Nested Sampling for Parameter Uncertainty and Variability

Why the Sampling Structure Matters

A PBPK prediction can vary for two fundamentally different reasons. The population-level parameter values may be uncertain because available data do not determine them exactly. Separately, individuals may differ around those population values because of biological variability. For example, uncertainty in a typical clearance parameter is not the same thing as the actual spread of clearance among people. A simulation intended to describe both needs to represent both sources, and to preserve their distinct roles.

The distinction matters especially when predicting several individuals under one model. A population-level parameter represents the same uncertain model quantity for every individual in that simulated population. Individual-level variability is sampled separately for each individual, conditional on that shared parameter draw. Resampling the population parameter independently for every person changes the dependence among predictions and can make a population mean appear more precise than it should.

The uncertainty distribution \(p_U(\theta)\) must have a stated interpretation: for example, it may be a posterior distribution, an elicited distribution for uncertain inputs, or another justified representation of parameter uncertainty. It is not automatically the distribution of values found across individuals. The individual distribution \(p(\eta\mid\theta)\) should represent the intended biological or physiological variability, with relevant dependencies included. These distributions and the PBPK model must be suitable for the population and prediction target being studied.

The Nested Prediction

For a single future individual, nested sampling approximates the predictive distribution by drawing \(\theta\) from \(p_U(\theta)\), then drawing that individual's \(\eta\) from \(p(\eta\mid\theta)\), and evaluating \(Y=g(\theta,\eta)\). For a simulated group, draw \(\theta\) once for the group and draw separate individual values \(\eta_j\) for its members. Conditional independence of individuals given \(\theta\) is a common assumption, but it is not automatic: shared exposures, family structure, or other dependencies may require a joint distribution for the individual-level variables.

The resulting predictive cumulative distribution function for one individual is

where \(\mathbb{I}\{\cdot\}\) equals 1 when the condition inside braces holds and 0 otherwise. In a simulation with \(M\) outer draws and \(N\) individuals per outer draw, write the draws as \(\theta^{(m)}\) and \(\eta^{(m,j)}\). The output for individual \(j\) under outer draw \(m\) is

The pooled values \(Y^{(m,j)}\) represent individual predictions across both uncertainty sources. In contrast, the inner average for each outer draw,

approximates the model-implied population mean conditional on \(\theta^{(m)}\). The collection of these outer-level averages describes how that conditional population mean changes across plausible parameter values, subject to finite inner-simulation error.

Why the Nesting Is Correct

The essential feature is that every member of a simulated group uses the same outer draw. To see why, consider two individuals whose individual-level variables are conditionally independent given \(\theta\). Conditional on \(\theta\), their outputs are independent under this assumption. But across the full nested simulation, they share uncertainty about \(\theta\). This yields the identity

when their conditional covariance given \(\theta\) is zero. The identity follows by writing each output as its conditional mean plus a residual: \(Y_j=E[Y_j\mid\theta]+(Y_j-E[Y_j\mid\theta])\). The conditional residuals have mean zero given \(\theta\), so their covariances with functions of \(\theta\) are zero. Taking the covariance leaves the covariance between the two conditional means. Thus shared parameter uncertainty induces dependence in joint predictions, even when individual variability is conditionally independent.

A one-stage simulation can be equivalent to nested sampling for a single individual's marginal prediction if it draws from exactly the same joint distribution \(p_U(\theta)p(\eta\mid\theta)\). The problem is not whether the code uses one loop or two. It is whether the sampling represents the intended joint structure—particularly whether a population parameter is shared across individuals when it should be.

Worked Consequences for a Simulated Group

Threshold Probabilities and Their Interpretation

Practical Checks and Common Failure Modes

Before interpreting simulation output, verify that the sampling design matches the prediction target. A prediction for one new individual requires one draw from the population-parameter uncertainty distribution and one conditional individual draw. A group prediction requires a shared outer draw for the group, followed by individual-level draws. If the target is the distribution of a population mean across plausible parameters, calculate an inner mean for each outer draw rather than treating every pooled individual output as a separate population mean.

• Check conditional distributions. For selected outer draws, inspect the range and plausibility of inner outputs. An unexpected shift can reveal a coding error, an incorrect conditional distribution, or a model response to \(\theta\) that was not anticipated.
• Check parameter sharing. Confirm that a population-level draw is reused where the simulated individuals represent one population under one uncertain model. Accidental redrawing per individual can suppress shared uncertainty in group summaries.
• Check for double-counting. If a population parameter distribution already includes between-individual variation, adding a separate individual variability distribution for the same source can exaggerate spread.
• Check the target of every summary. A pooled quantile describes individual predictions across the chosen mixture of sources; an outer distribution of inner means describes uncertainty in a conditional population summary. They answer different questions.
• Check support and dependence. Outer draws must be valid for the PBPK model, and conditional individual draws must respect required bounds and dependencies. Invalid or implausible inputs can produce misleading outputs even when the simulation loops are implemented correctly.

A common mistake is to combine parameter uncertainty and individual variability into one undifferentiated distribution, draw a value from it independently for each person, and interpret the resulting spread as biological heterogeneity. That may reproduce a single-individual marginal distribution in some cases, but it generally loses the shared-parameter structure for groups. The correction is to state what each distribution represents and preserve the conditional relationship in the joint simulation.

Nested Monte Carlo is not the only valid computational approach. A joint sampler can give the same result if it preserves the same joint distribution and shared outer draws. Deterministic integration or other sampling designs can also approximate the required integrals, especially when simulation cost is high. These alternatives change how the integrals are evaluated, not which uncertainty and variability structure the model should represent. No sampling design can compensate for unsupported distributions, a misspecified PBPK model, or an unclear prediction target.