Propagating Uncertainty in Joint Representations

Why Propagate Uncertainty in a Joint Representation?

A joint representation places cells in a common space using information from multiple modalities, such as RNA and surface protein measurements. A neighborhood graph derived from that representation summarizes which cells are close under the chosen analysis. But that graph is not a fixed biological object: changing the selected features, the cells used to construct it, or the relative influence of modalities can alter which cells are neighbors.

The aim here is to measure how sensitive particular cell relationships are to such choices. This is a stability analysis, not a claim that the graph has a known probability of being biologically true. Its result depends on which perturbations are considered plausible and how they are sampled. The earlier material on weighted nearest neighbors supplies the basic setting: a joint neighborhood can reflect modality-specific information and cell-specific modality weights. This tutorial asks how much a stated neighborhood relationship persists when those inputs vary.

Eligibility matters. In a cell-resampling run, a cell may be absent, so an edge for that pair cannot be evaluated. A missing edge and an unevaluable pair are not the same outcome. We will therefore report both pair eligibility and edge persistence among eligible runs.

Define the Relationship and the Perturbation Distribution

Before calculating stability, specify the graph construction and keep its rules fixed across runs. For example, choose a value of \(k\), construct a \(k\)-nearest-neighbor graph from the joint representation, and define an undirected edge using the union rule: cells \(i\) and \(j\) are connected if either cell is among the other’s \(k\) nearest neighbors. A mutual-neighbor rule, which requires both directions, is also possible, but it answers a different question. Do not switch rules between perturbation runs.

Let \(Q\) denote a stated distribution over analysis perturbations. A draw \(q\sim Q\) might specify a feature subset, a cell resample, and a modality-weight setting. The full pipeline should be rerun as appropriate for that perturbation, including any preprocessing step whose result is meant to respond to the changed inputs. For a fixed pair \((i,j)\), define \(E_{ij}(q)=1\) if both cells are present and eligible for comparison, and \(E_{ij}(q)=0\) otherwise. When \(E_{ij}(q)=1\), let \(A_{ij}(q)=1\) if the specified edge is present and \(A_{ij}(q)=0\) if it is absent. \(A_{ij}(q)\) is not defined when the pair is ineligible.

When all cells are present in every run, coverage is 1 and persistence is simply the edge probability across perturbations. When cells are resampled, define the perturbation distribution for persistence conditional on both cells being eligible, or explicitly estimate persistence among the eligible runs. The conditioning convention must be stated because it changes the target being summarized.

Estimating Persistence and Its Simulation Error

Suppose \(B\) independent perturbations are drawn from a distribution in which the pair is eligible in every draw. Write \(A_{ij,b}\) for the edge indicator in run \(b\). Then the estimate of conditional persistence is the sample proportion of runs containing the edge. Independence of runs and a common perturbation distribution make the indicators identically distributed Bernoulli variables with success probability \(p_{ij}^{(Q)}\).

The expectation follows by linearity: each indicator has expectation \(p_{ij}^{(Q)}\), so the mean of \(B\) such indicators has expectation \(B^{-1}B p_{ij}^{(Q)}=p_{ij}^{(Q)}\). The displayed standard error estimates Monte Carlo variability from drawing a finite number of perturbations. It does not quantify every source of uncertainty in the data or establish a probability that the edge is biologically real. A normal-approximation interval based on this standard error is most credible when the number of runs and the expected counts of edge and non-edge outcomes are not small; otherwise, use a suitable binomial interval or report the counts directly.

If the \(B\) runs are sampled without conditioning on eligibility, let \(M_{ij}=\sum_{b=1}^{B}E_{ij,b}\). Estimate coverage by \(M_{ij}/B\), and estimate persistence among eligible runs by \(\sum_b E_{ij,b}A_{ij,b}/M_{ij}\), provided \(M_{ij}>0\). This ratio estimates the conditional probability, but with a random denominator it is not generally exactly unbiased in a finite sample. It is consistent as the number of independent runs grows when pair eligibility has positive probability. Do not code an ineligible run as an absent edge and average over all \(B\): that would estimate the joint probability \(P(E_{ij}=1,A_{ij}=1)=c_{ij}p_{ij}^{(Q)}\), not conditional persistence.

Resampling Cells: Keep Coverage Separate

Cell resampling changes the set of cells available to construct the representation and graph. If the pair is absent in a run, its edge status is undefined, not zero. Two summaries are informative: pair coverage and conditional edge persistence. Low coverage means the relationship is assessed in relatively few runs, so its persistence estimate may be imprecise and may describe a selected subset of runs.

One clean design is to draw cell-resampling runs from the original resampling procedure conditional on both cells being retained. If these eligible runs are independently sampled from that conditional distribution, the mean edge indicator is unbiased for persistence under that conditional distribution, by the calculation above. Alternatively, use all runs and calculate the edge fraction among eligible runs. In either case, state whether the resampling scheme is with or without replacement and whether the graph is constructed on the resampled cells or on a fixed reference set.

Varying Modality Weights and Comparing Perturbations

Modality-weight perturbations ask whether a relationship depends on the relative contribution of modalities. Specify the weight settings or a distribution over weights before examining the results. If the goal is to summarize an average over plausible settings, the weights in \(Q\) should reflect that goal; an arbitrary collection of settings does not automatically represent a meaningful probability distribution. It is often useful to report persistence separately by setting as well as the average, so that a high overall value does not hide a relationship that disappears under a scientifically plausible weighting.

Diagnostics, Comparisons, and Limitations

Stability summaries are most useful when they expose failure modes rather than compress them into one score. Check whether conclusions change when the feature-perturbation scheme, cell-resampling design, or plausible modality-weight range changes. Examine the number of eligible runs for every reported pair; small denominators make persistence estimates noisy. Also check whether results depend on the graph’s \(k\), distance metric, directed-to-undirected conversion, or tie handling. Those are analysis choices and should either be held fixed or included explicitly in the perturbation design.

A single fixed embedding provides a concise graph but no direct measure of sensitivity to analysis choices. Repeated perturbation provides that sensitivity assessment, at the cost of computation and dependence on the chosen perturbation distribution. A comparison of several hand-selected settings can be easier to interpret, but does not estimate an average probability unless a sampling distribution and weighting scheme are specified. Neither approach replaces biological validation or a statistical model for a separate scientific outcome.

The binomial standard-error calculation assumes independent perturbation runs. Runs that reuse random seeds, share resampled cells, or are generated in correlated batches may not provide \(B\) independent pieces of Monte Carlo information. In that case, the displayed standard error can be too small; use independent batches or a variance estimate that respects the dependence. Edge indicators for different cell pairs within a run are also dependent, so do not treat all pairwise edges as independent observations when making a broader inferential claim.

A useful report might say: “Under the prespecified feature and weight perturbations, this edge was present in 68 of 100 eligible runs (persistence 0.68); both cells were eligible in all 100 runs.” For resampling, add the eligibility count and make clear that the persistence denominator contains only runs in which both cells could be compared. Avoid calling a low persistence value proof that the relationship is absent: it can instead indicate sensitivity to the chosen analysis pipeline.