Estimate intercept and signal strength — signal

signal_strength estimates $(\beta_0, \gamma)$ using the estimated $\kappa_s$ and observed proportion of successes.

Usage

signal_strength(
  rho_prime = rho_prime_logistic,
  kappa_hat,
  intercept = FALSE,
  p0 = NA,
  verbose = FALSE,
  tol = 1e-04
)

Arguments

rho_prime: A function that computes the success probability $\rho'(t) = \mathrm{P}(Y=1 | X^\top \beta = t)$, here $\beta$ is the coefficient. The default is logistic model.
kappa_hat: Numeric. Estimated dimension where the data becomes linearly separable
intercept: Logical TRUE if the model contains an intercept.
p0: Numeric. Proportion of outcomes $Y=1$.
verbose: Should progress be printed? If TRUE, prints progress at each step.
tol: Numeric. Tolerance to be used in fsolve function.

Value

If the model does not contain an intercept, returns estimated gamma_hat. Otherwise, returns a list with two components

gamma_hat: Estimated signal strength.
b_hat: Estimated intercept.

Details

Assume that $Y$ depends on $X$ as $$ \mathrm{P}(Y=1\,|\,X) = \rho'(X^\top \beta + \beta_0), $$ and let the signal strength be $\gamma = \mathrm{Var}(X^\top \beta)^{1/2}$. The pair $(\beta_0, \gamma)$ satisfies

They are on the phase transition curve $\kappa(\beta_0, \gamma)$. $$ \hat{\kappa} \approx \kappa(\beta_0, \gamma) $$
The observed proportion of $Y=1$ should be close to the expected proportion. $$ p_0 \approx \mathrm{P}(Y = 1\,|\, \beta_0, \gamma) = \mathrm{E}{\mathrm{Ber}(\rho'(\beta_0 + \gamma Z)} $$ where $Z$ is a standard normal variable.

We solve the above system of two equations to obtain an estimate of $(\beta_0, \gamma)$

Examples

if (FALSE) {
# no signal case
# should return 0, returns 0.0127
signal <- signal_strength(kappa_hat = 0.5, intercept = FALSE)
signal$gamma_hat
}