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We characterise the behaviour of the maximum Diaconis-Ylvisaker priorpenalized likelihood estimator in high-dimensional logistic regression, wherethe number of covariates is a fraction $\kappa \in (0,1)$ of the number ofobservations $n$, as $n \to \infty$. We derive the estimator's aggregateasymptotic behaviour when covariates are independent normal random variableswith mean zero and variance $1/n$, and the vector of regression coefficientshas length $\gamma \sqrt{n}$, asymptotically. From this foundation, we deviseadjusted $Z$-statistics, penalized likelihood ratio statistics, and aggregateasymptotic results with arbitrary covariate covariance. In the process, we fillin gaps in previous literature by formulating a Lipschitz-smooth approximatemessage passing recursion, to formally transfer the asymptotic results fromapproximate message passing to logistic regression. While the maximumlikelihood estimate asymptotically exists only for a narrow range of $(\kappa,\gamma)$ values, the maximum Diaconis-Ylvisaker prior penalized likelihoodestimate not only exists always but is also directly computable using maximumlikelihood routines. Thus, our asymptotic results also hold for $(\kappa,\gamma)$ values where results for maximum likelihood are not attainable, withno overhead in implementation or computation. We study the estimator'sshrinkage properties and compare it to logistic ridge regression anddemonstrate our theoretical findings with simulations.

Diaconis-Ylvisaker prior penalized likelihood for $p/n \to \kappa \in (0,1)$ logistic regression