Distinguishing Cell-State Uncertainty From Fate Probability

Two Questions That Can Have Different Answers

A cell near a branchpoint may be difficult to place on either branch from its measured expression. Separately, a cell in a transitional state may have a genuine chance of reaching either terminal fate. These are different questions: how uncertain is the model’s assignment of the observed cell? and what is the probability of each biological outcome, given the cell’s state and context? Confusing them can turn uncertainty about measurement or model fit into an unsupported claim about cell fate.

The distinction depends partly on what a model’s branch label means. A latent label may identify a cell’s current trajectory segment, or it may represent the cell’s eventual terminal lineage. Those definitions are not interchangeable. The interpretation of any reported probability therefore begins by naming the random quantity and the information on which the probability is conditioned.

A posterior assignment is conditional inference about a latent label. Its uncertainty can arise because the cell’s measured features are compatible with multiple branches, because model parameters are uncertain, or because the model itself is only an approximation. A fate probability describes the distribution of a specified outcome under specified conditions. It may reflect stochastic biological processes, or it may be an operational conditional probability for a population of comparable cells. In either case, it is not automatically equal to uncertainty about a latent label.

The earlier material on soft lineage assignment and Bayesian inference for lineage fate probabilities provides the relevant modeling ideas: branch labels can be latent, and fate probabilities can be estimated probabilistically. Here the essential additional discipline is to state whether the latent label means current state or eventual fate, and to keep assignment and outcome probabilities distinct unless the model explicitly equates them.

Define the Random Quantity Before Interpreting a Probability

Suppose a fitted model assigns a cell probabilities of 0.55 to branch A and 0.45 to branch B. This statement is incomplete without the label definition. If \(Z_i\) denotes the cell’s current trajectory segment, the probabilities express uncertainty about where the cell belongs in the fitted representation. They do not say that the cell has a 55% biological chance of ending in fate A. If \(Z_i\) instead explicitly denotes the eventual fate \(F_i\), and the model is calibrated for the relevant state and context, the assignment probability can also serve as a model-based fate prediction. That interpretation requires those assumptions; it does not follow from the numbers alone.

In Bayesian notation, a fate probability may depend on unknown model parameters. If \(q_{ib}(\theta)=P(F_i=b\mid S_i,E_i,\theta)\), then the probability of fate \(b\) conditional on the available data is the posterior predictive probability:

This averaging gives a predictive probability for the outcome while accounting for posterior uncertainty in \(\theta\). The posterior distribution \(p(\theta\mid D)\) and the predictive probability are related but not identical: the first describes uncertainty about model parameters, while the second describes the model’s probability for a future fate outcome after that uncertainty is integrated out. A posterior probability for \(Z_i=b\) answers a different question unless \(Z_i\) has been defined as that outcome and the relevant conditioning is the same.

Transitional Cells and Competing Outcomes

A transitional cell makes the distinction especially important. It may be confidently located in a shared trunk or intermediate state, even though its eventual destination is not yet determined. Thus, high confidence about a cell’s present position does not imply high confidence about its future fate. Conversely, a cell can be hard to assign to a current branch while its fate probability, under a separate validated model, is quite unequal between outcomes.

One simple fate model treats transitions to two terminal fates as competing events. Assume that, conditional on the current state and context, the instantaneous transition rates are constant: \(\lambda_A\) per day to fate A and \(\lambda_B\) per day to fate B. Assume also that these are the only possible exits, that neither rate changes over time, and that the cell remains at risk until one event occurs. The probability of no transition by time \(t\) is \(e^{-(\lambda_A+\lambda_B)t}\). The probability of transitioning to A in a short interval after time \(t\) is approximately \(\lambda_A\,dt\), conditional on still being in the state. Integrating over all possible transition times yields the eventual probability of A:

The final equality follows because the integral of \(e^{-ct}\) from zero to infinity is \(1/c\) for \(c>0\). The probability for B is correspondingly \(\lambda_B/(\lambda_A+\lambda_B)\). These are fate probabilities under the stated constant-rate, two-outcome model, not posterior probabilities that a cell currently belongs to a branch.

Derivation: Why Better Assignment Need Not Change Fate Propensity

The source of assignment uncertainty can be seen directly from Bayes’ rule. For two possible labels A and B, posterior odds equal prior odds multiplied by the likelihood ratio:

If the observed features have nearly equal likelihood under A and B, the likelihood ratio is near one. The data then provide little information for distinguishing the labels, so the posterior assignment remains close to the prior odds. This is a statement about the information in the observed features under the model. It is not a derivation of the cell’s biological transition propensity.

A fate probability changes when information relevant to the fate model changes, or when the fate model’s parameters are updated. Improving measurement of a current label may sharpen a posterior assignment without changing the biological state, context, or fate model. If the improved measurement also provides validated information about future fate, then the fate prediction should be updated too. Whether it does so is a modeling question, not something implied by a more confident branch assignment.

Diagnostics, Alternatives, and Limitations

Before reporting a probability, check the label definition, the conditioning information, and the model’s biological interpretation. A posterior predictive check can reveal that a branching model fails to reproduce relevant features of the data, but satisfactory checks do not by themselves prove that its latent labels equal biological outcomes. Evidence about model fit and evidence about fate semantics are separate requirements.

• Check what the label represents. If it denotes a current trajectory segment, report it as a segment-assignment probability, not as the probability of terminal fate.
• Check what information conditions the probability. State whether the quantity is conditional on expression, current state, context, fitted parameters, or an outcome model. Probabilities with different conditioning information need not agree.
• Check calibration and outcome definitions. A fate probability needs a defined endpoint and a model supported by relevant outcome information. A soft clustering output alone does not establish calibration for future fate.
• Check assumptions about transitions. Constant competing rates are convenient but may fail if rates vary with time, cells can enter a third fate, or the model omits cell death or another exit. Then the two-rate formula is not justified.
• Inspect ambiguity rather than hiding it. If several labels have similar posterior probabilities, a hard label conceals uncertainty. If one label dominates, that still does not establish a biological fate probability unless the label and model support that interpretation.

Hard assignment replaces a probability vector with the most probable label. It is simple to display, but discards uncertainty and can make borderline cells look certain. Soft assignment preserves the posterior distribution over labels, but it remains a statement about those labels as defined by the model. Direct fate prediction instead targets eventual outcomes and should be based on an explicitly defined outcome model; a posterior predictive probability can account for uncertainty in that model’s parameters. None of these approaches removes the need to justify the meaning of the target quantity.

The principal limitation is that observational expression data may not identify either a unique latent branch assignment or an individual cell’s eventual fate. A high posterior probability can still be conditional on a misspecified model, while an estimated fate probability can be poorly calibrated or apply only to a different state or context. Finally, probability does not imply a fractional biological outcome: the cell either realizes a particular outcome under the defined endpoint or it does not. The probability describes uncertainty or outcome frequency under stated conditions, not a literal fraction of the cell.