Weakly Informative Priors for NLME Parameters

Why Prior Scale Matters

A prior for a population parameter should express what values are plausible before the current data are used, while allowing the data room to update that belief. In a nonlinear mixed-effects model, a prior that is too narrow can dominate a weakly informative likelihood; a prior that is extremely broad can permit parameter values that are scientifically implausible or make computation difficult. The goal here is to choose and assess prior scales for positive population parameters, such as clearance or volume, and unconstrained population parameters, such as a coefficient on a log scale.

The earlier material on priors for population parameters introduced the role of a prior in Bayesian estimation. The focus now is construction: start with scientifically interpretable ranges, translate those ranges into a distribution, and check the resulting prior on the parameter’s original scale. A weakly informative prior is not a universal distribution or a guarantee of negligible influence. It is a proper prior whose scale expresses broad, defensible information while discouraging extreme values that are not scientifically credible.

Definition: A weakly informative prior is a prior distribution that assigns substantial probability to a broad range of plausible parameter values, while reducing probability for extreme values that are not credible for the application. Its location, scale, and support must be justified in context. “Weakly informative” does not mean “uninformative,” nor does it mean that the prior will have little influence in every data set.

A prior scale is easiest to judge through its implied quantiles and tail probabilities. For example, ask whether a central 95% prior interval includes the range of values that subject-matter knowledge considers plausible. That check does not establish that the prior is correct: a central interval is only one summary, and values outside it still have positive probability for many distributions. It does reveal when a proposed scale is plainly inconsistent with the intended degree of prior uncertainty.

Positive Parameters: Calibrating on the Log Scale

A positive parameter \(\theta\), such as clearance in L/h, must have support only above zero. One convenient construction is to put a normal prior on its logarithm:

\log(\theta)\sim N(\mu,\tau^2) \qquad\Longleftrightarrow\qquad \theta\sim\operatorname{Lognormal}(\mu,\tau^2).

Here \(\mu\) is the prior mean of \(\log(\theta)\), and \(\tau\) is the prior standard deviation of \(\log(\theta)\). The median of \(\theta\) is \(\exp(\mu)\); its mean is \(\exp(\mu+\tau^2/2)\). These are different when \(\tau>0\), so the intended center must be stated. If the goal is to center prior beliefs on a typical value \(m\), set \(\mu=\log(m)\), making \(m\) the median rather than the arithmetic mean.

If the prior is intended to place about 95% of its mass between positive values \(L\) and \(U\), and the interval is symmetric on the log scale, choose

m=\sqrt{LU},\qquad \mu=\log(m),\qquad \tau=\frac{\log(U)-\log(L)}{2(1.96)} =\frac{\log(U/L)}{3.92}.

This follows because the central 95% interval for a normal variable is approximately \(\mu\pm1.96\tau\). Exponentiating its endpoints gives \([\exp(\mu-1.96\tau),\exp(\mu+1.96\tau)]\). The geometric midpoint \(\sqrt{LU}\) is the center of the proposed range on the log scale. This construction is suitable when equal multiplicative departures above and below the center are reasonable. It does not fit an arbitrary positive range if that range is asymmetric on the log scale.

Worked Example: A Broad Prior for Typical Clearance

Suppose a hypothetical analysis concerns typical clearance \(CL\), measured in L/h. Before seeing the analysis data, the modeler judges that values from 2 to 12 L/h should be comfortably covered, and chooses a lognormal prior with median 5 L/h and log-scale standard deviation \(\tau=0.45\). Check the central 95% prior interval.

For a lognormal prior, the interval is obtained by exponentiating the normal interval for \(\log(CL)\):

\left[ 5\exp(-1.96(0.45)),\; 5\exp(1.96(0.45)) \right] = [5\exp(-0.882),\;5\exp(0.882)].

Numerically, \(5\exp(-0.882)\approx2.07\) and \(5\exp(0.882)\approx12.08\). Thus the central 95% prior interval is approximately \([2.07,12.08]\) L/h. It covers the stated plausible range closely. This is a scale check, not proof that the prior is appropriate: the center, range, and lognormal shape still require scientific justification.

The prior median is 5 L/h, but the prior mean is higher: \(5\exp(0.45^2/2)=5\exp(0.10125)\approx5.53\) L/h. Reporting the median as though it were the mean would misstate the prior’s location.

Worked Example: Constructing a Prior From a Positive Range

For a hypothetical typical volume parameter \(V\) measured in L, suppose values from 5 to 40 L are considered plausible, with the intended central 95% prior interval spanning that range symmetrically on the log scale. Find a lognormal prior that achieves this calibration.

The median is the geometric midpoint: \(\sqrt{5(40)}=\sqrt{200}\approx14.14\) L. The required log-scale standard deviation is

\tau=\frac{\log(40)-\log(5)}{3.92} =\frac{\log(8)}{3.92} =\frac{2.07944}{3.92} \approx0.5305.

Therefore, specify \(\log(V)\sim N(\log(14.14),0.5305^2)\), or equivalently \(V\sim\operatorname{Lognormal}(\log(14.14),0.5305^2)\). The central 95% interval is \([14.14\exp(-1.96(0.5305)),14.14\exp(1.96(0.5305))]\), which is approximately \([5,40]\) L, with small differences possible from rounding. The prior is broad in multiplicative terms: 40 L is eight times 5 L. This may suit a parameter whose plausible uncertainty is naturally expressed as ratios rather than fixed additive differences.

The same probability distribution can be described on different scales, but its density changes with the transformation. If \(\eta=\log(\theta)\) has a normal density, then the density on the original positive scale is

f_\theta(\theta) = \frac{1}{\theta\tau\sqrt{2\pi}} \exp\left[-\frac{(\log(\theta)-\mu)^2}{2\tau^2}\right], \qquad \theta>0.

The factor \(1/\theta\) is the change-of-variables Jacobian. Consequently, a normal prior on \(\log(\theta)\) is not a normal prior on \(\theta\), and it is not a uniform prior on \(\theta\). The choice of scale is part of the prior specification, not a cosmetic computational decision.

Unconstrained Population Parameters

Some population parameters may take any real value. Examples include a coefficient for a covariate in a log-linear model, where a negative coefficient represents a decrease and a positive coefficient represents an increase. A normal prior is a common weakly informative starting point:

\beta\sim N(\beta_0,s_\beta^2).

The location \(\beta_0\) is the prior center and \(s_\beta\) is the prior standard deviation, in the units of \(\beta\). A proposed central 95% range \([A,B]\), symmetric about \(\beta_0\), corresponds to \(s_\beta=(B-A)/3.92\). If the parameter enters the model on a transformed scale, translate the range back to the scientific interpretation before deciding whether it is plausible.

Worked Example: A Prior for a Log-Linear Covariate Effect

Consider a hypothetical model in which \(\beta\) is the coefficient for a covariate measured in one standardized unit, and the covariate acts multiplicatively on typical clearance through \(\exp(\beta)\). Suppose a ratio between 0.5 and 2.0 per unit is considered a broad plausible range, with no prior preference for an increase rather than a decrease. Construct a normal prior for \(\beta\).

The ratio range corresponds to the coefficient range \([\log(0.5),\log(2.0)]=[-\log(2),\log(2)]\), centered at zero. Setting this as an approximate central 95% interval gives

s_\beta =\frac{\log(2)-[-\log(2)]}{3.92} =\frac{2\log(2)}{3.92} \approx0.3536.

Thus a suitable stated prior is \(\beta\sim N(0,0.3536^2)\). Its central 95% interval is approximately \([-0.6931,0.6931]\). Exponentiating those endpoints gives ratios \([\exp(-0.6931),\exp(0.6931)]\approx[0.5,2.0]\). Under this prior, the central 95% prior interval for the multiplicative effect runs from a 50% multiplier to a twofold multiplier per standardized covariate unit. This does not assert that the effect is likely to be exactly zero or that the data will support either endpoint.

Checking a Proposed Prior Scale

A scale check should start from the scientific quantity, not from a convenient default such as “use a standard deviation of one.” A value of one can be narrow for a parameter measured in L/h and very broad for a coefficient measured on a log-ratio scale. For each proposed prior, check the following:

Support: Does the prior rule out values the parameter cannot take? A lognormal prior is positive, whereas a normal prior on the original scale assigns some probability to negative values.
Center: Is the prior center stated as a mean, median, or other quantity? For skewed distributions these need not coincide.
Interval coverage: Does a central prior interval cover the plausible range on the relevant scientific scale?
Tail behavior: Does the prior place appreciable mass on values that are implausibly small, large, or associated with extreme multiplicative effects?
Units and transformation: Are the scale and any transformed interpretation consistent with how the parameter enters the model?

A central interval alone does not describe the whole prior. For a lognormal prior, values outside the interval remain possible, and the upper tail is unbounded. For a normal prior on an unconstrained coefficient, translating tail values through an exponential link can produce very large or small multiplicative effects. Inspecting such implications is a parameter-scale check; it is not a substitute for examining the model’s implications for observable data.

Alternatives, Diagnostics, and Limitations

A lognormal prior is useful for positive parameters when beliefs are naturally multiplicative and right-skewness is reasonable. A gamma prior is another positive-support option; its shape can differ from a lognormal, so matching a mean and variance does not make the two priors equivalent. A normal prior is natural for an unconstrained coefficient when a roughly symmetric scale is defensible. A Student-\(t\) prior can give an unconstrained parameter heavier tails than a normal prior, but those heavier tails may allow extreme effects that the modeler intended to discourage. In every case, compare distributions on the scientific scale, not only by looking at a parameterization’s location and scale.

The main diagnostic for a weakly informative prior is whether its stated plausibility claims match its implied distribution. If expert judgment says that a positive parameter is unlikely below \(L\) or above \(U\), yet the proposed prior’s central interval lies mostly outside \([L,U]\), the scale or center needs reconsideration. If a normal prior on a log-effect scale implies a central range of multiplicative effects far wider than intended, its standard deviation is too large for that intention. Conversely, if plausible values lie in remote tails, the prior may be too restrictive.

Prior sensitivity is also important: refit or re-evaluate the analysis under a small set of defensible alternative scales and compare the resulting inferences. Strong changes indicate that conclusions depend materially on prior specification, especially when the likelihood is weakly informative for that parameter. This does not identify one prior as objectively correct; it makes the dependence visible. No single generic scale works for all pharmacometric parameters, populations, measurement units, or model structures.

Common pitfall: “A normal prior with a large standard deviation is always weakly informative.” This reasoning ignores the parameter’s units, support, and transformation. A normal prior on a positive clearance parameter assigns probability to negative clearance, while a seemingly modest normal scale for a log-effect can imply a very wide range of multiplicative effects. Instead, choose a parameterization with appropriate support, translate a scientifically plausible range into prior quantiles, and inspect those quantiles on the original scientific scale.

State the parameter and scale.
Record its units, allowed values, and role in the model, including any transformation such as a log link.

Specify plausible values before using the current data.
Choose an interpretable center and a range that reflects prior scientific knowledge, not the estimate obtained from the analysis being fitted.

Translate the range into a distribution.
Use quantile relationships on the chosen scale, then state whether the center refers to a mean or median.

Check the implied values and sensitivity.
Compute relevant quantiles and tail implications on the scientific scale, and assess whether reasonable alternative scales materially change conclusions.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

A positive parameter has a proposed lognormal prior centered at median 10, with log-scale standard deviation 0.25. What is its approximate central 95% interval?
A normal prior for an unconstrained coefficient has standard deviation 0.5. What is its approximate central 95% interval about its prior mean, and why should that interval be interpreted in the coefficient’s units?
Why is a normal prior directly on a positive clearance parameter potentially unsuitable, even if its mean is positive?
A proposed prior’s central 95% interval excludes values that experts regard as plausible. What should be checked or changed?

Solutions. (1) The interval is \([10\exp(-1.96(0.25)),10\exp(1.96(0.25))]=[10\exp(-0.49),10\exp(0.49)]\approx[6.13,16.32]\), in the parameter’s units. (2) It is \(\beta_0\pm1.96(0.5)=\beta_0\pm0.98\); the coefficient’s units determine what those numerical departures mean. (3) A normal distribution has support over the whole real line and therefore assigns positive probability to negative clearance. (4) Revisit the center, scale, and parameterization; verify that the plausible range and prior quantiles are expressed on the same scale, then choose a defensible prior that covers the plausible values and document the choice.