# ProxNest: Proximal nested sampling for high-dimensional Bayesian model selection

`ProxNest` is an open source, well tested and documented Python implementation of the proximal nested sampling algorithm (Cai et al. 2022) which is uniquely suited for sampling from very high-dimensional posteriors that are log-concave and potentially not smooth (e.g. Laplace priors). This is achieved by exploiting tools from proximal calculus and Moreau-Yosida regularisation (Moreau 1962) to efficiently sample from the prior subject to the hard likelihood constraint. The resulting Markov chain iterations include a gradient step, approximating (with arbitrary precision) an overdamped Langevin SDE that can scale to very high-dimensional applications.

# Basic Usage

The following is a straightforward example application to image denoising (Phi = I), regularised with Daubechies wavelets (DB6).

```# Import relevant modules.
import numpy as np
import ProxNest

params = params    # Parameters of the prior resampling optimisation problem.
options = options  # Options associated with the sampling strategy.

# Construct your forward model (phi) and wavelet operators (psi).
phi = ProxNest.operators.sensing_operators.Identity()
psi = ProxNest.operators.wavelet_operators.db_wavelets(["db6"], 2, (dim, dim))

# Define proximal operators for both your likelihood and prior.
proxH = lambda x, T : ProxNest.operators.proximal_operators.l1_projection(x, T, delta, Psi=psi)
proxB = lambda x, tau: ProxNest.optimisations.l2_ball_proj.sopt_fast_proj_B2(x, tau, params)

# Write a lambda function to evaluate your likelihood term (here a Gaussian)
LogLikeliL = lambda sol : - np.linalg.norm(y-phi.dir_op(sol), 'fro')**2/(2*sigma**2)

# Perform proximal nested sampling
BayEvi, XTrace = ProxNest.sampling.proximal_nested.ProxNestedSampling(
np.abs(phi.adj_op(data)), LogLikeliL, proxH, proxB, params, options
)
```

At this point you have recovered the tuple BayEvi and dict Xtrace which contain

```Live = options["samplesL"] # Number of live samples
Disc = options["samplesD"] # Number of discarded samples

# BayEvi is a tuple containing two values:
BayEvi = 'Estimate of Bayesian evidence (float).'
BayEvi = 'Variance of Bayesian evidence estimate (float).'

# XTrace is a dictionary containing the np.ndarrays:
XTrace['Liveset'] = 'Set of live samples (shape: Live, dim, dim).'
XTrace['LivesetL'] = 'Likelihood of live samples (shape: Live).'

XTrace['DiscardPostMean'] = 'Posterior mean solution (shape: dim, dim)'
```

from which one can perform e.g. Bayesian model comparison.

# Referencing

A BibTeX entry for `ProxNest` is:

```@article{Cai:ProxNest:2021,
author = {Cai, Xiaohao and McEwen, Jason~D. and Pereyra, Marcelo},
title = {"High-dimensional Bayesian model selection by proximal nested sampling"},
journal = {ArXiv},
eprint = {arXiv:2106.03646},
year = {2021}
}
```