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Compute Fisher information matrix

Source: R/fisher_information.R
fisher_information.Rd

Compute the Fisher information matrix for a model specification at given parameter values. Supports both observed information (negative Hessian) and expected information (outer product of gradients estimator).

Usage

fisher_information(
  spec,
  y,
  par,
  type = c("observed", "expected"),
  method = c("hessian"),
  cl = NULL
)

Arguments

spec

A model_spec object

y

Numeric vector of observed data

par

Named numeric vector of parameter values at which to evaluate

type

Type of information: "observed" (default) uses the negative Hessian at the data; "expected" uses the outer product of gradients (OPG) estimator computed from per-observation score contributions

method

Computation method: "hessian" (default) uses numerical differentiation

cl

Optional parallel cluster from setup_cluster(). Only used for type = "expected" to parallelize per-observation score computation.

Value

An S3 object of class fisher_info containing:

matrix

The Fisher information matrix

eigenvalues

Eigenvalues (sorted decreasing)

eigenvectors

Corresponding eigenvectors

condition

Condition number

rank

Numerical rank

par

Parameter values used

par_names

Parameter names

type

Type of information computed

Details

The Fisher information matrix characterizes the curvature of the log-likelihood surface. Key properties:

  • Positive definite: all parameters locally identifiable

  • Singular (rank deficient): some parameters not identifiable

  • Near-singular (high condition number): some parameters weakly identified

Two types of information are available:

  • Observed information (type = "observed"): The negative Hessian of the log-likelihood. This is the default and most common choice.

  • Expected information (type = "expected"): Estimated using the outer product of gradients (OPG), where scores are summed over observations. The score is the gradient of the log-likelihood for each observation. Under regularity conditions, this converges to the same limit as observed information but may differ in finite samples. Requires the log-likelihood to be additive over observations.

Examples

spec <- model_spec(
  loglik_fn = function(y, mu, sigma) {
    sum(dnorm(y, mean = mu, sd = sigma, log = TRUE))
  },
  par_names = c("mu", "sigma"),
  par_bounds = list(sigma = c(1e-6, Inf))
)

set.seed(123)
y <- rnorm(100, mean = 5, sd = 2)

# Observed information (negative Hessian)
info_obs <- fisher_information(spec, y, par = c(mu = 5, sigma = 2))
print(info_obs)

# Expected information (OPG estimator)
info_exp <- fisher_information(spec, y, par = c(mu = 5, sigma = 2),
                               type = "expected")
print(info_exp)