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22.5 Posteriors with unbounded parameters

In some cases, the posterior density will not grow without bound, but parameters will grow without bound with gradually increasing density values. Like the models discussed in the previous section that have densities that grow without bound, such models also have no posterior modes.

Separability in logistic regression

Consider a logistic regression model with \(N\) observed outcomes \(y_n \in \{ 0, 1 \}\), an \(N \times K\) matrix \(x\) of predictors, a \(K\)-dimensional coefficient vector \(\beta\), and sampling distribution \[ y_n \sim \textsf{Bernoulli}(\operatorname{logit}^{-1}(x_n \beta)). \] Now suppose that column \(k\) of the predictor matrix is such that \(x_{n,k} > 0\) if and only if \(y_n = 1\), a condition known as “separability.” In this case, predictive accuracy on the observed data continue to improve as \(\beta_k \rightarrow \infty\), because for cases with \(y_n = 1\), \(x_n \beta \rightarrow \infty\) and hence \(\operatorname{logit}^{-1}(x_n \beta) \rightarrow 1\).

With separability, there is no maximum to the likelihood and hence no maximum likelihood estimate. From the Bayesian perspective, the posterior is improper and therefore the marginal posterior mean for \(\beta_k\) is also not defined. The usual solution to this problem in Bayesian models is to include a proper prior for \(\beta\), which ensures a proper posterior.