \(P(Z_i=k\mid Y)\) is estimated by the fraction of draws assigning sample \(i\) to \(k\). The first summarizes uncertainty about a feature-level model state; the second summarizes uncertainty about an individual sample’s cluster membership. Neither one can substitute for the other.

Diagnostics and Choice of Summary

A profile table can look decisive even when its posterior summary is not trustworthy. Check convergence and effective sample size for the parameters or indicators being reported, not just for a global model statistic. A trace plot with long periods in one state, poor mixing between modes, or very few visits to a cluster can make means, quantiles, and feature probabilities unstable. For a rare cluster, inspect cluster occupancy across draws; a nominal cluster-specific mean may be poorly informed when few samples are assigned there.

Check that labels refer to the same clusters across draws before averaging cluster-indexed parameters. If labels are not aligned, averaging “cluster 1” across draws can combine different biological profiles. Also check units, transformations, and the event definition for binary rates. When a profile is calculated from assigned observations rather than taken directly from a model parameter, note that its denominator can vary across draws. Empty or nearly empty clusters may make such summaries undefined or highly variable.

Common alternatives answer different questions. A hard assignment followed by a within-cluster sample average is simple, but it discards assignment uncertainty and generally understates uncertainty if treated as fixed. A posterior mean of a model parameter uses all draws but should be accompanied by an uncertainty summary and assignment-aware interpretation. Posterior predictive summaries describe replicated or future observations generated from the fitted model; they include observation-level variation as well as parameter uncertainty, so they are not the same as a credible interval for a cluster mean. Co-clustering probabilities summarize whether pairs of samples share a cluster, rather than the modality-specific feature profile.

A further limitation is that a cluster profile is conditional on the chosen model, prior, feature scale, and number of clusters. Posterior summaries do not establish that a cluster is a natural biological subtype, that a feature causes membership, or that the model is correctly specified. Interpret profiles as model-based descriptions of the observed data, and report uncertainty rather than ranking clusters solely by point estimates.