MorphZ

MorphZ estimates the Bayesian marginal likelihood, or evidence, from posterior samples. It is a post-processing tool: it does not run its own sampler, and it places no requirements on which sampler produced the posterior draws. The same workflow can be used with nested sampling, parallel-tempered MCMC, HMC, NUTS, or any other sampler that provides posterior samples.

Given posterior samples, the prior, and the likelihood, MorphZ returns the log-evidence estimate together with a bridge-sampling relative-error diagnostic. The target integral is

z(X \mid \mathcal{M}) = \int_{\Theta}
  L(X \mid \boldsymbol{\theta}, \mathcal{M})
  \pi(\boldsymbol{\theta} \mid \mathcal{M})
  d\boldsymbol{\theta}.

How It Works

MorphZ has two main components.

Morph Approximation

MorphZ first builds a tractable approximation to the posterior, called the Morph approximation, by factorising the full posterior into a product of low-dimensional joint factors over disjoint blocks of parameters, plus marginals over leftover singleton parameters:

\mathcal{M}_{B_L}(\boldsymbol{\theta}) =
  \prod_{b \in B_L} P_b(\boldsymbol{\theta}_b)
  \prod_{s \in S(B_L)} P_s(\theta_s).

Block membership is chosen by maximising the sum of block total correlations, an information-theoretic measure of joint dependence inside each block. This selects the product-of-blocks approximation that is closest to the posterior at the chosen order. The optimisation is solved with Seeded Greedy Maximisation (SGM), and block and singleton densities are fit with multivariate Gaussian-kernel KDEs.

Optimal Bridge Sampling

MorphZ then uses the Morph approximation as the proposal distribution inside an optimal bridge-sampling estimator. To avoid reuse bias, it splits the supplied posterior samples into one batch for fitting the Morph approximation and a separate batch for bridge sampling.

For each call, MorphZ produces:

• logz, the log-evidence estimate.
• A bridge-sampling relative mean-squared error diagnostic, computed from the same posterior and proposal samples.

Features

• Morph approximation object.
• Total correlation computation for higher-order correlations.
• Multiprocessing support for non-vectorized likelihoods.
• Mutual information plot.

Installation

Python 3.10+ is recommended.

pip install morphz

From source (editable):

pip install -e .

Core Usage

To estimate log_z, provide posterior samples and a log-posterior function. If you already have log-posterior values for those same samples, pass them as log_posterior_values to avoid recomputing them.

import morphZ

log_z= morphZ.evidence(
    samples,
    log_posterior_function=lp_fn,
    log_posterior_values=log_prob,
    n_resamples=5000,
    n_estimations=1,
    morph_type="2_group",
    output_path="./example/",
)

Run The Examples

Interactive notebooks live in examples/:

• examples/eggbox.ipynb — eggbox likelihood (dynesty nested sampling)
• examples/gaussian shell.ipynb — Gaussian shell (dynesty nested sampling)
• examples/peak_sampling.ipynb — sharply peaked posterior
• examples/numpyro_gaussian_shell.ipynb — Gaussian shell with NUTS via NumPyro
• examples/jaxns_gaussian_shell.ipynb — Gaussian shell with nested sampling via JAXNS
• examples/numpyro_morphz_lnz.ipynb — log-evidence comparison across morph types (NumPyro/NUTS)

Citation

If you use MorphZ, please cite:

@article{1554-y6ns,
  title   = {Enhancing evidence estimation through informed probability density approximation},
  author  = {Zahraoui, El Mehdi and Maturana-Russel, Patricio and Vajpeyi, Avi
             and van Straten, Willem and Meyer, Renate and Gulyaev, Sergei},
  journal = {Phys. Rev. D},
  volume  = {113},
  issue   = {8},
  pages   = {083014},
  year    = {2026},
  month   = {Apr},
  publisher = {American Physical Society},
  doi     = {10.1103/1554-y6ns},
  url     = {https://link.aps.org/doi/10.1103/1554-y6ns}
}

API Highlights

• Morphs: Morph_Indep, Morph_Pairwise, Morph_Group, Morph_Tree.
• Bandwidths: select_bandwidth, compute_and_save_bandwidths.
• Evidence: evidence, bridge_sampling_ln (lower‑level), compute_bridge_rmse.
• Total correlation: Nth_TC.compute_and_save_tc.

Notes:

• If you pass a numeric kde_bw (e.g., 0.9) the library skips bandwidth JSONs.
• pair/group proposals will compute and cache MI.json/params_*_TC.json on first use.

Dependencies

• Core: numpy, scipy, matplotlib, corner, networkx, emcee, statsmodels, scikit-learn
• Optional: pandas (CSV labels), pygraphviz (nicer tree layout), scikit-sparse (optional exception type)

Development

• Build wheels/sdist: python -m build
• Check metadata: twine check dist/*
• Tests live in tests/

Versioning & Release

Versioning is derived from git tags via setuptools_scm.

• Tag a release: git tag vX.Y.Z && git push --tags
• CI: publishes to TestPyPI on pushes to main/master; to PyPI on v* tags.
• Uses PyPI/TestPyPI Trusted Publishing (OIDC). You can also use API tokens if preferred.

License

BSD-3-Clause. See LICENSE for details.