solve_kappa: Compute the phase transition curve

In zq00/glmhd: Estimates the inflation and variance of the MLE from a high-dimensional GLM

Compute the phase transition curve

Description

solve_kappa computes the problem dimension \kappa where the phase transition occurs in binary regression, given \beta and \gamma_0. solve_beta and solve_gamma computes \beta_0 and \gamma_0 on the phase transition curve given the other one and \kappa.

Usage

solve_kappa(rho_prime, beta0, gamma0)

solve_beta(rho_prime, kappa, gamma0, verbose = FALSE)

solve_gamma(rho_prime, kappa, beta0, verbose = FALSE)

Arguments

rho_prime	Function. Success probability \rho(t) = \mathrm{P}(Y=1\,|\, X^\top \beta = t)
beta0	Numeric. Intercept value.
gamma0	Numeric. Signal strength.
kappa	Numeric. Problem dimension on the phase transition curve.
verbose	Print progress if TRUE.

Details

When covariates are multivariate Gaussian, the phase transition dimension can be characterized as following.

\kappa > h_{\mathrm{MLE}}(\beta_0, \gamma_0) \implies \lim_{n,p\to\infty} \mathrm{P}(\text{MLE exists}) = 0

\kappa < h_{\mathrm{MLE}}(\beta_0, \gamma_0) \implies \lim_{n,p\to\infty} \mathrm{P}(\text{MLE exists}) = 1.

The function h is defined to be

h_{\mathrm{MLE}}(\beta_0, \gamma_0) = \min_{t_0, t_1 \in \mathbb{R}} \mathbb{E}\left[(t_0 Y + t_1 V - Z)_+^2 \right],

where X\sim\mathcal{N}(0,1) and \mathrm{P}(Y=1|X) = 1- \mathrm{P}(Y=-1|X) = \rho'(\beta_0 + \gamma_0 X) . Z\sim\mathcal{N}(0,1) and is independent of X,Y. The phase transition curve is thus \kappa(\beta_0, \gamma_0). It also depends on the success probability \rho'.

Value

Numeric. Problem dimension \kappa (\beta or \gamma) on the phase transition curve.

References

The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression Emmanuel J. Candes and Pragya Sur, Ann. Statist., Volume 48, Number 1 (2020), 27-42.

Examples

## Not run: 
# when Y is independent of X, should return 0.5 for logistic model
# should return 0.5
rho_prime_logistic <- function(t) 1 / (1 + exp(-t))
solve_kappa(rho_prime_logistic, 0, 0)

## End(Not run)

zq00/glmhd documentation built on April 7, 2023, 7:45 a.m.