Multi-Null Non-Parametric (MN²) Jensen–Shannon Distance (JSd) Hypothesis Test (MN-squared test)

mn-squared implements a multi-null non-parametric (MN²) Jensen–Shannon distance (JSd) based hypothesis test for multinomial data. Given multiple candidate null distributions ($\mathbf{p}_{\ell}$) and observed histograms ($\mathbf{h}$), it computes p-values using an exact or Monte-Carlo CDF backend and returns decisions that control per-null significance levels and the overall family-wise error rate (FWER).

Status: Stable (v1.0) — public API is stable.

Installation

pip install mn-squared

Python ≥ 3.10, NumPy ≥ 1.24.

Quickstart

from mn_squared import MNSquaredTest

test = MNSquaredTest(evidence_size=100, prob_dim=3, cdf_method="mc_multinomial", mc_samples=50_000, seed=1234)
test.add_nulls([0.5, 0.3, 0.2], target_alpha=0.05)
test.add_nulls([0.4, 0.4, 0.2], target_alpha=0.01)

histograms = [[55, 22, 23], [40, 39, 21], [0, 3, 97]]
p_vals = test.infer_p_values(histograms)
decisions = test.infer_decisions(histograms)

print("p-values:", p_vals)
print("decisions:", decisions)

Concepts

• JSd statistic: divergence between empirical histogram ($\mathbf{h}/n$) and a reference ($\mathbf{p}$).
• Multi-null setting: several candidate vectors$. A query, $\mathbf{h}$, is either assigned to the “least-rejected” null or rejects all
• Backends:
  • exact: enumerate histograms in $\Delta'_{k,n}$ to compute the empirical CDF (ECDF); complexity $O(n^{k-1})$ for fixed $k$.
  • mc_multinomial: draw $\mathbf{H}\sim \text{Multinomial}(n, \mathbf{p})$; ECDF converges by strong law of large numbers (SLLN).
  • mc_normal: CLT proxy with $\mathcal{N}(n\mathbf{p}, n(\mathrm{diag}(\mathbf{p})-\mathbf{p}\mathbf{p}^\top))$.
• Error control: per-null $\alpha$; overall family-wise error rate (FWER); worst-case ($\beta$) at a query $\mathbf{q}$.

Public API

from mn_squared import MNSquaredTest, available_cdf_backends

Advanced imports:

mn_squared.cdf_backends     # ExactCDFBackend, NormalMCCDFBackend, MultinomialMCCDFBackend
mn_squared.null_structures  # IndexedHypotheses, NullHypothesis

Performance & numerics

• Exact backend is intended for small-to-moderate ((n,k)) due to combinatorial growth.
• Monte-Carlo backends scale well and are recommended for larger regimes.
• Validations use a small floating tolerance for simplex / integer-like checks.
• MC results are deterministic under a fixed seed.

Project layout

mn_squared/
  cdf_backends/        # CDF backends (exact, MC multinomial, MC normal)
  null_structures/     # NullHypothesis & IndexedHypotheses containers
  _validators.py       # Shared validation helpers (implemented)
  core.py              # MNSquaredTest orchestrator (stub)
tests/                 # Unit + property tests and backend contract tests
docs/                  # Documentation (Sphinx)
experiments/           # Optional: benchmarking / comparison scripts (if present)

Versioning & license

• Versioning: SemVer.
• License: Apache-2.0.

Citation

There is an associated preprint describing the methodology being written up. In the meantime, if you use this project in research, please cite:

@software{mn_squared,
  title = {mn-squared: Multi-Null Non-Parametric Jensen–Shannon Distance Hypothesis Test in Python},
  author = {ALGES},
  year = {2026},
  url = {https://github.com/alges/mn-squared}
}