Why “Uniform” Needs a Parameterization
A prior described as “uniform” sounds neutral: it may seem to give every possible value of a clinical-trial event probability equal weight. But uniformity depends on the scale on which the parameter is measured. A prior that has constant density for a probability \(p\) does not have constant density for its logit, and the reverse is also true. The resulting distributions can assign different prior probabilities to clinically important ranges.
This matters whenever investigators choose priors for a trial endpoint or fit a model using a transformed parameter. The transformation changes the density through its Jacobian. It does not change the underlying probability distribution if the prior is correctly transformed; choosing a new uniform density after changing coordinates, however, generally does change that distribution.
The logit maps probabilities in \((0,1)\) to all real values. It is increasing, so larger logits correspond to larger event probabilities; \(\eta=0\) corresponds to \(p=0.5\). The derivative of the inverse transformation is
That derivative is the Jacobian needed to compare densities. A density describes probability per unit of its own parameter scale. Since equal-sized intervals on the logit scale do not correspond to equal-sized intervals on the probability scale, the numerical density changes under transformation.
Transforming a Uniform Probability Prior
Suppose \(p\) has a uniform prior on \((0,1)\), with density \(\pi_p(p)=1\). To find the density of \(\eta=\operatorname{logit}(p)\), use the change-of-variables rule: multiply the density of \(p\) by the absolute derivative \(dp/d\eta\).
This is a proper density on the real line. Its mass is largest at \(\eta=0\), where \(p=0.5\), and tends toward zero in either tail. Thus a uniform probability prior induces a logit prior concentrated relatively more around moderate probabilities than around probabilities near zero or one. It is not uniform on the logit scale.
Worked Example: Probability Mass on the Logit Scale
Suppose \(p\sim\operatorname{Uniform}(0,1)\). What is the probability that the logit lies between 0 and \(\log(3)\)?
Because the logit is increasing, this event is equivalent to \(0.5<p<0.75\): the inverse logit of 0 is \(0.5\), and the inverse logit of \(\log(3)\) is \(3/(1+3)=0.75\). Under a uniform prior on \(p\), the probability is the length of that interval:
The same result follows from the induced logit density. Its cumulative distribution function is \(P(\eta\le z)=\operatorname{logit}^{-1}(z)\); therefore the probability is \(0.75-0.5=0.25\). This calculation shows how to express the probability-scale prior on the logit scale without replacing it by a different prior.
Transforming a Uniform Logit Prior
Now suppose instead that the density is constant on the logit scale. For a proper version, let \(\eta\sim\operatorname{Uniform}(-A,A)\), where \(A>0\) is a chosen finite bound. The probability support is then \((\operatorname{logit}^{-1}(-A),\operatorname{logit}^{-1}(A))\), not the full open interval \((0,1)\). To find the density on the probability scale, use the reverse Jacobian, \(d\eta/dp=1/[p(1-p)]\).
for probabilities in that support, and zero outside it. This density rises toward the support boundaries because \(p(1-p)\) gets smaller there. A bounded uniform logit prior therefore assigns more density per unit of probability near its lower and upper probability limits than near the center.
A constant density over the entire real line, sometimes called a flat or uniform logit prior, is not a proper prior: its integral is infinite. It cannot be normalized to have total probability one. It is an improper prior, which may sometimes produce a proper posterior after data are observed, but that must be checked for the particular likelihood and data. Treating an improper prior as though it assigned prior probabilities to events is invalid.
Worked Example: Comparing Priors With the Same Probability Support
Consider two proper priors, both restricted to \(0.1\le p\le0.9\). The first is uniform in \(p\). The second is uniform in \(\eta\) between the logits of 0.1 and 0.9. Compare the prior probability that \(p<0.2\).
For the uniform probability prior, the support has length \(0.9-0.1=0.8\), while the part below 0.2 has length \(0.2-0.1=0.1\). Thus
For the uniform logit prior, the logit support is \([-\log(9),\log(9)]\), with total width \(2\log(9)\approx4.394\). The logit of 0.2 is \(-\log(4)\approx-1.386\). The portion of the logit interval below that threshold has width \(\log(9)-\log(4)\approx0.8109\). Therefore
The priors have identical probability support but assign different probabilities to the range \(p<0.2\): approximately 12.5% versus 18.45%. Neither calculation is an error. Each prior is uniform on a different scale.
What Happens to a Binomial Posterior?
The prior choice can affect posterior inference, especially when the trial data are limited. Suppose \(Y\) is the number of participants with an event among \(n\) independent participants, each with the same event probability \(p\). Then \(Y\mid p\sim\operatorname{Binomial}(n,p)\), so the likelihood, up to a factor that does not depend on \(p\), is \(p^y(1-p)^{n-y}\), where \(y\) events were observed. This model assumes independent outcomes and a common event probability for all participants in the group.
With a uniform prior on \(p\), the posterior density is proportional to \(p^y(1-p)^{n-y}\), giving a \(\operatorname{Beta}(y+1,n-y+1)\) posterior. With a flat, improper prior on \(\eta\), the prior density on \(p\) is proportional to \(1/[p(1-p)]\). Multiplying by the likelihood gives a posterior density proportional to \(p^{y-1}(1-p)^{n-y-1}\). This is a proper \(\operatorname{Beta}(y,n-y)\) posterior only when \(y>0\) and \(n-y>0\). If either count is zero, the posterior is improper.
Worked Example: Posterior Means Under Two Prior Choices
In a hypothetical trial group, 4 of 20 participants experience the event. Compare the posterior mean event probability under (a) a uniform prior on \(p\), and (b) a flat, improper prior on the logit. The binomial assumptions above are taken to hold for this calculation.
(a) Uniform probability prior. The posterior is \(\operatorname{Beta}(y+1,n-y+1)=\operatorname{Beta}(5,17)\). The mean of a \(\operatorname{Beta}(a,b)\) distribution is \(a/(a+b)\), so
(b) Flat logit prior. Since there are both events and non-events, the posterior is proper and equals \(\operatorname{Beta}(y,n-y)=\operatorname{Beta}(4,16)\). Its mean is
The posterior means are 22.73% and 20.00%, respectively. These are posterior summaries under two different priors, not contradictory estimates under an identical prior. With only 20 participants, the prior choice remains visible in the result. The flat-logit calculation also depends on the data including at least one event and one non-event; the same improper prior would not produce a proper posterior if all 20 participants had the same outcome.
Checking the Choice and Its Limits
A useful prior check is to translate the prior into the scale and ranges that matter for the trial. For example, calculate prior probabilities for clinically meaningful event-rate ranges, inspect the implied distribution of \(p\), and verify whether the prior support permits plausible values. A prior density plot on only one scale can obscure its implications on another. Compare probability statements, not just the visual flatness of density curves: density heights use different units on the \(p\) and \(\eta\) scales.
State whether uniformity is intended for the event probability \(p\), the logit \(\eta\), or another quantity.
A uniform density over a finite interval can be proper; a constant density over all real logits is improper. Record any bounds and their rationale.
Use the Jacobian to find the implied density on the other scale, then calculate prior probabilities for ranges relevant to the clinical question.
For an improper prior, establish that the likelihood makes the posterior integrable for the observed data. Do not assume that a finite-looking estimate guarantees a proper posterior.
The main alternatives are to specify a distribution directly on the probability scale, or to specify a distribution on the logit scale with deliberate finite bounds or a proper spread. These choices may be motivated by prior clinical knowledge, plausible event-rate ranges, or the consequences of the prior for key trial decisions. A transformed density should be carried through the Jacobian when re-expressing the same prior; independently declaring a new density to be uniform is a new prior choice.
No uniform prior is universally noninformative. A flat density on one scale encodes a particular distribution of probability mass, and a change of parameterization exposes that choice. Bounds for a proper logit-uniform prior also matter: changing \(A\) changes both the probability support and the prior probabilities within it. The appropriate choice depends on the parameterization and the scientific question, rather than on the word “uniform” alone.
Common Mistake
Wrong reasoning: “A uniform prior on the logit and a uniform prior on the event probability both look flat, so they express the same lack of prior information.”
Correction: They are flat with respect to different scales and generally imply different probability distributions. For the same prior distribution expressed in two coordinates, transform its density with the Jacobian. If instead you choose a fresh uniform density after changing coordinates, recognize that you have changed the prior. Also distinguish a proper bounded uniform logit prior from the improper constant density on the entire real line.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- Under a uniform prior on \(p\in(0,1)\), what is the density induced on \(\eta=\operatorname{logit}(p)\), and where is it largest?
- A proper prior is uniform in \(\eta\) on a finite interval. What factor appears in its density on the \(p\) scale, and what does it imply near the ends of the supported probability interval?
- Why can a constant prior density over all real logits not be treated as a proper probability distribution?
- For \(n=12\) and \(y=0\), is the posterior under a flat, improper logit prior proper? Explain.
Solutions. (1) The induced density is \(p(1-p)\), which is largest at \(p=0.5\), or \(\eta=0\). (2) The transformed density includes the factor \(1/[p(1-p)]\); it is relatively larger near the finite support boundaries than near the center. (3) A constant density integrated over the real line has infinite total mass, so it cannot be normalized to integrate to one. (4) No. With \(y=0\), the transformed posterior is proportional to \(p^{-1}(1-p)^{11}\), which is not integrable near \(p=0\); equivalently, the posterior is improper because the data contain no events.