Bayesian Prior Sensitivity Analysis

Why Examine Prior Sensitivity?

A Bayesian calibration combines a likelihood, which represents how the observations relate to model parameters, with a prior, which represents information or constraints specified before those observations are used. In a mechanistic QSP model, prior choices may encode plausible parameter ranges, pathway knowledge, or expected correlations. Even when each prior is scientifically defensible, different choices can produce meaningfully different posterior conclusions—especially when the data weakly identify a parameter or a model prediction.

Prior sensitivity analysis asks how posterior inferences change when the prior is replaced by other scientifically credible alternatives while the model, observed data, and likelihood are held fixed. The point is not to identify a prior that gives a preferred answer. It is to establish which conclusions are supported across plausible prior assumptions and which depend on those assumptions.

The posterior under prior \(\pi_j\) is proportional to the likelihood times that prior:

A valid comparison therefore changes the prior and recalibrates the model under each alternative. Reusing a posterior from one prior while merely relabeling it as another prior is not a sensitivity analysis. Nor is changing the observation-error model, data, or ODE system at the same time: that would confound prior sensitivity with other sources of change.

What to Compare and What Must Stay Fixed

Before recalibrating, specify the scientifically defensible alternatives and the quantities to compare. Alternatives might reflect different expert estimates of a rate constant, a wider or narrower plausible range, or a different credible correlation between parameters. They should be motivated by biological knowledge or a transparent elicitation rationale—not selected after observing which one yields the most appealing result.

For each prior, compare more than a single parameter estimate. Useful summaries include posterior medians or means, credible intervals, dependence between parameters, and posterior distributions of quantities calculated from the calibrated model. A parameter can shift while an important prediction remains nearly unchanged; conversely, a small parameter shift can produce a large change in a nonlinear prediction. This is particularly relevant when posterior geometry is extended, correlated, or otherwise difficult to summarize by marginal intervals alone.

The comparison relies on common conditions: the same model structure and parameterization, the same data and observation-error assumptions, and a proper, well-defined posterior under every prior considered. The numerical procedure must also characterize each posterior adequately. If a computation fails, or explores only one part of a posterior, apparent differences between priors may instead be computational artifacts. A prior predictive check can reveal whether a proposed prior implies implausible model behavior before conditioning on the observed data.

A Normal Example: Changing the Prior Location

The effect of a prior-location change can be derived exactly in a simple model. Let \(\theta\) be a dimensionless log-rate parameter and suppose the hypothetical observation is \(y=0.80\). Assume the likelihood is \(Y\mid\theta\sim N(\theta,0.36)\), so the observation standard deviation is \(0.60\). Compare two normal priors with the same standard deviation \(0.40\): \(N(0.50,0.16)\) and \(N(1.00,0.16)\). These alternatives represent different prior centers, not claims about real QSP data.

To justify the result, multiply the likelihood and prior densities and retain the terms involving \(\theta\). Their exponent is minus one half of

where \(C\) does not depend on \(\theta\). Completing the square gives a normal density with the stated precision, variance, and mean. This derivation also shows that the posterior mean is a precision-weighted average of the observation and prior means.

Prior Sensitivity in a Derived Prediction

QSP interpretation often concerns a model output rather than a parameter in isolation. Let \(\theta=\log(k/(1\,\mathrm{h}^{-1}))\), where \(k\) is a positive elimination rate in \(\mathrm{h}^{-1}\). The fraction remaining after one hour in a first-order elimination model is \(q(\theta)=\exp[-\exp(\theta)]\). Because this function decreases as \(\theta\) increases, a posterior interval for \(\theta\) must have its endpoints reversed when transformed to an interval for \(q\).

When a derived quantity is not a simple monotone transformation, evaluate it across the joint posterior distribution of the parameters. Transforming separate marginal intervals independently can miss parameter dependence and need not produce an interval with the intended posterior coverage. In practice, the prediction should be calculated for each adequately characterized posterior draw or integration point, and the resulting distribution summarized.

Changing Prior Scale and Diagnosing Sensitivity

A different question is whether posterior conclusions depend on how concentrated a prior is, even when its center is fixed. A narrow prior can dominate a weakly informative likelihood; a broad prior may allow the data to move the parameter further. The following example isolates prior scale by holding the prior mean fixed.

A practical comparison should inspect posterior locations and spreads, correlations, and predictions under each prior. Also ask whether the observed data substantially update each prior and whether the alternative priors imply reasonable prior predictive behavior. Large posterior changes can be scientifically important, but they do not by themselves identify the cause: weak identifiability, nonlinear parameter dependence, and model structure can all contribute. The earlier material on posterior geometry in nonlinear ODE models is relevant when a single summary for each parameter obscures such structure.

Approaches, Failure Modes, and Limitations

Prior sensitivity analysis has limits. It cannot make an incorrect mechanistic model correct, resolve all non-identifiability, or determine which scientifically plausible prior is true from the posterior comparison alone. Its conclusions are conditional on the alternatives considered, the likelihood, and the model. If all tested priors share the same unsupported assumption, the analysis may appear robust while overlooking that shared weakness.

Several practical checks help distinguish a genuine sensitivity result from a flawed comparison:

• Keep the comparison controlled. Use the same data, mechanistic equations, parameter definitions, likelihood, and numerical accuracy under every prior.
• Check posterior adequacy. Confirm that each calculation represents the posterior distribution rather than an optimizer's point estimate or an inadequately explored region.
• Compare relevant predictions. Report changes in biological outputs as well as changes in parameter summaries.
• Use prior predictive reasoning. Reject alternatives that imply biologically implausible behavior unless the implausibility is itself the subject of investigation.
• State the scope. A conclusion of robustness means robustness across the priors actually examined, not across every possible prior.

A common mistake is to call the prior that gives the closest fit to the observed data the “best” prior and then report only that result. The data were already used in the posterior calculation, so choosing a prior by its posterior fit can make the analysis outcome-driven and conceal dependence on the prior. Instead, justify plausible alternatives using information independent of the calibration result, perform the recalibrations consistently, and report the range of conclusions across them.