Plausibility-Constrained Sampling of Parameter Sets

Why constrain sampled parameter sets?

A PBPK uncertainty analysis samples values for multiple inputs and evaluates the model for each set. Sampling each parameter from a marginal distribution can produce combinations that are individually in range but physiologically impossible together. For example, two tissue fractions may each be below one while their sum exceeds the fraction of body mass available to those tissues. Such sets are not merely unusual values: they violate a relationship that the model is meant to respect.

The earlier material on sampling parameter sets and representing correlated parameter uncertainty establishes why the joint distribution matters. Here the key question is how to sample while also enforcing a known constraint, and what distribution the resulting samples represent. A constraint should be stated in terms of the quantities and physiology it governs; an arbitrary cutoff can silently replace the intended uncertainty model with a different one.

This definition makes an important distinction. A hard constraint specifies which combinations are impossible or inadmissible under the model definition. A soft plausibility judgment, such as “this combination seems unlikely,” usually belongs in the joint distribution or in a sensitivity analysis, rather than being turned into a hard rejection rule without justification. Constraints may encode nonnegativity, a sum limit, a conservation relationship, or a known algebraic dependence. They do not by themselves establish that the remaining values or their probabilities are scientifically well supported.

Three ways to enforce constraints

There are three common strategies. Their results are not interchangeable unless they are constructed to target the same joint distribution.

• Joint distribution: Specify a distribution whose support already lies inside the admissible set. This can represent dependence directly and can be efficient, but requires defensible assumptions about the joint distribution.
• Conditional sampling: Sample from a specified distribution conditional on meeting the constraint. This is useful when the conditional distribution can be sampled correctly. Conditioning changes probabilities; it is not simply “sample one parameter, then truncate another” in every model.
• Rejection sampling: Draw from a proposal distribution and retain only draws satisfying the constraint. If the proposal is the intended unconstrained distribution, the accepted draws follow that distribution conditional on acceptance. Rejection is straightforward but can be inefficient when few draws pass.

If the constraint set has probability \(a=P(\boldsymbol{\theta}\in\mathcal{C})\) under the proposal, then the expected number of accepted draws from \(M\) independent proposals is \(Ma\). The expected number of proposals per accepted draw is \(1/a\). This calculation concerns computational efficiency; a high acceptance rate does not prove that the chosen proposal or constraint is scientifically appropriate.

Conditional sampling must preserve the target

A tempting alternative is to draw \(f_1\) from its original distribution, then draw \(f_2\) from its distribution truncated to values that satisfy the constraint. This always generates admissible pairs, but in general it does not generate the original joint distribution conditional on admissibility. For that conditional distribution, the probability of drawing \(f_1=x\) must also account for how much probability of \(f_2\) remains admissible at that \(x\).

For a joint density \(p(x,y)\) and constraint \(C\), the conditional density of the first component is proportional to \(p_X(x)P(C\mid X=x)\). Once \(x\) is drawn from this reweighted density, draw \(y\) from its conditional distribution given \(x\) and \(C\). This factorization follows directly from the definition of conditional probability:

The following example uses the same simple proposal as above, so the weighting can be seen explicitly. It also shows why an algorithm can obey a constraint while still targetting the wrong distribution.

Build relationships into a joint distribution

When parameters form a composition, a joint distribution can be more natural than independent marginal distributions followed by repair. Consider \(K\) nonnegative fractions \(f_1,\ldots,f_K\) that together exhaust a defined whole, so \(\sum_{k=1}^K f_k=1\). A Dirichlet distribution is supported on this simplex and therefore respects nonnegativity and the exact sum by construction. Its parameters must still be chosen to reflect the intended uncertainty and dependence; satisfying the sum alone does not guarantee that its marginal shapes are suitable.

An exact sum constraint is not appropriate if the modeled fractions cover only part of a total. In that case, include a residual component for the remainder, or use a distribution and constraint that represent the actual accounting relationship. Forcing a set of partial tissue fractions to sum to one would be a modeling error, even though the resulting samples are mathematically valid.

Check the target, not only the constraint

After implementing a constraint, inspect both whether every sample is admissible and whether the retained or generated values have the intended distribution. Useful checks include the acceptance rate, the number of failed draws, marginal summaries before and after conditioning, pairwise relationships, and whether boundary values accumulate unexpectedly. For a joint construction, verify the constraint numerically for every sampled set and check that the implied marginal means and variability agree with the specification.

When the acceptance rate is very low, rejection may be computationally wasteful and may make a finite simulation vulnerable to too few retained draws. A direct joint model or a correctly derived conditional sampler may be preferable. Conversely, a convenient joint distribution should not be selected only because it enforces a constraint: compare its implied dependence and marginal behavior with the scientific information available.

A common mistake is to sample all parameters independently, discard invalid combinations, and then report the original marginal distributions as if they still described the accepted sets. The accepted distribution is conditional on passing the rule, so its marginals and dependence can change. The correction is to define the target distribution explicitly, state whether the constraint is part of that target or a conditioning event, and summarize the actual distribution used for model runs.

Another failure mode is repairing a draw by changing one value after a constraint violation—for example, reducing \(f_2\) until \(f_1+f_2=1\). This deterministic repair can create a new, often unexamined dependence and concentrate samples on a boundary. Unless the repair transformation is itself the specified sampling model, it should not be treated as equivalent to conditioning or to sampling from a joint distribution.

Limitations and interpretation

A plausible parameter set is not necessarily a probable one, and a mathematically constrained sample is not proof that its input distributions are well calibrated. The constraint only removes or excludes combinations according to its definition. If the assumed constraint is too strict, valid uncertainty may be omitted; if it is incomplete, implausible combinations may remain. Report the constraint, the distribution before conditioning, the sampling method, the acceptance rate when relevant, and summaries of the resulting parameter sets.

Plausibility-constrained sampling is most useful when it makes the relationship between uncertain inputs explicit and preserves the distinction between uncertainty about parameters and variability in individuals. It should not be used to conceal a conflict among inputs or to force model outputs into an expected range. The resulting PBPK predictions depend on both the model and the constrained distribution chosen to represent its uncertain inputs.