Posterior Co-Clustering Probabilities

Why Summarize Clustering With Pairwise Probabilities?

A clustering model can describe each sample with a latent cluster label. But the numeric values of those labels are arbitrary: relabeling every cluster leaves the partition unchanged. When posterior draws assign different numbers to the same underlying groups, comparing the labels directly can therefore be misleading. This issue is known as label switching; the earlier material on Label Switching and Identifiability in Mixture Posteriors explains why label-specific summaries can fail.

A label-invariant question is whether two samples belong to the same cluster. Its posterior probability can be estimated directly from posterior draws, regardless of how clusters are numbered in each draw. The resulting pairwise probabilities summarize posterior evidence about shared membership. They are useful summaries, but they do not automatically specify a valid single partition.

The estimate is the proportion of retained draws in which the pair is together. The event \(z_i=z_j\) is unchanged if the cluster labels in any draw are permuted. Thus, this estimate is unaffected by label switching. The matrix with entries \(\hat{p}_{ij}\) is often called a posterior similarity matrix. Its diagonal entries equal 1, it is symmetric, and every entry lies between 0 and 1. A value near 1 indicates strong posterior support for the pair sharing a cluster; a value near 0 indicates little support.

Estimating Pairwise Probabilities From Draws

For each pair, the calculation is an average of indicator values. No alignment of cluster labels across draws is needed. The draws must, however, represent the posterior distribution of interest: the model and sampling procedure must be appropriate, and the retained draws must adequately explore that distribution. If posterior computation has not explored relevant partitions, the empirical proportions can be misleading even though the formula is correct.

The estimate is a Monte Carlo approximation to a posterior probability, not the probability that a particular fixed partition is correct. With independent draws, the approximate Monte Carlo standard error of \(\hat{p}_{ij}\) is \(\sqrt{\hat{p}_{ij}(1-\hat{p}_{ij})/M}\). Posterior simulation draws are often dependent, so this independent-draw expression can understate Monte Carlo error; dependence must be considered when judging numerical precision.

Constraints Implied by Partitions

Pairwise co-clustering events cannot vary arbitrarily. In every draw, if A is together with B and B is together with C, then A is together with C. This transitivity yields a useful probability constraint. Write \(E_{AB}\) for the event that A and B share a cluster, and define \(E_{BC}\) and \(E_{AC}\) similarly. The probability of the intersection of two events is at least the sum of their probabilities minus 1. Since \(E_{AB}\cap E_{BC}\) is contained in \(E_{AC}\),

The same argument applies to any three samples and any ordering of the pair. This is a necessary coherence condition, not a complete characterization of every possible posterior similarity matrix. It can nonetheless expose impossible proposed values or arithmetic errors.

Choosing a Representative Partition

One principled summary is to choose a valid partition whose pairwise same-cluster indicators are close to the posterior similarity estimates. For a candidate partition \(c\), let \(I(c_i=c_j)\) indicate whether the candidate puts samples \(i\) and \(j\) together. The least-squares summary associated with Dahl’s method selects a posterior-draw partition that minimizes the sum of squared differences between its pair indicators and the posterior similarity matrix. A broader version minimizes over all possible partitions, whether sampled or not.

The minimization balances pairwise evidence while respecting the requirement that a partition be transitive. A minimum need not be unique; if several partitions attain the same minimum, each is an equally good summary under this criterion. This is a decision summary, not a replacement for the full posterior distribution over partitions.

Alternatives, Diagnostics, and Limitations

Different summaries answer different questions. A pairwise threshold makes the rule easy to communicate but can violate transitivity. A least-squares summary uses all pairwise probabilities jointly and returns a valid partition, but its result depends on the chosen distance and may be non-unique. Another option is the maximum a posteriori partition: the partition with the greatest posterior probability. In practice, selecting the most frequent sampled partition estimates this mode among the sampled partitions. It can be difficult to estimate reliably when posterior mass is spread across many distinct partitions, and it need not best represent average pairwise co-clustering.

Several checks help identify problems. Confirm that the estimated matrix is symmetric, has ones on the diagonal, and has entries between 0 and 1. Check transitivity-based bounds for triples; violations beyond rounding indicate an error in pair counting or data handling. Inspect whether pairwise estimates are stable across adequately explored posterior draws or chains. A long run of nearly identical indicators may reflect dependence or poor exploration rather than overwhelming posterior certainty. These checks diagnose the estimated summary, but they do not establish that the underlying clustering model is appropriate for the multi-omics data.

Co-clustering probabilities also have limits. They describe posterior beliefs conditional on the specified model, prior, and observed data; they do not demonstrate biological causation or establish that a cluster has a particular biological meaning. Pairwise summaries can conceal uncertainty about the number of clusters and about competing global arrangements. A representative partition should therefore be reported as a convenient summary alongside the pairwise probabilities or other relevant posterior information, not as though it were the sole plausible answer.