# 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
# Load your data and set parameters.
data = np.load(<path to your data.npy>)
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[0] = 'Estimate of Bayesian evidence (float).'
BayEvi[1] = '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['Discard'] = 'Set of discarded samples (shape: Disc, dim, dim).'
XTrace['DiscardL'] = 'Likelihood of discarded samples (shape: Disc).'
XTrace['DiscardW'] = 'Weights of discarded samples (shape: Disc).'
XTrace['DiscardPostProb'] = 'Posterior probability of discarded samples (shape: Disc)'
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}
}
```