API Documentation

The algebraic.mle package provides an algebra over Maximum Likelihood Estimators (MLEs). It enables manipulation and inference on fitted models through a consistent, composable interface.

Core Class: `mle`

The central abstraction is the mle S3 class, which wraps fitted model results with statistical properties.

Constructors

Function	Description
`mle()`	Create an MLE object from components (theta, variance, log-likelihood)
`mle_numerical()`	Wrap results from `optim()` or similar numerical optimizer
`mle_boot()`	Create a bootstrap MLE for uncertainty quantification
`mle_weighted()`	Create a weighted MLE from multiple estimates

Creating an MLE

library(algebraic.mle)

# From optim() results
result <- optim(par = c(0, 1), fn = negloglik, hessian = TRUE)
fit <- mle_numerical(result)

# From explicit components
fit <- mle(
  theta.hat = c(mu = 2.5, sigma = 1.2),
  loglike = -150.3,
  sigma = solve(-result$hessian),  # Variance-covariance matrix
  nobs = 100
)

Statistical Properties

The mle class provides methods for common statistical operations:

Point Estimates

Function	Description
`params(mle)`	Extract parameter estimates (θ̂)
`coef(mle)`	Alias for `params()`
`loglik_val(mle)`	Extract log-likelihood value

Uncertainty Quantification

Function	Description
`vcov(mle)`	Variance-covariance matrix of estimates
`se(mle)`	Standard errors (sqrt of diagonal of vcov)
`confint(mle)`	Confidence intervals (default 95%)
`bias(mle)`	Bias estimate (if available)

Model Diagnostics

Function	Description
`aic(mle)`	Akaike Information Criterion
`nobs(mle)`	Number of observations
`summary(mle)`	Print comprehensive summary

Working with Estimates

Confidence Intervals

# 95% confidence interval (default)
confint(fit)

# 99% confidence interval
confint(fit, level = 0.99)

# From variance-covariance matrix
confint_from_sigma(theta.hat, sigma, level = 0.95)

Information Criteria

# Compare models using AIC
aic(model1)
aic(model2)

# Lower AIC indicates better fit (penalizing complexity)

Marginal and Transformed Estimates

Extracting Marginals

# Extract marginal distribution for a single parameter
marginal_mu <- marginal(fit, "mu")

# Work with the marginal
confint(marginal_mu)

Transformations

The delta method enables inference on transformations of parameters:

# If fit has parameters (mu, sigma), compute mean lifetime = mu
# For exponential rate lambda, mean = 1/lambda
expectation(fit)  # Uses the appropriate transformation

Bootstrap Methods

When asymptotic normality is questionable, use bootstrap:

# Create bootstrap MLE
boot_fit <- mle_boot(data, fit_function, R = 1000)

# Bootstrap confidence intervals
confint(boot_fit, type = "bca")  # Bias-corrected accelerated
confint(boot_fit, type = "perc") # Percentile method

# Check for bias
bias(boot_fit)

Integration with likelihood.model

algebraic.mle is designed to work with likelihood.model:

library(likelihood.model)
library(algebraic.mle)

# Build a likelihood model
model <- likelihood_contr_model(
  obs_type = "right_censored",
  distribution = weibull_distribution()
)

# Fit using MLE
fit <- mle(model, data = my_data, start = c(shape = 1, scale = 1))

# Use algebraic.mle methods on the result
confint(fit)
aic(fit)
summary(fit)

Full Reference

For complete API documentation with all function signatures: