Quantifying Uncertainty in Sample Cluster Assignments

Why Quantify Assignment Uncertainty?

A cluster label is often reported as if it were certain: sample 12 belongs to cluster 2. In a Bayesian integrative clustering model, however, the observed omics data generally support several possible assignments. A sample may be assigned to cluster 2 in most posterior draws but to cluster 1 or 3 in others. Reporting only the most probable label conceals that uncertainty, which can matter when interpreting cluster profiles or deciding whether a sample has a stable classification.

This tutorial summarizes that uncertainty using posterior assignment probabilities and entropy. These are conditional summaries: they describe uncertainty in the latent assignment given the observed data and the specified model. They do not establish that the model is correctly specified, nor do they imply that cluster membership causes any clinical or molecular outcome.

Posterior Assignment Probabilities

The probabilities integrate uncertainty in the latent assignment and, in a Bayesian model, uncertainty in model parameters under the posterior. In practice, posterior draws often estimate them by counting assignments. If \(M\) posterior draws have comparable cluster labels, then

where \(I(\cdot)\) equals 1 when the statement is true and 0 otherwise. With equally weighted draws, this is the fraction of draws assigning sample \(i\) to cluster \(k\). If draws have unequal weights, use the corresponding normalized weighted fraction instead.

A necessary condition is that cluster \(k\) means the same thing across the draws being counted. Mixture-model labels can switch names without changing the represented partition. Thus, counting raw labels across posterior draws can produce meaningless probabilities if labels have not been aligned or otherwise given a consistent interpretation. An alignment procedure can match labels to a reference partition, but when the match is ambiguous, the resulting label-specific probabilities depend on that choice. Posterior co-clustering probabilities, which concern whether two samples share a cluster, avoid this label-naming problem; they answer a related but different question and do not by themselves provide probabilities for membership in named clusters.

The summaries below assume a specified set of \(K\) mutually exclusive and exhaustive cluster labels with comparable meaning across the posterior draws used. If the number of clusters varies, or if a cluster has no stable correspondence across draws, the analyst must define a meaningful reference or membership event before reporting label-specific probabilities. The formula for entropy does not solve that correspondence problem.

What Entropy Measures

Entropy is low when most posterior probability is concentrated on one cluster and high when probability is spread over several clusters. Its minimum, 0 nats, occurs when one cluster has probability 1 and all others have probability 0. Its maximum, \(\log(K)\) nats, occurs when all \(K\) clusters have equal probability \(1/K\). Normalization expresses entropy relative to this maximum for the chosen number of clusters; it does not make different models or different cluster definitions automatically comparable.

A useful justification follows by comparing the assignment distribution \(q_i=(q_{i1},\ldots,q_{iK})\) with the uniform distribution \(u_k=1/K\). The Kullback-Leibler divergence from \(q_i\) to the uniform distribution is

Because \(D_{\mathrm{KL}}(q_i\|u)\geq0\), it follows that \(H_i\leq\log(K)\). Also, each term \(-q_{ik}\log(q_{ik})\) is nonnegative for \(0\leq q_{ik}\leq1\), so \(H_i\geq0\). Equality at the upper bound requires \(q_i\) to be uniform; equality at zero requires a probability of 1 on a single cluster.

The quantity \(\exp(H_i)\) can also be reported as the effective number of clusters for that sample. It is the number of equally probable clusters that would have the same entropy. It is a convenient transformation, not a literal count of clusters to which the sample belongs.

Entropy Alongside Other Assignment Summaries

Entropy is complementary to, rather than a replacement for, the maximum posterior probability. The maximum reports the probability of the chosen label; entropy uses the entire assignment distribution. A simple margin, defined as the largest probability minus the second largest, provides another summary of how separated the leading assignment is from its closest competitor. No single number fully describes the posterior distribution.

Diagnostics, Interpretation, and Limitations

Before interpreting assignment uncertainty, check that each probability vector is valid: all entries must be between 0 and 1, and their sum should be 1 up to numerical rounding. Check how labels were made comparable across draws, especially when a model allows empty components or a varying number of clusters. If a component is absent in a draw, its assignment probability for that draw is zero only when that component has a defined, comparable identity in the chosen representation. An arbitrary label match should not be treated as a substantive result.

Entropy describes uncertainty in the modeled latent assignment conditional on the data and model. It does not measure whether the data fit the model, whether cluster labels are biologically meaningful, or whether a sample truly has a “mixed” biological identity. A high entropy may reflect genuinely overlapping evidence, weak information for that sample, an overly flexible cluster representation, or difficulties in the model or its label alignment. These possibilities require substantive and model-based investigation; entropy alone cannot distinguish them.

• Do not call a sample certain from its MAP label alone. Report the leading probability or the full probability vector when the uncertainty matters.
• Do not compare raw entropy values across different \(K\) without context. The maximum possible entropy changes with \(K\); normalized entropy helps describe the fraction of that maximum but does not eliminate differences in model definitions.
• Do not confuse posterior uncertainty with Monte Carlo error. More posterior draws can reduce simulation error in estimated probabilities, but they do not necessarily make the underlying posterior assignment more concentrated.
• Do not ignore label switching. Frequencies of arbitrary component numbers are not valid assignment probabilities unless the labels have a coherent correspondence across draws.

In practice, a useful report gives the maximum-probability cluster, its posterior probability, and either the full assignment vector or an entropy summary. When comparing samples, state the number of clusters and whether entropy is raw or normalized. For decisions that depend on a threshold, specify the threshold and its purpose rather than treating one probability cutoff as universal.