Detect linear reparameterization — detect_linear

Check whether two model specifications are related by a linear reparameterization (affine transformation of parameters).

Usage

detect_linear_reparam(spec_a, spec_b, y, n_test = 50, tol = 1e-04)

Arguments

spec_a: First model_spec object
spec_b: Second model_spec object
y: Numeric vector of observed data
n_test: Number of parameter points to test (default 50)
tol: Tolerance for detecting linear relationship

Value

An S3 object of class linear_reparam containing:

is_linear: Logical; whether a linear relationship was detected
transformation: If linear, the estimated transformation matrix and offset
r_squared: R-squared value of the linear fit
residuals: Residuals from the linear fit

Details

This function tests whether there exists a linear transformation theta_B = A * theta_A + b such that the two models produce the same likelihood for all data.

Examples of linear reparameterizations:

Normal(mu, sigma) vs Normal(mu, sigma^2): not linear (square transformation)
Exponential(rate) vs Exponential(mean): linear (mean = 1/rate is not linear)
Location-scale: linear in location, but log-linear in scale

Examples

# Two models with different parameterizations
exp_spec <- model_spec(
  loglik_fn = function(y, lambda) sum(dexp(y, rate = lambda, log = TRUE)),
  par_names = "lambda",
  par_bounds = list(lambda = c(0.1, 10)),
  name = "Exponential"
)

gamma_spec <- model_spec(
  loglik_fn = function(y, rate) sum(dgamma(y, shape = 1, rate = rate, log = TRUE)),
  par_names = "rate",
  par_bounds = list(rate = c(0.1, 10)),
  name = "Gamma(1)"
)

set.seed(123)
y <- rexp(50, rate = 2)

result <- detect_linear_reparam(exp_spec, gamma_spec, y)
print(result)