Why Input Distributions Matter
Global sensitivity analysis asks how model outputs vary over a specified collection of plausible inputs. The parameter ranges and probability distributions define that collection: they determine which parameter values are explored, how often different regions are sampled, and therefore what “sensitivity” means for the analysis. A parameter ranking is not a property of the model alone; it is conditional on the input space and its probability measure.
For a deterministic QSP model, write the output of interest as \(Y=f(\boldsymbol{\theta})\), where \(\boldsymbol{\theta}=(\theta_1,\ldots,\theta_p)\) is the vector of uncertain inputs. An input distribution is a joint probability distribution \(\pi(\boldsymbol{\theta})\) over a specified feasible region. It must describe both the range of each input and, when relevant, how inputs vary together. A finite range alone does not say how probability is allocated within it.
The distinction from a calibrated posterior is important. A posterior distribution combines a prior with observed data under a likelihood or other calibration procedure. A sensitivity analysis may instead use scientifically justified ranges, an elicited distribution, a posterior distribution, or another explicitly chosen measure. Each answers a different question. If posterior draws are used, conclusions concern sensitivity over parameter combinations considered plausible after conditioning on those data and assumptions—not over every biologically plausible parameter combination.
Before choosing a distribution, specify the quantity and its units, the model’s valid domain, the source of uncertainty, and any dependencies among inputs. For example, a rate constant must be positive; a fraction must lie between zero and one; and fractions that partition a total may also need to sum to one. Bounds should reflect scientific knowledge and model validity, not be chosen merely to make sampling convenient.
Bounds, Scale, and Probability Weight
A bounded uniform distribution is a simple choice when every value in a stated interval is considered equally plausible on the chosen scale. For \(\theta\sim U(a,b)\), its density is \(1/(b-a)\) for \(a\leq\theta\leq b\), and zero outside that interval. Uniformity is not neutral: it is a substantive assumption about equal probability per unit of the selected scale. Uniformity on the original scale and uniformity on the logarithmic scale put different weights on parameter values.
A logarithmic scale is often useful for positive parameters that can plausibly vary over orders of magnitude, such as a rate constant or potency. If \(\log_{10}(\theta)\sim U(L,U)\), then the parameter is sampled uniformly in its base-10 logarithm between \(10^L\) and \(10^U\). This is sometimes called a log-uniform distribution. It assigns equal probability to intervals of equal width in log space, not equal probability to intervals of equal width in the original units.
The density formula follows by change of variables: \(f_\theta(\theta)=f_Z(\log_{10}\theta)\left|d(\log_{10}\theta)/d\theta\right|\), and \(d(\log_{10}\theta)/d\theta=1/(\theta\ln(10))\). The resulting density is larger per unit of \(\theta\) at smaller values, even though equal-width intervals on the log scale have equal probability. A lognormal distribution is another option for positive inputs when a distribution with multiplicative spread is plausible; if it is truncated to enforce bounds, the truncation changes its probability weights and should be stated.
Worked Example: Sampling a Rate Constant on a Log Scale
Suppose a hypothetical degradation rate constant \(k\), in day\(^{-1}\), is judged plausible from \(0.001\) to \(0.1\) day\(^{-1}\), with uncertainty spanning two orders of magnitude. Choose \(\log_{10}(k)\sim U(-3,-1)\). To transform a uniform draw \(u=0.25\) from \(U(0,1)\), use \(\log_{10}(k)=-3+u[-1-(-3)]=-2.5\), so \(k=10^{-2.5}=0.003162\) day\(^{-1}\).
The median is \(10^{(-3-1)/2}=0.01\) day\(^{-1}\), because the median is the midpoint on the log scale. The arithmetic mean is not that midpoint. From the density, or by integrating the inverse transformation, \(\mathrm{E}[k]=(0.1-0.001)/\ln(0.1/0.001)=0.099/\ln(100)=0.02150\) day\(^{-1}\).
For comparison, a linear-scale uniform distribution on the same endpoints has mean \((0.001+0.1)/2=0.05050\) day\(^{-1}\). Its median is also \(0.05050\) day\(^{-1}\), and it places half its probability above that value. Thus the same numerical bounds yield substantially different sampled values and weights under the two distributions. The log-scale choice is defensible here only if equal multiplicative changes are more meaningful than equal additive changes.
A distribution may also encode a central tendency and uncertainty differently from uniform sampling. For example, a triangular distribution can place most probability near a best-supported value while retaining finite bounds. A truncated normal distribution can represent approximately symmetric uncertainty around a mean, provided truncation and any resulting asymmetry are handled explicitly. These alternatives do not remove the need to justify the bounds or scale.
Joint Distributions and Biological Constraints
Specifying a distribution for each parameter separately is not enough when parameters are dependent or constrained together. If inputs are treated as independent, the joint density is the product of their marginal densities, and every combination in the product of their supports is allowed. This may be inappropriate when, for example, two fractions share a fixed total, or when one parameter is biologically tied to another.
The joint distribution must then assign probability only to feasible combinations and encode the intended dependence. A convenient sampling algorithm is not itself a justification for that distribution. Rejection sampling, transformation to constrained coordinates, and direct sampling from a joint distribution can generate feasible inputs, but they may induce different probability weights unless designed to target the same distribution.
Worked Example: Two Fractions Constrained by a Shared Total
Suppose \(f_1\) and \(f_2\) are hypothetical fractions of a cell population assigned to two mutually exclusive states, with the remainder in a third state. Biological feasibility requires \(f_1\geq0\), \(f_2\geq0\), and \(f_1+f_2\leq1\). Independent uniform draws from \([0,1]\) would often violate the final condition.
One simple choice is a uniform distribution over the feasible triangle, with joint density \(2\) on that triangle and zero elsewhere. The area of the triangle is \(1/2\), so the density integrates to \(2(1/2)=1\). Its marginal means can be derived directly: \(\mathrm{E}[f_1]=2\int_0^1\int_0^{1-f_1} f_1\,df_2\,df_1=2\int_0^1 f_1(1-f_1)\,df_1=1/3\). By symmetry, \(\mathrm{E}[f_2]=1/3\); the expected remainder is therefore \(1-1/3-1/3=1/3\).
An equivalent way to generate this particular distribution is to draw \(U_1,U_2\) independently from \(U(0,1)\) and retain only pairs with \(U_1+U_2\leq1\). The accepted pairs are uniform on the triangle because the original density is constant there. This induces negative dependence: \(\mathrm{E}[f_1f_2]=2\int_0^1\int_0^{1-f_1} f_1f_2\,df_2\,df_1=1/12\), so \(\mathrm{Cov}(f_1,f_2)=1/12-(1/3)(1/3)=-1/36\). The constraint therefore affects not just feasible values, but how the inputs vary together. Another biological model might support a different distribution on the same triangle; uniformity over the feasible region is a choice, not a biological law.
Constraints can also be imposed through parameterization. If fractions must sum exactly to one, one can define a joint distribution directly on the simplex rather than sample fractions independently and discard invalid combinations. A Dirichlet distribution is one standard family for such vectors, with its concentration parameters controlling how probability is distributed over the simplex. The family and its parameters should reflect the intended biology; the fact that it enforces the constraint does not establish that its probability weights are appropriate.
A Practical Specification and Its Checks
A defensible specification makes the assumed input space reproducible and interpretable. Record each parameter’s name, units, bounds, scale, distribution, and scientific rationale. For a multivariate input, also record dependencies, constraints, and the exact sampling procedure. If the inputs are drawn from a posterior, state which posterior and what conditioning data and model assumptions produced it.
Confirm its units, whether it is positive or bounded, and whether it is a primitive parameter or a derived quantity.
Use evidence, biological constraints, and the model’s valid domain. Do not interpret chosen bounds as confidence limits unless they were constructed as such.
Decide whether additive or multiplicative changes are meaningful, then specify a distribution consistent with that choice.
Encode dependencies and constraints, then inspect sampled values and combinations before expensive model runs.
Useful diagnostics include histograms or quantiles on both original and transformed scales, checks of bounds and algebraic constraints, and pairwise plots for dependent inputs. Evaluate whether sampled combinations produce invalid states, numerical failures, or implausible biology. A large fraction of rejected draws can indicate that independent marginal distributions are poorly matched to a constrained joint space. Conversely, a sample that technically satisfies a constraint may still cluster in biologically implausible regions if its probability weights are poorly chosen.
Changing an input distribution changes the regions that the model explores and the probability assigned to each region. This can change output variability and the apparent importance of individual parameters. Comparing alternative plausible distributions is therefore a useful robustness check, especially when bounds, log scaling, or dependence are weakly informed. Such comparisons should be reported as sensitivity to the assumed input specification, rather than as evidence that one distribution is uniquely correct.
Common Pitfalls and Limitations
- Using a convenient range without a rationale. Wider bounds are not automatically more conservative; they may include invalid model behavior. Tie bounds to scientific knowledge and document exclusions.
- Calling every bounded input uniform. Uniformity assigns equal probability per unit of the selected scale. State whether that scale is linear or logarithmic and why.
- Assuming independent inputs by default. Independent sampling can violate constraints or create implausible combinations. Specify a joint distribution when biology requires dependence.
- Treating parameter uncertainty as all output variability. If the model includes stochastic processes or measurement noise, distinguish those sources from uncertainty in the parameters being sampled.
- Interpreting bounds as a calibrated interval. A range selected for analysis is not automatically a confidence interval or a posterior credible interval.
No input distribution can compensate for a structurally inadequate model or weak biological evidence. Some parameters may be poorly characterized, and multiple distributions may be defensible. In such cases, transparent assumptions and sensitivity checks across alternatives are more informative than implying unjustified precision.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A positive parameter spans \(0.01\) to \(1\). What does uniformity on the base-10 log scale imply about the probability of \(0.01\) to \(0.1\) versus \(0.1\) to \(1\)?
- For \(X\sim U(2,8)\), calculate \(P(X>5)\). State why this result depends on the assumed distribution.
- Two fractions must sum to no more than one. Why can independent \(U(0,1)\) sampling not be used without modification?
- If a calibrated posterior is used as the GSA input distribution, what population of parameter values do the conclusions describe?
Solutions. (1) The intervals have equal width on the base-10 log scale, so each has probability \(1/2\), though they do not have equal width on the original scale. (2) \(P(X>5)=(8-5)/(8-2)=3/6=0.5\), using a uniform density on \([2,8]\); another distribution on the same bounds could give a different probability. (3) Independent draws have positive probability of violating the constraint: for example, the square \([0,1]^2\) includes the region \(x_1+x_2>1\), which has area \(1/2\). The joint sampling scheme must restrict draws to feasible combinations and specify their probability weights. (4) The results describe sensitivity over parameter values represented by that posterior, conditional on its data, likelihood or approximation, prior, and model assumptions—not all conceivable biologically plausible values.