Distribution-Inference

Reproducible distribution fitting, model discrimination, and likelihood-based uncertainty quantification across Python, R, MATLAB, and Fortran.

Distribution-Inference is research software for inferring a probability model from one or more observed samples. It fits domain-compatible candidate distributions by maximum likelihood, calibrates goodness-of-fit through a refitted parametric Bootstrap, ranks statistically admissible models, and quantifies parameter uncertainty with both profile-likelihood and Wald confidence intervals. Four independent language implementations execute one canonical statistical specification and emit a shared result schema.

Figure 1. Statistical workflow implemented independently by all four computational backends.

Methodological scope

For observations $x_1,\ldots,x_n$ and a candidate density or probability mass function $f(x\mid\theta)$, parameters are estimated by

$$ \hat\theta=\arg\max_\theta \ell(\theta), \qquad \ell(\theta)=\sum_{i=1}^{n}\log f(x_i\mid\theta). $$

Each successful fit reports K-S, AIC, small-sample corrected AIC, and BIC:

$$ \mathrm{AIC}=2k-2\ell(\hat\theta),\qquad \mathrm{BIC}=k\log n-2\ell(\hat\theta). $$

Because the model parameters are estimated from the tested sample, the K-S p-value is calibrated by parametric Bootstrap with parameter re-estimation in every replicate. Continuous and discrete samples are analyzed separately; the discrete statistic is evaluated on both sides of each probability-mass jump. The recommended model is the minimum-BIC candidate among those not rejected at the configured K-S level. If every model is rejected, the lowest-BIC result is reported only as a diagnostic fallback.

Parameter uncertainty is evaluated at 90%, 95%, and 99% by default:

• Profile-likelihood LR: nuisance parameters are re-optimized for every fixed value of the parameter of interest.
• Wald: the observed information matrix is inverted on an unconstrained parameter scale and transformed back to the canonical scale.

Supported probability models

Data class	Distribution	Canonical parameters
Continuous, real	Normal	mu, sigma
Continuous, positive	Lognormal	mu_log, sigma_log
Continuous, nonnegative	Exponential	rate
Continuous, positive	Gamma	shape, scale
Continuous, nonnegative	Weibull	shape, scale
Continuous, bounded	Beta	alpha, beta, known bounds
Continuous, real	Gumbel maximum	loc, scale
Continuous, constrained support	GEV	loc, scale, xi
Discrete, nonnegative integer	Poisson	lambda

The GEV convention is

$$ F(x)=\exp\left[-\left(1+\xi\frac{x-\mathrm{loc}}{\mathrm{scale}}\right)^{-1/\xi}\right], $$

with the Gumbel limit at $\xi\rightarrow0$. The canonical convention takes precedence over language-library aliases or sign conventions.

Cross-language benchmark

Figure 2. Model evidence, cross-language numerical spread, confidence intervals, and the selected-model fit for the bundled synthetic benchmark. The figure is generated from machine-readable outputs by scripts/generate_readme_figures.py.

On the bundled data, every implementation selects the same model:

Data set	Selected model	Scientific role
normal_sample	Normal	real-valued symmetric observations
positive_sample	Lognormal	positive right-skewed observations
beta_sample	Beta	bounded observations on (0,1)
count_sample	Poisson	nonnegative event counts

Closed-form model likelihoods agree to numerical precision across the four backends. Iterative Beta, Gumbel, Gamma, Weibull, and GEV fits are compared with model-specific tolerances documented in the statistical specification.

Reproducible installation

Create the project environment from the repository root:

conda env create -f environment.yml
conda activate distribution-inference

The environment contains both Python and R dependencies. MATLAB and gfortran are detected but are not installed automatically. If Python and R already reside in separate environments, pass -PythonEnvironment and -REnvironment on PowerShell, or --python-env and --r-env on Shell.

Running an analysis

Windows PowerShell:

.\run_all.ps1 -Input examples\samples.csv `
  -DatasetConfig examples\datasets.csv `
  -Output results\analysis

Linux or a research server:

./run_all.sh \
  --input examples/samples.csv \
  --dataset-config examples/datasets.csv \
  --output results/analysis

All repository paths remain relative to the project root. Production runs use 1000 Bootstrap replicates; use --bootstrap 99 or -Bootstrap 99 for development.

Input and output contract

The sample CSV contains one independent data set per numeric column. Blank cells are omitted column-wise. The optional data-set configuration defines:

dataset,data_type,beta_lower,beta_upper,distributions
beta_sample,continuous,0,1,beta;normal;gumbel

data_type accepts auto, continuous, or discrete. Automatic classification treats an entirely nonnegative integer sample as discrete; override it for rounded continuous measurements.

Each backend writes the same files below its result directory:

• summary.json: settings, selection status, and data-set metadata.
• candidates.csv: convergence, likelihood, K-S, AIC/AICc/BIC, and selection status.
• parameters.csv: canonical MLEs and selected-model standard errors.
• confidence_intervals.csv: LR and Wald limits with explicit status fields.

The shared reporting layer adds comparison.csv, comparison_differences.csv, report.md, fitted-density plots, and Q-Q plots.

Repository organization

src/python/    Python reference implementation and reporting
src/r/         Independent R implementation
src/matlab/    Independent MATLAB implementation
src/fortran/   Self-contained Fortran 2008 numerical implementation
spec/          Canonical parameterization and inferential rules
examples/      Synthetic benchmark samples and data-set configuration
tests/         Deterministic unit and command-line tests
scripts/       Setup, compilation, validation, and figure generation

Verification and development

pytest
./scripts/build_fortran.sh debug
./scripts/smoke_test.sh
python scripts/generate_readme_figures.py

PowerShell equivalents are provided for setup, Fortran compilation, and smoke testing. Generated results and compiler products are ignored by version control.

Statistical limitations

• Bootstrap precision is finite and depends on the requested number of replicates; interpret p-values near the selection threshold cautiously.
• Beta support bounds are treated as known, not estimated from sample minima and maxima.
• Profile intervals can be unbounded or numerically unavailable near support boundaries; such cases are reported rather than silently replaced.
• Automatic model selection is conditional on the supplied candidate set and does not establish that the data-generating mechanism is uniquely identified.
• The software supports reproducible research workflows, but domain-specific assumptions and study design remain the analyst's responsibility.