Induced Priors Under Parameter Transformations

Why the Parameter Scale Matters

A QSP parameter may be constrained to be positive, such as a degradation rate, or to lie between zero and one, such as a fractional pathway occupancy. A transformation can represent such a parameter on an unconstrained scale: logarithms map positive values to the real line, and logits map probabilities strictly between zero and one to the real line. This can make prior specification convenient, but a prior on the transformed parameter is not the same distribution as a prior with a similar-looking shape on the original parameter.

The distinction is a change in probability density, not merely a change in labels. A density must account for how intervals expand or contract under the transformation. The Jacobian supplies that adjustment. Consequently, a normal prior on a log parameter induces a lognormal prior on the positive parameter, while a normal prior on a logit parameter induces a logistic-normal prior on a probability. Neither conclusion says that every prior on the original scale must be lognormal or logistic-normal.

The Change-of-Variables Rule

Definition: Let $Y=g(X)$ be a differentiable, one-to-one transformation between continuous scalar parameters, with differentiable inverse $X=h(Y)$. If $X$ has density $f_X$, then the density of $Y$ is $$ f_Y(y)=f_X(h(y))\left|\frac{dh(y)}{dy}\right|. $$ The absolute derivative is the Jacobian factor. It adjusts density so that probability is preserved under the transformation.

The same rule can be written in the opposite direction. If a prior density $f_Y$ is specified on the transformed parameter $Y=g(X)$, its induced density on $X$ is $$ f_X(x)=f_Y(g(x))\left|\frac{dg(x)}{dx}\right|. $$ These are two uses of the same rule; which expression applies depends on which scale was assigned the prior. For a vector of parameters, the derivative is replaced by the absolute determinant of the Jacobian matrix of the inverse transformation. If a transformation acts separately on each component, that determinant is the product of the componentwise absolute derivatives.

The basic justification is conservation of probability. For a small interval around $x$, its image around $y=g(x)$ has width approximately $|g'(x)|\,dx$. The probability assigned to either interval must agree. Thus $f_X(x)\,dx=f_Y(y)\,dy$, which gives the formula above. Equivalently, integrating the transformed density over any transformed region returns the probability of its inverse image. This requires a one-to-one transformation on the support under consideration; if a transformation is not one-to-one, contributions from all inverse branches must be added.

Log Transformations for Positive Parameters

For a positive parameter $\theta$, define the log-scale variable $Z=\log(\theta/\theta_{\mathrm{ref}})$, where $\theta_{\mathrm{ref}}>0$ is a fixed reference value in the same units as $\theta$. The ratio inside the logarithm is dimensionless. The inverse is $\theta=\theta_{\mathrm{ref}}\exp(Z)$, and $$ \left|\frac{dZ}{d\theta}\right|=\frac{1}{\theta}. $$

If a prior is assigned to $Z$, its induced density for $\theta$ is therefore $$ f_\theta(\theta)=f_Z\!\left(\log(\theta/\theta_{\mathrm{ref}})\right)\frac{1}{\theta}, \qquad \theta>0. $$ For $Z\sim N(\mu,\sigma^2)$, this becomes $$ f_\theta(\theta)=\frac{1}{\theta\,\sigma\sqrt{2\pi}} \exp\!\left[-\frac{\{\log(\theta/\theta_{\mathrm{ref}})-\mu\}^2}{2\sigma^2}\right],\qquad \theta>0. $$ This is a lognormal density. Its support is positive, and the factor $1/\theta$ is essential; leaving it out does not give the density induced by the stated normal prior. The density is expressed per unit of $\theta$. Changing the unit used to report $\theta$ changes the numerical density value, but not the probabilities assigned to corresponding intervals.

Worked Example: A Normal Prior on a Log Rate

Suppose a hypothetical QSP model has a degradation rate $k$, measured in $\mathrm{day}^{-1}$. Set $k_{\mathrm{ref}}=0.1\ \mathrm{day}^{-1}$, and assign $Z=\log(k/k_{\mathrm{ref}})\sim N(0,0.5^2)$. Find the induced density at $k=0.2\ \mathrm{day}^{-1}$.

The transformed value is $z=\log(0.2/0.1)=\log 2=0.6931$. Substituting into the change-of-variables formula gives $$ f_k(0.2)=\frac{1}{0.2(0.5)\sqrt{2\pi}} \exp\!\left[-\frac{(0.6931-0)^2}{2(0.5)^2}\right]. $$

The exponent is $-0.6931^2/0.5=-0.9609$, so the exponential is approximately $0.3825$. The denominator is $0.2(0.5)(2.5066)=0.2507$. Thus $f_k(0.2)\approx 1.526$ per $\mathrm{day}^{-1}$, equivalently in days as the reciprocal parameter unit. This is a density value, not the probability that $k$ equals $0.2$. Probabilities require integration over intervals.

Logit Transformations for Fractional Parameters

For $0<p<1$, define $Z=\operatorname{logit}(p)=\log\{p/(1-p)\}$. Its inverse is the logistic function $p=\exp(z)/\{1+\exp(z)\}$. Differentiation gives $$ \frac{dZ}{dp}=\frac{1}{p(1-p)}. $$ Consequently, a prior specified for $Z$ induces $$ f_p(p)=f_Z\!\left(\log\{p/(1-p)\}\right)\frac{1}{p(1-p)},\qquad 0<p<1. $$ The Jacobian is not constant, so the density shape generally changes. However, if $f_Z(z)=f_Z(-z)$, then the induced density is symmetric about $p=0.5$: the identity $\operatorname{logit}(1-p)=-\operatorname{logit}(p)$ and equality $p(1-p)=(1-p)p$ imply $f_p(p)=f_p(1-p)$. Symmetry is therefore preserved in this particular sense when the logit-scale density is symmetric about zero; a bell-shaped form need not be preserved.

Worked Example: Symmetry and Shape Under a Logit-Normal Prior

Let a hypothetical pathway fraction have $Z=\operatorname{logit}(p)\sim N(0,2^2)$. Compare the induced density at $p=0.5$ and $p=0.01$, and assess its symmetry.

At $p=0.5$, the logit is zero and the Jacobian is $1/\{0.5(0.5)\}=4$. Hence $$ f_p(0.5)=\frac{1}{2\sqrt{2\pi}}\times 4\approx 0.7979. $$ At $p=0.01$, $z=\log(0.01/0.99)\approx-4.5951$, and $p(1-p)=0.0099$. Therefore $$ f_p(0.01)= \frac{1}{2\sqrt{2\pi}}\exp\!\left(-\frac{(-4.5951)^2}{8}\right)\frac{1}{0.0099} \approx \frac{0.01424}{0.0099} \approx 1.438. $$ The density at $0.01$ exceeds its value at $0.5$, so the induced density is not bell-shaped with a central maximum. Since the normal density for $Z$ is symmetric about zero, $f_p(0.01)=f_p(0.99)$; this density is symmetric about one half. These density values do not mean that the prior assigns more probability to the single value $0.01$ than to $0.5$, since single points have probability zero. Probabilities compare intervals and depend on their widths.

Specifying a Prior on One Scale or the Other

There are two distinct modeling choices. One may specify a prior directly for the original parameter, such as a beta distribution for $p$, or specify a prior for its transformed representation, such as a normal distribution for $\operatorname{logit}(p)$. The induced prior on the other scale follows from the Jacobian. These choices should not be treated as interchangeable descriptions unless their densities are related by the change-of-variables formula.

For instance, a uniform prior on $p\in(0,1)$ has density $f_p(p)=1$. Transforming it to $Z=\operatorname{logit}(p)$ gives $$ f_Z(z)=f_p(p)\left|\frac{dp}{dz}\right|=p(1-p) =\frac{\exp(z)}{\{1+\exp(z)\}^2}. $$ It is not uniform on the logit scale. Conversely, a constant density over the entire real line for $Z$ would induce a density proportional to $1/\{p(1-p)\}$ on $p$. That function is not integrable on $(0,1)$, so it cannot be normalized to a proper probability density. Calling it a “uniform prior on the logit scale” does not make it a proper prior.

Worked Example: Converting a Uniform Fraction Prior to the Logit Scale

Suppose a hypothetical fractional occupancy $p$ is assigned a uniform prior on $(0,1)$. What density does this imply for $Z=\operatorname{logit}(p)$, and what is its value at $z=0$?

The original density is $f_p(p)=1$. The inverse transformation is $p=\exp(z)/\{1+\exp(z)\}$, with derivative $dp/dz=p(1-p)$. Thus $$ f_Z(z)=1\times p(1-p)=\frac{\exp(z)}{\{1+\exp(z)\}^2}. $$ At $z=0$, $p=0.5$, and $f_Z(0)=0.5(0.5)=0.25$. The density is not constant: it approaches zero as $z$ tends to either positive or negative infinity. This is consistent with a uniform prior on the original fraction, which puts equal probability on equal-width intervals of $p$, not equal-width intervals of $z$.

Checking the Transformation and Choosing an Approach

A direct prior on the original scale can be easier to explain in biological terms, particularly when credible ranges or bounds are elicited in those units. A prior on a transformed scale can be convenient for positive or bounded parameters and can encode multiplicative variation for positive parameters. Neither approach is automatically preferable: the relevant question is which distribution on plausible parameter values expresses the intended assumptions. The introductory material on eliciting priors from biological knowledge motivates stating those assumptions in interpretable terms; the Jacobian calculation makes the resulting scale explicit.

When checking a proposed transformation, verify the following before using it in calibration:

Support: a log transformation requires $\theta>0$, and a logit transformation requires $0<p<1$. Exact boundary values need separate modeling if they have positive prior probability.
Direction: determine which density was specified first. If the prior is on $Z$ and the target is $p$, use $|dZ/dp|$; if converting a density on $p$ to $Z$, use $|dp/dZ|$.
Normalization: the transformed density should integrate to one over its support. The change-of-variables rule guarantees this when the starting density is normalized and the transformation is one-to-one.
Units: densities carry inverse units of their variable. Use a dimensionless ratio inside a logarithm, and interpret density values relative to the unit of the original parameter.
Joint dependence: for several jointly distributed parameters, transform the joint density with the full Jacobian determinant. Transforming marginal densities separately and multiplying them is valid only when the joint prior actually factors into independent marginals.

A common error is to assign a normal density to $z=\log(\theta/\theta_{\mathrm{ref}})$, substitute $z$ into the normal formula, and call the result a density for $\theta$. That omits $1/\theta$. The correction is to multiply by the absolute derivative $|dz/d\theta|=1/\theta$. Similarly, for a logit-normal prior on $p$, omitting $1/\{p(1-p)\}$ fails to preserve probability. A useful diagnostic is to integrate the proposed density numerically over its full support and check that the result is approximately one. Also inspect the tails and symmetry implied by the stated prior; a correct transformation can yield a shape that is less visually familiar on the original scale.

Transformations alone do not resolve whether a prior is biologically appropriate, whether parameters are identifiable from observations, or whether a model describes the data adequately. They specify how probability is represented across scales. The prior should still be assessed on the scale of scientific interpretation, and boundary behavior or extreme values should be considered explicitly.

Key takeaway: A prior distribution belongs to a parameter scale. To express that same prior on another scale, evaluate the density at the inverse-transformed value and multiply by the absolute Jacobian. A transformed-scale prior is a substantive choice because its induced original-scale distribution may have a different shape.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $Z=\log(\theta/\theta_{\mathrm{ref}})$ has a density $f_Z$, what Jacobian factor is required for the density of positive $\theta$?
A normal prior on $Z=\operatorname{logit}(p)$ is symmetric about zero. What symmetry does its induced density have on $p$, and why?
What density on $Z=\operatorname{logit}(p)$ is induced by a uniform prior on $p\in(0,1)$?
Why is a constant density over all real values of $Z=\operatorname{logit}(p)$ not a proper prior when transformed to $p$?

Solutions. (1) The inverse transformation is $Z=\log(\theta/\theta_{\mathrm{ref}})$, so the factor is $|dZ/d\theta|=1/\theta$, giving $f_\theta(\theta)=f_Z(\log(\theta/\theta_{\mathrm{ref}}))/\theta$. (2) The density is symmetric about $p=0.5$: $\operatorname{logit}(1-p)=-\operatorname{logit}(p)$, the logit-scale density has equal values at opposite arguments, and the Jacobian denominator $p(1-p)$ is unchanged by replacing $p$ with $1-p$. (3) Since $dp/dz=p(1-p)$, the induced density is $f_Z(z)=p(1-p)=\exp(z)/\{1+\exp(z)\}^2$. (4) The implied density is proportional to $1/\{p(1-p)\}$, whose integral diverges near both zero and one, so no finite normalizing constant exists.