degen

two models the data can't tell apart

Detection of observational equivalence and non-identifiability in parametric models, from their log-likelihood surfaces.

Hand it your model specifications. degen tells you which ones the data can actually distinguish: it compares likelihood surfaces to find observationally equivalent models, reads the Fisher information eigenspectrum to find parameters the likelihood is flat in, and groups models into equivalence classes by union-find. Two parameterizations that collapse to the same distribution will fit equally well forever, and the usual diagnostics will not say so.

library(degen)

# are these two models telling the data apart, or just telling stories?
pair   <- equivalence_pair(model_spec_exponential(),
                           model_spec_gamma(parameterization = "shape_rate"))
result <- compare_surfaces(pair, y = rexp(200, rate = 2))
result

# is a single model's likelihood flat in some direction?
info <- fisher_information(model_spec_gamma(parameterization = "shape_rate"),
                           y = rgamma(200, 2, 3), par = c(shape = 2, rate = 3))
identifiability_check(info)

The question convergence checks skip

Standard diagnostics ask whether a fit converged, predicts, and leaves clean residuals. None of them ask whether the model structure was distinguishable from its alternatives in the first place. When two specifications encode the same distribution over observable data, no amount of data separates them; when a parameter sits in a flat direction of the likelihood, it is unestimable. Both show up downstream as convergence failures, ridge-like surfaces, or answers that flip between software packages, and diagnosing them by hand means comparing likelihood functions term by term.

degen does that comparison for you, and it can do it two ways: numerically, by searching the likelihood surfaces for a separating point, and symbolically, by checking exact parameter equivalence and detecting linear reparameterizations. The numerical verdict is evidence (no counterexample found in the searched region); the symbolic verdict is a proof of equivalence when it succeeds.

What it works on

• model_spec(): define a model from any log-likelihood function, with named parameters and bounds. Ten built-ins are ready (model_spec_normal(), model_spec_gamma(), model_spec_poisson(), model_spec_weibull(), and the rest), and model_spec_from_fit() pulls a spec straight out of a fitted lm or glm.
• compare_surfaces() / equivalence_pair(): test two models for observational equivalence, by grid search or optimization, in both directions.
• check_exact_equivalence() / detect_linear_reparam(): the symbolic path, for when you want a proof rather than evidence.
• equivalence_classes() / compare_all_pairs(): group a list of models into empirical equivalence classes (union-find), or build the full pairwise comparison matrix.
• fisher_information() / identifiability_check() / null_directions(): observed or expected information, eigenvalue decomposition, and the parameter combinations the data cannot pin down.
• profile_likelihood() / profile_ci(): profile curves and the confidence intervals read off them.
• diagnose_model(): the single-call report covering all of the above.

Numerical or symbolic?

	compare_surfaces()	check_exact_equivalence()
Needs data?	Yes	No
Returns	Evidence (no counterexample in searched region)	Proof of equivalence when it succeeds
Handles arbitrary log-likelihoods?	Yes	Within the supported algebra
Best for	Any pair of specifications	Suspected exact reparameterizations

Run compare_surfaces() to see whether the data can tell two models apart; reach for the symbolic path when you suspect they are exactly the same model wearing different parameters.

Custom log-likelihoods

Any function that maps data and named parameters to a scalar log-likelihood is a model:

# a model degen has never seen, defined in three lines
spec <- model_spec(
  loglik_fn  = function(y, mu, sigma) sum(dnorm(y, mu, sigma, log = TRUE)),
  par_names  = c("mu", "sigma"),
  par_bounds = list(sigma = c(1e-6, Inf))
)

info <- fisher_information(spec, y = rnorm(200, 1, 2), par = c(mu = 1, sigma = 2))
identifiability_check(info)

This is the case degen is built for: mixture models with label switching, mechanistic models with competing functional forms, hierarchical models whose variance components trade off. Wherever model structure carries meaning, it tells you how much of that structure the data actually support.

Robustness and Bayesian models

An equivalence verdict should not hinge on a tolerance setting. sensitivity_analysis() sweeps tolerances and grid sizes, bootstrap_equivalence() and cross_validate_equivalence() resample the test, and prior_sensitivity() checks how far a prior is doing the identifying. For fitted Bayesian models, compare_posteriors() and extract_samples() work on MCMC draws from brms, rstan, and rstanarm. Results export through export_results() (JSON, CSV, markdown) and to_latex().

Installation

install.packages("pak")                   # development version
pak::pak("gcol33/degen")

Documentation

• Introduction
• Model Specification
• Comparing Models
• Identifiability Diagnostics
• Case Studies
• Function Reference

Support

"Software is like sex: it's better when it's free." — Linus Torvalds

I'm a PhD student who builds R packages in my free time because I believe good tools should be free and open. I started these projects for my own work and figured others might find them useful too.

If this package saved you some time, buying me a coffee is a nice way to say thanks. It helps with my coffee addiction.

License

MIT (see the LICENSE.md file)

Citation

@software{degen,
  author = {Colling, Gilles},
  title = {degen: Detect Observational Equivalence and Non-Identifiability in Parametric Models},
  year = {2025},
  url = {https://github.com/gcol33/degen}
}