Type:	Package
Title:	Likelihood-Based Statistical Inference in the Fisherian Tradition
Version:	1.0.0
Description:	Facilitates building likelihood models in the Fisherian tradition following Richard Royall (1997, ISBN:978-0412044113) "Statistical Evidence: A Likelihood Paradigm". Defines generic methods for working with likelihoods (loglik(), score(), hess_loglik(), fim()) and provides functions for pure likelihood-based inference (support(), relative_likelihood(), likelihood_interval(), profile_loglik()).
License:	MIT + file LICENSE
Encoding:	UTF-8
Depends:	algebraic.mle (≥ 2.0.0)
Imports:	algebraic.dist, generics, stats, numDeriv, boot
URL:	https://github.com/queelius/likelihood.model, https://queelius.github.io/likelihood.model/
BugReports:	https://github.com/queelius/likelihood.model/issues
Suggests:	mvtnorm, testthat (≥ 3.0.0), knitr, rmarkdown
VignetteBuilder:	knitr
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-03-17 17:53:00 UTC; spinoza
Author:	Alexander Towell [aut, cre]
Maintainer:	Alexander Towell <lex@metafunctor.com>
Repository:	CRAN
Date/Publication:	2026-03-19 05:00:02 UTC

likelihood.model: Likelihood-Based Inference in the Fisherian Tradition

Description

The likelihood.model package provides a framework for likelihood-based inference. The package is organized in layers:

Core Concept (core-generics.R): The likelihood_model "concept" – an abstract interface that any model can implement. At minimum, implement loglik(). Optionally provide score() and hess_loglik() for analytical derivatives; defaults use numerical differentiation via numDeriv.

Core Infrastructure:

• fisher_mle / fisher_boot: Result objects from MLE fitting, with methods for coef(), vcov(), confint(), algebraic.mle::se(), stats::AIC(), stats::BIC(), summary().

• fit(): Default MLE solver using optim(). Models can override this with closed-form solutions (see exponential_lifetime for an example).

• Fisherian inference: support(), relative_likelihood(), likelihood_interval(), profile_loglik(), evidence() – pure likelihood-based inference without probability statements.

• lrt(): Likelihood ratio test for nested models.

Example Implementation: exponential_lifetime: Exponential with right-censoring support. Demonstrates closed-form MLE (no optim needed), analytical FIM, and rdata() for Monte Carlo validation.

For contribution-based models with heterogeneous observation types, see the companion package likelihood.contr.

Author(s)

Maintainer: Alexander Towell lex@metafunctor.com (ORCID)

See Also

Useful links:

• https://github.com/queelius/likelihood.model

• https://queelius.github.io/likelihood.model/

• Report bugs at https://github.com/queelius/likelihood.model/issues

Compute sufficient statistics for the exponential model

Description

Compute sufficient statistics for the exponential model

Usage

.exponential_sufficient_stats(df, model)

Arguments

Value

List with d (number of exact obs), total_time (sum of all times), n (total observations)

Retrieve the assumptions the likelihood model makes about the data.

Description

Retrieve the assumptions the likelihood model makes about the data.

Usage

assumptions(model, ...)

Arguments

Value

A list of assumptions

Assumptions for exponential_lifetime

Description

Assumptions for exponential_lifetime

Usage

## S3 method for class 'exponential_lifetime'
assumptions(model, ...)

Arguments

Value

Character vector of assumptions

Compute bootstrap bias estimate

Description

Estimates bias from bootstrap replicates: bias = mean(bootstrap estimates) - original estimate

Usage

## S3 method for class 'fisher_boot'
bias(x, theta = NULL, ...)

Arguments

Value

Bias estimate vector

Bias for fisher_mle

Description

Estimates the bias of the MLE. Without a model and true parameter value, returns zeros (asymptotic bias is zero under regularity conditions). When both theta and model are provided, performs a Monte Carlo simulation study to estimate finite-sample bias.

Usage

## S3 method for class 'fisher_mle'
bias(x, theta = NULL, model = NULL, n_sim = 1000, ...)

Arguments

Value

Bias estimate vector

Extract coefficients from fisher_mle object

Description

Extract coefficients from fisher_mle object

Usage

## S3 method for class 'fisher_mle'
coef(object, ...)

Arguments

Value

Named numeric vector of parameter estimates

Confidence intervals from bootstrap

Description

Computes bootstrap confidence intervals using various methods.

Usage

## S3 method for class 'fisher_boot'
confint(
  object,
  parm = NULL,
  level = 0.95,
  type = c("perc", "bca", "norm", "basic"),
  ...
)

Arguments

Value

Matrix with columns for lower and upper bounds

Compute confidence intervals for fisher_mle parameters

Description

Computes asymptotic Wald confidence intervals based on the estimated variance-covariance matrix.

Usage

## S3 method for class 'fisher_mle'
confint(object, parm = NULL, level = 0.95, ...)

Arguments

Value

Matrix with columns for lower and upper bounds

Deviance for likelihood models

Description

Computes the deviance, which is useful for model comparison.

Usage

## S3 method for class 'fisher_mle'
deviance(object, null_model = NULL, ...)

Arguments

Details

When called with a single model, returns -2 * logL (the deviance relative to a saturated model).

When called with two models, returns the deviance difference: D = 2 * (logL_full - logL_reduced)

Under the null hypothesis that the reduced model is correct, D is asymptotically chi-squared with df = p_full - p_reduced.

Value

Deviance value

Likelihood-Based Evidence

Description

Computes the strength of evidence for theta1 vs theta2: E(theta1, theta2) = logL(theta1) - logL(theta2)

Usage

evidence(model, ...)

## S3 method for class 'likelihood_model'
evidence(model, data, theta1, theta2, ...)

Arguments

Details

Positive values favor theta1, negative values favor theta2.

Conventional interpretation (following Royall):

• |E| > log(8) ~ 2.08: Strong evidence

• |E| > log(32) ~ 3.47: Very strong evidence

Value

Evidence value (log likelihood ratio)

Exponential lifetime model with right-censoring support

Description

A likelihood model for the Exponential(lambda) distribution with optional right-censoring. This is a reference implementation demonstrating:

• Closed-form MLE: Overrides fit() to compute lambda_hat = d/T directly, bypassing optim entirely.

• Analytical derivatives: score, Hessian, and FIM in closed form.

• Right-censoring: Natural handling via the sufficient statistic (d, T) where d = number of exact observations and T = total time.

• rdata() method: For Monte Carlo validation and FIM estimation.

The log-likelihood is:

\ell(\lambda) = d \log \lambda - \lambda T

where d is the number of exact (uncensored) observations and T is the total observation time (sum of all times, whether censored or not).

Usage

exponential_lifetime(ob_col, censor_col = NULL)

Arguments

Value

An exponential_lifetime likelihood model object

Examples

# Uncensored exponential data
model <- exponential_lifetime("t")
df <- data.frame(t = rexp(100, rate = 2))
mle <- fit(model)(df)
coef(mle)  # should be close to 2

# Right-censored data
model_c <- exponential_lifetime("t", censor_col = "status")
df_c <- data.frame(
  t = c(rexp(80, 2), rep(0.5, 20)),
  status = c(rep("exact", 80), rep("right", 20))
)
mle_c <- fit(model_c)(df_c)
coef(mle_c)

Fisher information matrix method

Description

This function calculates the Fisher information matrix (FIM), an expectation over the data-generating process (DGP). The FIM is a crucial concept in statistics because it provides information about the precision of estimates and the amount of information that data carries about an unknown parameter.

Usage

fim(model, ...)

Arguments

Details

FIM is a function of the parameters, and is used to compute the standard errors of the parameters. It is also used to compute the covariance matrix of the parameters, which is in turn used to compute standard errors of the parameters.

Additionally, FIM is used to compute the Cramer-Rao lower bound (CRLB), the inverse of the FIM. CRLB represents the lower limit of the variance that an unbiased estimator can attain. This is used to compute the asymptotic relative efficiency (ARE) of an estimator of the parameters, which is the ratio of the variance of the estimator to the CRLB.

The function computes FIM(x)(theta), the FIM of the likelihood model x, is based on the following formulas:

FIM(x)(theta) = E[-loglik_hessian(x)(ob,theta)]
FIM(x)(theta) = E[score(x)(ob,theta) %*% t(score(x)(ob,theta))]

where the expectation is taken with respect to ob ~ DGP. The first formula is the expected hessian of the log-likelihood function, and the second formula is the expected outer product of the score function. The two formulas are equivalent.

Value

Function that computes the FIM given a parameter vector and sample size

Fisher information matrix for exponential_lifetime

Description

Returns the analytical FIM: n / lambda^2 (1x1 matrix).

Usage

## S3 method for class 'exponential_lifetime'
fim(model, ...)

Arguments

Value

A function(theta, n_obs, ...) computing the FIM

Default FIM method using Monte Carlo estimation

Description

Computes the Fisher information matrix by Monte Carlo simulation using the negative expected Hessian approach. For each of n_samples replicates, generates a single-observation dataset via rdata, computes -hess_loglik(single_obs, theta), and averages. The result is n_obs * I_1(theta) where I_1(theta) is the per-observation FIM.

Usage

## S3 method for class 'likelihood_model'
fim(model, ...)

Arguments

Details

This default requires the model to implement rdata and hess_loglik (or loglik, since hess_loglik falls back to numerical differentiation).

Value

Function that takes (theta, n_obs, n_samples, ...) and returns FIM matrix

Bootstrap MLE Estimate

Description

Creates a fisher_boot object representing bootstrap-based inference for maximum likelihood estimates.

Usage

fisher_boot(boot_result, original_mle)

Arguments

Value

An object of class c("fisher_boot", "fisher_mle", "mle_fit", "boot")

Maximum Likelihood Estimate (Fisherian)

Description

Creates a fisher_mle object representing a maximum likelihood estimate with methods for standard inference. This class emphasizes the Fisherian approach to likelihood-based inference.

Usage

fisher_mle(
  par,
  vcov = NULL,
  loglik_val,
  hessian = NULL,
  score_val = NULL,
  nobs = NULL,
  converged = TRUE,
  optim_result = NULL
)

Arguments

Value

An object of class c("fisher_mle", "mle_fit")

Fisherian Likelihood Inference

Description

Functions for pure likelihood-based inference in the Fisherian tradition. These emphasize the likelihood function itself as the primary object of inference, without requiring probability statements about parameters.

Fit a model

Description

Re-exported from generics. See generics::fit() for details.

Closed-form MLE for exponential_lifetime

Description

Computes the MLE directly as lambda_hat = d / T, bypassing optim. This demonstrates that specialized models can provide exact solutions.

Usage

## S3 method for class 'exponential_lifetime'
fit(object, ...)

Arguments

Value

A solver function that takes (df, par, ...) and returns a fisher_mle object. The par argument is accepted but ignored since the MLE is computed in closed form.

Default MLE solver for subclasses of likelihood_model.

Description

Note that likelihood_model is not a class, but a concept, that other likelihood models implement. They should add likelihood_model to their class definition, and then they can use this function to compute the MLE.

Usage

## S3 method for class 'likelihood_model'
fit(object, ...)

Arguments

Details

This function uses the optim function to find the MLE of the parameters of a likelihood model. It uses the loglik and score methods to compute the log-likelihood and score function, respectively.

There are a few interesting options for the control argument:

• method: The optimization method to use. The default is Nelder-Mead, which is a derivative-free method. Other options like include BFGS are gradient-based methods, which may be preferable if you provide a score function (rather than using the default finite-difference). There is also the SANN method, which is a simulated annealing method. This method is useful for multimodal likelihood functions, where the MLE may be sensitive to the initial guess. The SANN method is a more global method, but it is slower and may require some tweaking. Regardless, if you do use SANN, you should follow it up with a local search method like Nelder-Mead to refine the solution.

Value

An MLE solver (function) that returns an MLE object and accepts as arguments:

• df: The data frame

• par: The initial guess for the parameters

• control: Control parameters for the optimization algorithm

• ...: Additional arguments to pass into the likelihood model's constructed functions from loglik, score, and hess_loglik.

Hessian of log-likelihood method

Description

This function returns the hessian of the log-likelihood function of a model

Usage

hess_loglik(model, ...)

Arguments

Value

A function to compute the hessian of the log-likelihood given a data frame and parameters

Hessian of the log-likelihood for exponential_lifetime

Description

Returns a function computing the 1x1 Hessian: -d/lambda^2.

Usage

## S3 method for class 'exponential_lifetime'
hess_loglik(model, ...)

Arguments

Value

A function(df, par, ...) computing the Hessian matrix

Default method to compute the hessian of the log-likelihood.

Description

In case a hess_loglik method is not provided, this function will be used. It computes the hessian of the log-likelihood function using numerical differentiation.

Usage

## S3 method for class 'likelihood_model'
hess_loglik(model, control = list(), ...)

Arguments

Value

A function(df, par, ...) that computes the Hessian matrix of the log-likelihood evaluated at par given data df.

Check if an object is a likelihood model

Description

Tests whether an object satisfies the likelihood_model concept by checking if "likelihood_model" is in its class hierarchy. To be a likelihood model, at a minimum the object must implement loglik(). For optimal results, it may also provide score() and hess_loglik().

Usage

is_likelihood_model(x)

Arguments

Value

Logical indicating whether x is a likelihood model

Likelihood Interval

Description

Computes the likelihood interval for a parameter: LI(k) = {theta : R(theta) >= 1/k} = {theta : S(theta) >= -log(k)}

Usage

likelihood_interval(x, ...)

## S3 method for class 'fisher_mle'
likelihood_interval(x, data, model, k = 8, param = NULL, ...)

Arguments

Details

Unlike confidence intervals, likelihood intervals make no probability statements about the parameter. They simply identify the set of parameter values that are well-supported by the data.

Common choices for k:

• k = 8: 1/8 likelihood interval (~95% CI equivalent)

• k = 15: 1/15 likelihood interval (~99% CI equivalent)

• k = 32: 1/32 likelihood interval (~99.9% CI equivalent)

For multivariate parameters, specify param to get a profile likelihood interval for that parameter (profiling over the others).

Value

Matrix with lower and upper bounds for each parameter

Extract log-likelihood from fisher_mle object

Description

Extract log-likelihood from fisher_mle object

Usage

## S3 method for class 'fisher_mle'
logLik(object, ...)

Arguments

Value

A logLik object

Log-likelihood method

Description

This function returns the log-likelihood function of a model

Usage

loglik(model, ...)

Arguments

Value

A log-likelihood function to compute the log-likelihood given a data frame and parameters.

Log-likelihood for exponential_lifetime

Description

Returns a function computing ell(lambda) = d * log(lambda) - lambda * T.

Usage

## S3 method for class 'exponential_lifetime'
loglik(model, ...)

Arguments

Value

A function(df, par, ...) computing the log-likelihood

Default loglik method

Description

Provides a clear error when a model class does not implement loglik.

Usage

## S3 method for class 'likelihood_model'
loglik(model, ...)

Arguments

Value

Never returns; always errors.

Likelihood ratio test

Description

Computes the likelihood ratio test statistic and p-value for nested models.

Usage

lrt(null, alt, data, null_par, alt_par, dof = NULL, ...)

Arguments

Value

An lrt_result object with components stat, p.value, and dof

Mean squared error for fisher_mle

Description

Computes MSE = Var + Bias^2 (scalar) or Vcov + bias %*% t(bias) (matrix). Under regularity conditions, asymptotic bias is zero, so MSE equals the variance-covariance matrix. When model is provided, uses Monte Carlo bias estimation via bias.fisher_mle().

Usage

## S3 method for class 'fisher_mle'
mse(x, theta = NULL, ..., model = NULL, n_sim = 1000)

Arguments

Value

MSE matrix or scalar

Extract number of observations from fisher_mle object

Description

Extract number of observations from fisher_mle object

Usage

## S3 method for class 'fisher_mle'
nobs(object, ...)

Arguments

Value

Number of observations

Number of parameters in fisher_mle

Description

Number of parameters in fisher_mle

Usage

## S3 method for class 'fisher_mle'
nparams(x)

Arguments

Value

Integer count of parameters

Extract observed data from fisher_mle

Description

fisher_mle objects do not store observed data by design, so this always returns NULL.

Usage

## S3 method for class 'fisher_mle'
obs(x)

Arguments

Value

Always NULL. fisher_mle objects do not store the observed data.

Observed Fisher information matrix from fisher_mle

Description

Returns the negative Hessian of the log-likelihood evaluated at the MLE, which estimates the Fisher information matrix.

Usage

## S3 method for class 'fisher_mle'
observed_fim(x, ...)

Arguments

Value

A matrix, or NULL if the Hessian was not computed

Extract parameter estimates from fisher_mle

Description

Extract parameter estimates from fisher_mle

Usage

## S3 method for class 'fisher_mle'
params(x)

Arguments

Value

Named numeric vector of parameter estimates

Print fisher_boot object

Description

Print fisher_boot object

Usage

## S3 method for class 'fisher_boot'
print(x, ...)

Arguments

Value

The fisher_boot object, invisibly

Print fisher_mle object

Description

Print fisher_mle object

Usage

## S3 method for class 'fisher_mle'
print(x, ...)

Arguments

Value

The fisher_mle object, invisibly

Print likelihood interval

Description

Print likelihood interval

Usage

## S3 method for class 'likelihood_interval'
print(x, ...)

Arguments

Value

The likelihood_interval object, invisibly

Print method for likelihood models

Description

Print method for likelihood models

Usage

## S3 method for class 'likelihood_model'
print(x, show.loglik = FALSE, ...)

Arguments

Value

The likelihood model object, invisibly

Print method for likelihood ratio test results

Description

Print method for likelihood ratio test results

Usage

## S3 method for class 'lrt_result'
print(x, ...)

Arguments

Value

The lrt_result object, invisibly

Print profile log-likelihood

Description

Print profile log-likelihood

Usage

## S3 method for class 'profile_loglik'
print(x, ...)

Arguments

Value

The profile_loglik object, invisibly

Print summary of fisher_mle

Description

Print summary of fisher_mle

Usage

## S3 method for class 'summary_fisher_mle'
print(x, ...)

Arguments

Value

The summary_fisher_mle object, invisibly

Profile Log-Likelihood

Description

Computes the profile log-likelihood for a subset of parameters. For each value of the parameters of interest, the remaining (nuisance) parameters are optimized out.

Usage

profile_loglik(x, ...)

## S3 method for class 'fisher_mle'
profile_loglik(
  x,
  data,
  model,
  param,
  grid = NULL,
  n_grid = 50,
  range_mult = 4,
  ...
)

Arguments

Details

The profile likelihood is useful for:

• Visualizing the likelihood surface

• Computing likelihood intervals

• Eliminating nuisance parameters

Value

A data frame with parameter values and profile log-likelihood

Random data generation method

Description

Returns a function that generates random data from the model's data-generating process (DGP) at a given parameter value.

Usage

rdata(model, ...)

Arguments

Details

This is used by the default fim method for Monte Carlo estimation of the Fisher information matrix.

Value

Function that takes (theta, n, ...) and returns a data frame

Random data generation for exponential_lifetime

Description

Generates exponential data, optionally with right-censoring at a fixed censoring time.

Usage

## S3 method for class 'exponential_lifetime'
rdata(model, ...)

Arguments

Value

A function(theta, n, censor_time, ...) that returns a data frame

Default rdata method

Description

Provides a clear error when a model class does not implement rdata. The rdata method is required by the default fim.likelihood_model() for Monte Carlo FIM estimation.

Usage

## S3 method for class 'likelihood_model'
rdata(model, ...)

Arguments

Value

Never returns; always errors.

Relative Likelihood

Description

Computes the relative likelihood (likelihood ratio) for theta: R(theta) = L(theta) / L(theta_hat) = exp(S(theta))

Usage

relative_likelihood(x, ...)

## S3 method for class 'fisher_mle'
relative_likelihood(x, theta, data, model, ...)

Arguments

Details

The relative likelihood is always between 0 and 1, with maximum 1 at the MLE. Common cutoff values:

• R >= 0.15 (k=8): roughly equivalent to 95% confidence

• R >= 0.10 (k=10): more conservative

• R >= 0.05 (k=20): very conservative

Value

Relative likelihood value(s): L(theta)/L(theta_hat)

Bootstrap sampler for fisher_boot

Description

Returns a function that resamples from the bootstrap distribution.

Usage

## S3 method for class 'fisher_boot'
sampler(x, ...)

Arguments

Value

Function that takes n and returns n x p matrix of resampled estimates

Asymptotic sampler for fisher_mle

Description

Returns a function that samples from the asymptotic normal distribution of the MLE: N(theta_hat, vcov).

Usage

## S3 method for class 'fisher_mle'
sampler(x, ...)

Arguments

Value

Function that takes n and returns n x p matrix of samples

Estimate the sampling distribution of the MLE for a likelihood model.

Description

We use the bootstrap method. In other words, we treat the data as an empirical distribution and sample from it to get a new dataset, then we fit the model to that dataset and return the MLE. We do this R times and return the R MLEs.

Usage

## S3 method for class 'likelihood_model'
sampler(x, df, par, ..., nthreads = 1L)

Arguments

Details

This is the default method, but if you want to use a different method, you should define your own method for your likelihood model.

Value

A function that returns a bootstrapped sampling distribution of an MLE (fisher_boot object).

Score method

Description

This function returns the score function of a model

Usage

score(model, ...)

Arguments

Value

A function to compute the score given a data frame and parameters

Score for exponential_lifetime

Description

Returns a function computing d/lambda - T.

Usage

## S3 method for class 'exponential_lifetime'
score(model, ...)

Arguments

Value

A function(df, par, ...) computing the score vector

Default score method

Description

In case a score method is not provided, this function will be used. It computes the score by numerical differentiation of the log-likelihood.

Usage

## S3 method for class 'likelihood_model'
score(model, control = list(), ...)

Arguments

Value

A function to compute the score given a data frame and parameters

Extract score vector from fisher_mle

Description

Extract score vector from fisher_mle

Usage

## S3 method for class 'fisher_mle'
score_val(x, ...)

Arguments

Value

The score vector at the MLE (should be near zero)

Extract standard errors from fisher_mle

Description

Computes standard errors as the square root of the diagonal of the variance-covariance matrix.

Usage

## S3 method for class 'fisher_mle'
se(x, ...)

Arguments

Value

Numeric vector of standard errors, or NA values if vcov is NULL

Summarize fisher_mle object

Description

Summarize fisher_mle object

Usage

## S3 method for class 'fisher_mle'
summary(object, ...)

Arguments

Value

A summary_fisher_mle object

Support Function (Log Relative Likelihood)

Description

Computes the support for parameter value theta relative to the MLE: S(theta) = logL(theta) - logL(theta_hat)

Usage

support(x, ...)

## S3 method for class 'fisher_mle'
support(x, theta, data, model, ...)

Arguments

Details

The support function is always <= 0, with maximum at the MLE. Values of theta with support > -2 are considered well-supported. Values with support > -log(8) ~ -2.08 correspond roughly to a 95% likelihood interval.

Value

Support value(s): logL(theta) - logL(theta_hat)

Extract variance-covariance matrix from fisher_mle object

Description

Extract variance-covariance matrix from fisher_mle object

Usage

## S3 method for class 'fisher_mle'
vcov(object, ...)

Arguments

Value

Variance-covariance matrix