Selecting Distributions for Uncertain Model Inputs

Why Distribution Choice Matters

In probabilistic decision analysis, uncertain model inputs are assigned distributions and sampled to propagate uncertainty through a model. The resulting prediction is not determined by an input’s mean alone: its spread, skewness, bounds, and dependence on other inputs can affect both predicted outcomes and the probability of meeting a decision criterion. Choosing a distribution is therefore part of specifying the uncertainty model, not a neutral software setting.

A useful starting point is the support: the set of values a distribution can take. Clearance cannot be negative; a fraction must lie between zero and one. A distribution that assigns appreciable probability to impossible values is usually inappropriate unless the model explicitly handles those values. Support is necessary but not sufficient: different distributions with the same support and similar moments can still imply different probabilities in the tails, where a development decision may depend on them.

Definition: A distribution for an uncertain model input specifies how probability is assigned to the input’s possible values. Its support, shape, and parameters should represent the uncertainty relevant to the decision. This uncertainty may describe limited knowledge about an input; it should not be confused with patient-to-patient variability or residual error unless the model and decision explicitly treat those as the same quantity.

Distribution parameters should be grounded in available evidence: estimates of location and spread, plausible ranges, expert-elicited quantiles, or other defensible information. A convenient default is not evidence. In particular, assigning a uniform distribution between two bounds says that equal-width intervals within those bounds have equal probability; it does not merely say that values outside the bounds are implausible.

Positive Inputs: Lognormal and Gamma Choices

For a strictly positive quantity such as clearance, exposure, or a positive scale parameter, lognormal and gamma distributions are common candidates. Both avoid negative values, but their shapes differ. A lognormal variable is the exponential of a normally distributed quantity; a gamma distribution has a shape and scale that control its skewness and spread. Neither is automatically correct just because the input is positive.

Definition: A positive random variable \(X\) is lognormal if \(X=\exp(Z)\), where \(Z\sim N(\mu,\sigma^2)\). Its median is \(\exp(\mu)\), its arithmetic mean is \(m=\exp(\mu+\sigma^2/2)\), and its coefficient of variation is \(c=\sqrt{\exp(\sigma^2)-1}\). A gamma distribution with shape \(k\) and scale \(\theta\), written \(X\sim\operatorname{Gamma}(k,\theta)\), has mean \(k\theta\) and variance \(k\theta^2\).

For a lognormal distribution specified by arithmetic mean \(m\) and coefficient of variation \(c\), solving the mean and coefficient-of-variation equations gives \(\sigma^2=\ln(1+c^2)\) and \(\mu=\ln(m)-\sigma^2/2\). For a gamma distribution specified by the same quantities, \(c^2=1/k\), so \(k=1/c^2\) and \(\theta=m/k\). These are parameter conversions, not tests that either distribution describes the underlying uncertainty well.

Worked Example: A Positive Input With an Elicited Mean and Spread

Suppose a hypothetical decision model uses clearance \(X\), in liters per hour. An elicitation gives an arithmetic mean of 8 L/h and a coefficient of variation of 0.50. Consider a lognormal representation, and check its behavior against a normal distribution with the same mean and standard deviation.

For the lognormal, convert the mean and coefficient of variation to log-scale parameters:

\sigma^2=\ln(1+0.50^2)=\ln(1.25)=0.22314355\ldots, \qquad \sigma=\sqrt{\ln(1.25)}=0.4723807\ldots

\mu=\ln(8)-\frac{\ln(1.25)}{2} =1.9678698\ldots

The lognormal has the specified arithmetic mean and coefficient of variation. Its median is \(\exp(\mu)\approx 7.156\) L/h, below its mean because the distribution is right-skewed. For a decision event defined here as clearance exceeding 12 L/h, standardize on the log scale:

P(X>12) =1-\Phi\!\left(\frac{\ln(12)-\mu}{\sigma}\right) =1-\Phi(1.0945\ldots) \approx 0.1368.

The corresponding normal distribution has mean 8 L/h and standard deviation \(0.50(8)=4\) L/h. It assigns probability \(\Phi(-2)\approx 0.0228\) to negative clearance, which is physically impossible under this input definition. It also gives \(P(X>12)=1-\Phi(1)\approx 0.1587\), rather than the lognormal value. The lognormal avoids negative values, but choosing it still requires evidence that its right-skewed shape is plausible.

Same Moments, Different Decision Probabilities

Matching a mean and variance does not determine a distribution. The following comparison makes that limitation explicit. For the same hypothetical clearance mean of 8 L/h and coefficient of variation of 0.50, the gamma parameter conversion gives \(k=1/(0.50^2)=4\) and \(\theta=8/4=2\) L/h. A gamma distribution with integer shape \(k=4\) has a simple survival probability, making the comparison transparent.

Worked Example: Comparing Lognormal and Gamma Tails

Continue with the hypothetical clearance input and the event \(X>12\) L/h. The lognormal probability from the preceding calculation is approximately 0.1368. Under the gamma model, \(X\sim\operatorname{Gamma}(4,2)\), so the threshold in scale units is \(12/2=6\).

For shape 4, the gamma survival probability at \(x\) is \(\exp(-x/\theta)\sum_{j=0}^{3}(x/\theta)^j/j!\). Substituting the threshold gives:

P(X>12) =e^{-6}\left(1+6+\frac{6^2}{2!}+\frac{6^3}{3!}\right) =e^{-6}(1+6+18+36) =61e^{-6} \approx 0.1512.

The gamma and lognormal distributions have the same mean, 8 L/h, and variance, 16 \((\mathrm{L/h})^2\), yet their probabilities of exceeding 12 L/h differ: about 15.12% versus 13.68%. This difference is a consequence of shape, not a discrepancy in the stated mean or variance. If a decision depends on this tail event, the choice should be informed by evidence about plausible tails, or explored as a distributional assumption when that evidence is weak.

This comparison is not a rule that gamma distributions always have heavier tails than lognormal distributions. Tail ordering depends on the parameter values and the threshold. It does show why reporting only an input mean and standard deviation can hide uncertainty relevant to a decision.

Bounded Inputs and Proportions

An uncertain proportion \(p\) is constrained to the interval from zero to one. The beta distribution is a flexible continuous choice on that interval. Its two positive shape parameters control its mean and concentration. A beta distribution can be symmetric or skewed, but a standard beta assigns probability only to values strictly between zero and one. It is therefore not a complete representation when an input can genuinely equal zero or one with positive probability.

Definition: If \(p\sim\operatorname{Beta}(a,b)\), with \(a>0\) and \(b>0\), its density is proportional to \(p^{a-1}(1-p)^{b-1}\) on \(0<p<1\). Its mean is \(a/(a+b)\), and its variance is \(ab/[(a+b)^2(a+b+1)]\). For a quantity known to lie between \(L\) and \(U\), a beta-distributed fraction can be rescaled as \(X=L+(U-L)p\).

Worked Example: Probability That an Uncertain Proportion Exceeds a Target

Suppose a hypothetical decision model uses \(p\), an uncertain response fraction, and the decision-relevant event is \(p>0.75\). Assume the uncertainty is represented by \(p\sim\operatorname{Beta}(2,2)\). This distribution is symmetric around 0.50, assigns values only between zero and one, and has mean \(2/(2+2)=0.50\). Its variance is \(4/[16(5)]=0.05\), so its standard deviation is approximately 0.2236.

The density is \(6p(1-p)\). Its cumulative distribution function on this interval is \(3p^2-2p^3\). Therefore:

P(p>0.75) =1-\left[3(0.75)^2-2(0.75)^3\right] =1-(1.6875-0.84375) =0.15625.

Under this specified uncertainty distribution, the probability that the response fraction exceeds 0.75 is 15.625%. This is a model-based probability conditional on the assumed beta distribution, not a statement that 15.625% of individual patients respond. If the input were a patient-level response indicator or a proportion estimated from a particular sample, the sampling model and the decision target would need to be specified accordingly.

A beta distribution is useful when uncertainty concerns a continuous fraction and a unimodal or otherwise beta-shaped density is plausible. A simple uniform distribution is the special case \(\operatorname{Beta}(1,1)\), but it implies a flat density over the entire interval. If a proportion can be exactly zero or one because of a genuine structural state, consider a model with explicit probability at those endpoints rather than forcing all uncertainty into a continuous beta density. For a bounded input whose limits are not zero and one, rescaling the beta distribution preserves the limits but does not establish that its shape is appropriate.

A Practical Selection and Checking Process

The distribution choice should follow the meaning and evidence for the input, then be checked in the context of the decision model. A disciplined process keeps convenient parameterization from being mistaken for scientific justification.

Define what is uncertain.
State whether the input is an uncertain population parameter, a future individual value, a proportion, or another quantity. Do not combine these interpretations without a model that supports the combination.

Enforce meaningful support.
Check whether possible values are positive, bounded, discrete, or allowed to include endpoints. Reject or explicitly modify choices that assign material probability to impossible values.

Calibrate to evidence.
Use defensible summaries or quantiles to set parameters, and check that the resulting distribution reproduces those summaries. If only bounds are known, make clear that further assumptions about shape are still required.

Inspect implications for the decision.
Check central values, tails, and relevant thresholds after propagation. When shape is uncertain, compare plausible alternatives while holding agreed summaries fixed where possible.

Diagnostics include checking simulated draws for impossible values, confirming that the simulated mean and spread match their targets, and inspecting whether tails or boundary behavior are plausible. Also check the propagated output: a valid input distribution can still interact with the model in a way that creates extreme or numerically unstable predictions. When inputs are dependent, selecting appropriate marginal distributions does not by itself preserve their joint dependence; the joint uncertainty representation must remain coherent.

A common mistake is to choose a distribution solely because it is familiar, then interpret its simulated decision probability as if that probability were supported independently of the choice. For example, saying “the probability of exceeding the target is 15%” without identifying the input distribution hides an assumption that can materially affect the result. A more accurate conclusion names the assumed distribution and, when the shape is not well established, reports how the decision probability changes across plausible choices.

No one family is best for every input. Normal distributions can be reasonable for quantities whose possible values are effectively unbounded around a central value, but are often unsuitable for strictly positive or bounded quantities. Lognormal and gamma distributions respect positivity but imply different shapes. Beta distributions respect fractional bounds but cannot put probability at exact endpoints. Uniform and triangular distributions can encode range and mode information, but only when those assumptions have a defensible interpretation. Distribution selection is limited by the quality of the uncertainty evidence; greater mathematical flexibility cannot compensate for unsupported inputs.

Key takeaway: Match the distribution’s support to the input, calibrate its shape to available evidence, and check how plausible alternatives affect the propagated decision. A distribution is an explicit assumption about uncertainty, not merely a way to generate random numbers.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A positive input has mean 10 units and coefficient of variation 0.25. What are the lognormal parameters \(\mu\) and \(\sigma^2\) for the parameterization used here?
Why might a normal distribution be unsuitable for a strictly positive input, even when its mean and standard deviation match elicited summaries?
For \(p\sim\operatorname{Beta}(3,3)\), what is the mean, and can this distribution assign positive probability to \(p=0\) or \(p=1\)?
Two candidate distributions have the same mean and variance but give different probabilities of exceeding a decision threshold. What does that imply for reporting the decision probability?

Solutions. (1) \(\sigma^2=\ln(1+0.25^2)=\ln(1.0625)\approx0.06062\), and \(\mu=\ln(10)-\sigma^2/2\approx2.272\). (2) It can assign probability to negative values, which may be impossible; the amount assigned there should be checked, not ignored. (3) The mean is \(3/(3+3)=0.50\). A standard beta distribution has continuous support on \(0<p<1\), so it assigns no point probability to either endpoint. (4) The probability depends on distributional shape as well as moments; report the assumed distribution and examine plausible alternatives when shape is uncertain.