Why Boundary Means Need Care
A response family may allow a mean at the edge of its parameter space, while a chosen inverse link produces only values strictly inside that space for finite predictors. This distinction matters whenever a model is asked to represent an exact boundary mean: a value that can be approached arbitrarily closely may still not be attained by any finite linear predictor.
For a generalized linear model, the linear predictor is \(\eta_i=\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}\), and the mean is obtained by applying the inverse link, \(\mu_i=h(\eta_i)=g^{-1}(\eta_i)\). In this tutorial, the limit behavior of common inverse links is the focus. In particular, “finite” means that the predictor is a real number, not \(+\infty\) or \(-\infty\).
The distinction is not merely about terminology. For a Bernoulli response, the mean parameter \(p\) can include 0 and 1, corresponding to degenerate response distributions. But the inverse logit gives a strictly interior probability for every finite predictor. Thus the family’s boundary values and the link’s finite-predictor range need not be the same set.
Limits of Common Inverse Links
The logit, probit, and complementary log-log links all map a real predictor to a probability strictly between 0 and 1. Their detailed shapes differ, but their limits at the two ends of the real line show why neither exact probability endpoint is attained at a finite predictor.
| Link | Inverse link \(h(\eta)\) | Finite-predictor range | Limits at the ends |
|---|---|---|---|
| Logit | \(\dfrac{1}{1+\exp(-\eta)}\) | \((0,1)\) | \(h(\eta)\to0\) as \(\eta\to-\infty\); \(h(\eta)\to1\) as \(\eta\to+\infty\) |
| Probit | \(\Phi(\eta)\) | \((0,1)\) | \(h(\eta)\to0\) as \(\eta\to-\infty\); \(h(\eta)\to1\) as \(\eta\to+\infty\) |
| Complementary log-log | \(1-\exp\{-\exp(\eta)\}\) | \((0,1)\) | \(h(\eta)\to0\) as \(\eta\to-\infty\); \(h(\eta)\to1\) as \(\eta\to+\infty\) |
| Log | \(\exp(\eta)\) | \((0,\infty)\) | \(h(\eta)\to0\) as \(\eta\to-\infty\); \(h(\eta)\to\infty\) as \(\eta\to+\infty\) |
| Identity | \(\eta\) | \(\mathbb{R}\) | No finite endpoint limits are needed to attain a finite value |
For the logit inverse, \(\exp(-\eta)>0\) at every finite \(\eta\), so \(0<1/\{1+\exp(-\eta)\}<1\). The probit inverse is a normal cumulative distribution function: at any finite \(\eta\), positive normal density remains both below and above \(\eta\), so \(0<\Phi(\eta)<1\). For the complementary log-log inverse, \(\exp(\eta)>0\), hence \(0<\exp\{-\exp(\eta)\}<1\), which again gives a value strictly between 0 and 1. These inequalities establish the ranges; the limits show how their endpoints are approached.
Proof. The range assumption gives \(a<h(\eta)<b\) for every \(\eta\in\mathbb{R}\), proving the first statement. To prove the assertion for \(b\), fix any real number \(M\). Strict increase gives \(h(M)<b\). Since \(h(\eta_k)\to b\), eventually \(h(\eta_k)>h(M)\). Strict increase then implies \(\eta_k>M\). This holds for every real \(M\), which is exactly \(\eta_k\to+\infty\). For the lower endpoint, fix any \(M\), so \(h(M)>a\). Eventually \(h(\eta_k)<h(M)\), and strict increase implies \(\eta_k<M\). Since this holds for every \(M\), \(\eta_k\to-\infty\). \(\square\)
The theorem applies to the three probability inverse links in the table. For instance, obtaining probabilities closer and closer to 1 under the logit link requires predictors tending to \(+\infty\); it does not identify any finite predictor that gives probability exactly 1. The same conclusion holds for probability 0 and the other two links, with the direction reversed for the lower endpoint.
Worked Example: Interior Probabilities Under the Logit Link
For the inverse logit, \(p=1/\{1+\exp(-\eta)\}\). For a target strictly between 0 and 1, solve for its finite predictor:
For \(p=0.2\), this gives \(\eta=\log(0.2/0.8)=\log(0.25)\approx-1.386\). For \(p=0.5\), it gives \(\eta=\log(1)=0\). For \(p=0.8\), it gives \(\eta=\log(0.8/0.2)=\log(4)\approx1.386\). All three targets have finite predictors.
At \(p=0\) or \(p=1\), the odds \(p/(1-p)\) are respectively 0 or undefined because the denominator is 0. There is no finite logarithm that supplies the required predictor. The limits are \(\eta\to-\infty\) for \(p\to0\) and \(\eta\to+\infty\) for \(p\to1\).
Worked Example: Probit Probabilities Near a Boundary
Under the probit link, \(p=\Phi(\eta)\), where \(\Phi\) is the standard normal cumulative distribution function. For an interior target \(p\), the predictor is \(\eta=\Phi^{-1}(p)\). For example, \(\Phi^{-1}(0.95)\approx1.645\), so a predictor of approximately 1.645 gives a probability of 0.95. By symmetry, a predictor of approximately \(-1.645\) gives probability 0.05.
At every finite \(\eta\), the standard normal density is positive on both \((-\infty,\eta)\) and \((\eta,\infty)\). Consequently, \(\Phi(\eta)>0\) and \(1-\Phi(\eta)>0\), so neither probability endpoint is attained. To see the limit explicitly, for any \(K>0\), if \(\eta>K\), then \(1-\Phi(\eta)<1-\Phi(K)\); as \(K\to\infty\), the latter tends to 0. Thus \(\Phi(\eta)\to1\) as \(\eta\to+\infty\). The lower-end limit follows by symmetry: \(\Phi(-\eta)=1-\Phi(\eta)\). An exact target of 1 or 0 therefore requires a predictor tending to the corresponding infinite limit, not a finite probit value.
Boundary Behavior for Positive Means
The log inverse link has a different boundary pattern. For a positive mean \(\mu\), the inverse relation is \(\eta=\log(\mu)\), which is finite for every \(\mu>0\). But \(\mu=0\) has no finite log, and \(\exp(\eta)\) remains positive for every finite \(\eta\). Zero is therefore approached as \(\eta\to-\infty\), while there is no finite upper boundary: the mean grows without bound as \(\eta\to+\infty\).
Worked Example: A Poisson Mean Approaching Zero
Consider a Poisson response with mean \(\mu=\exp(\eta)\). A target mean of 2 has predictor \(\eta=\log(2)\approx0.693\), and a target mean of 0.01 has predictor \(\eta=\log(0.01)\approx-4.605\). Both targets are positive and have finite predictors.
For the target \(\mu=0\), the equation \(\exp(\eta)=0\) has no real solution, because the exponential is positive for every real input. Instead, if \(\eta=-10\), the fitted mean is \(\exp(-10)\approx0.0000454\), a small but still positive number. More generally, \(\exp(\eta)\to0\) as \(\eta\to-\infty\). Under a Poisson distribution with mean zero, the response is zero with probability one, so zero is a meaningful boundary mean for the family even though the log inverse link does not attain it at a finite predictor.
The identity inverse link provides a useful contrast. It maps a finite predictor \(\eta\) directly to the mean \(\mu=\eta\), so a boundary value such as \(\mu=0\) can be attained at \(\eta=0\). Whether an identity-link model is valid for a particular response family also depends on keeping its fitted means in the family’s permissible mean space. Attainability at the link level does not itself guarantee that every possible set of model coefficients gives valid means.
Worked Example: Comparing Identity and Log Links at Zero
Suppose the target mean is \(\mu=0\). With the identity inverse link, \(h(\eta)=\eta\), the predictor \(\eta=0\) gives the target exactly. With the log inverse link, \(h(\eta)=\exp(\eta)\), no finite \(\eta\) gives zero.
For a sequence of finite log-link predictors \(\eta_k=-k\), where \(k\) is a positive integer, the corresponding means are \(\mu_k=\exp(-k)\). Each is positive, and \(\mu_k\to0\) as \(k\to\infty\). This sequence illustrates the distinction precisely: the boundary can be approached arbitrarily closely with finite predictors, but reaching it in the limit requires predictors whose magnitude diverges. By contrast, the identity link reaches zero without taking any limit.
When a Boundary Target Is Only a Limit
An exact boundary mean can also affect likelihood-based fitting. Consider \(n\) independent Bernoulli observations, all equal to 1, in an intercept-only model with the logit link. Here the common fitted probability is \(p=1/\{1+\exp(-\eta)\}\), and the likelihood as a function of the finite predictor is
For every finite \(\eta\), \(p<1\), so \(L(\eta)<1\). As \(\eta\to+\infty\), \(p\to1\) and \(L(\eta)\to1\). Since a Bernoulli likelihood cannot exceed 1, the likelihood’s supremum is 1, but no finite \(\eta\) attains it. Thus the data’s boundary-favoring pattern does not make the logit model’s finite fitted probability equal to 1; rather, the likelihood can keep increasing as the predictor grows.
This is a basic illustration of a broader practical point: fitted values from a finite coefficient vector and finite covariates have finite predictors, but fitting procedures may encounter estimates that grow very large when the data favor a boundary. A value reported as 0.9999 may be a numerical approximation to a probability near 1, not an exact finite-predictor probability of 1.
Interpreting the Distinction
When discussing a fitted mean near an endpoint, keep three questions separate: which means the response family allows, which means the inverse link produces at finite predictors, and which means are approached only as predictor limits. These questions have different answers for different links. The logit, probit, and complementary log-log links have the same open probability range, whereas the identity link can attain a finite boundary directly. The log link approaches zero but has no finite upper endpoint.
This distinction also clarifies what “infinite predictor” means in a model. For a fixed observation, a predictor \(\eta_i=\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}\) is finite when the covariates and coefficients are finite. A boundary mean that requires \(\eta_i\to+\infty\) or \(-\infty\) is not an ordinary finite fitted value, even if a sequence of finite coefficient values produces means arbitrarily close to that boundary.
Check Your Understanding
Use the inverse-link ranges and limits to answer the following questions.
- Under the logit inverse link, what happens to the predictor as a fitted probability approaches 0? What happens as it approaches 1?
- Why does a finite probit predictor give a probability strictly between 0 and 1?
- For a log-link model, which boundary mean is approached as the predictor tends to \(-\infty\), and can any finite predictor attain it?
- Give one reason the identity inverse link differs from the log inverse link when the target mean is zero.
- In the all-success Bernoulli example, why is the likelihood’s supremum not attained at a finite logit predictor?