Posterior mode and normal approximation for logistic regression

The N(0, prior_sd^2) prior keeps the Hessian invertible under separation. The approximation is centred at the MODE and scaled by the inverse Hessian. An intercept column is prepended; do not include one.

Usage

Bayeslogit(X, y, prior_sd = 10, iters = 50, tol = 1e-12)

Arguments

X

Predictors, WITHOUT an intercept column.

y

Binary responses in {0, 1}.

prior_sd

Standard deviation of the coefficient prior.

iters

Maximum Newton steps.

tol

Convergence tolerance on the maximum coefficient change.

Value

List with estimate, se, log_posterior, iterations, converged, n, p.

Details

Formula: Newton step beta <- beta + (X'WX + I/s^2)^-1 (X'(y - p) - beta/s^2), W = diag(p_i(1 - p_i)); Var(beta) ~= (X'WX + I/s^2)^-1

References

Gelman, Carlin, Stern, Dunson, Vehtari & Rubin (2013), Bayesian Data Analysis, 3rd edition, Section 4.1 and Chapter 16.

Examples

set.seed(1)
X <- cbind(1, matrix(rnorm(40), 20, 2))
y <- rbinom(20, 1, 0.5)
Bayeslogit(X, y)
#> $estimate
#> [1] -0.08970193 -0.08970193 -0.19423151  0.47370852
#> 
#> $se
#> [1] 7.0749519 7.0749519 0.5262784 0.5848783
#> 
#> $log_posterior
#> [1] -13.27426
#> 
#> $iterations
#> [1] 5
#> 
#> $converged
#> [1] 1
#> 
#> $n
#> [1] 20
#> 
#> $p
#> [1] 4
#> 
#> $method
#> [1] "Logistic regression posterior mode, BDA3 Section 4.1"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"