BAT to Python (batty)

A small python interface to the Bayesian Analysis Toolkit (BAT.jl) https://github.com/bat/BAT.jl

• Please check out the minimal example to get started below
• To understand how to define a prior + likelihood, please read this
• For experimental support of gradients, see this

Quick Start

Installation

There are two parts to an installation, one concerning the python side, and one the julia side:

• Python: pip install batty

• Julia: import Pkg; Pkg.add.(["PyJulia", "DensityInterface", "Distributions", "ValueShapes", "TypedTables", "ArraysOfArrays", "ChainRulesCore", "BAT"])

Minimal Example

The code below is showing a minimal example:

• using a gaussian likelihood and a uniform prior
• generating samples via Metropolis-Hastings
• plotting the resulting sampes
• estimating the integral value via BridgeSampling

%load_ext autoreload
%autoreload 2

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from batty import BAT_sampler, BAT, Distributions, jl

sampler = BAT_sampler(llh=lambda x : -0.5 * x**2, prior_specs=Distributions.Uniform(-3, 3))
sampler.sample();
sampler.corner();

Usage

Using Different Algotihms

There are a range of algorihtms available within BAT, and those can be further customized via arguments. Here are just a few examples:

MCMC Sampling:

results = {}

• Metropolis-Hastings:

results['Metropolis-Hastings'] = sampler.sample(strategy=BAT.MCMCSampling(nsteps=10_000, nchains=2))

• Metropolis-Hastings with Accept-Reject weighting:

results['Accept-Reject Weighting'] = sampler.sample(strategy=BAT.MCMCSampling(mcalg=BAT.MetropolisHastings(weighting=BAT.ARPWeighting()), nsteps=10_000, nchains=2))

• Prior Importance Sampling:

results['Prior Importance Sampling'] = sampler.sample(strategy=BAT.PriorImportanceSampler(nsamples=10_000))

• Sobol Sampler:

results['Sobol Quasi Random Numbers'] = sampler.sample(strategy=BAT.SobolSampler(nsamples=10_000))

• Grid Sampler:

results['Grid Points'] = sampler.sample(strategy=BAT.GridSampler(ppa=1000))

Plotting the different results:

fig = plt.figure(figsize=(9,6))
bins=np.linspace(-3, 3, 100)
for key, item in results.items():
    plt.hist(item.v, weights=item.weight, bins=bins, density=True, histtype="step", label=key);
plt.legend()

<matplotlib.legend.Legend at 0x7f069fcd5ee0>

Specifying Priors and Likelihoods

Priors are specified via Julia Distributions, multiple Dimensions can be defined via a dict, where the key is the dimension name and the value the distribution, or as a list in case flat vectors with paraeter names are used.

Below the example with parameter names

s = np.array([[0.25, 0.4], [0.9, 0.75]])
prior_specs = {'a' : Distributions.Uniform(-3,3), 'b' : Distributions.MvNormal(np.array([1.,1.]), jl.Array(s@s.T))}

The log-likelihood (llh) can be any python callable, that returns the log-likelihood values. The first argument to the function is the object with paramneter values, here x. If the prior is simple (i.e. like in the example in the beginning, x is directly the parameter value). If the prior is specified via a dict, then x contains a field per parameter with the value. Any additional args to the llh can be given in the sampler, such as here d for data:

def llh(x, d):
    return -0.5 * ((x.b[0] - d[0])**2 + (x.b[1] - d[1])**2/4) - x.a

Or alternatively without parameter names (this will result in a flat vector):

# prior_specs = [Distributions.Uniform(-3,3), Distributions.MvNormal(np.array([1.,1.]), jl.Array(s@s.T))]
# def llh(x, d):
#     return -0.5 * ((x[1] - d[0])**2 + (x[2] - d[1])**2/4) - x[0]

d = [-1, 1]

sampler = BAT_sampler(llh=llh, prior_specs=prior_specs, llh_args=(d,))

Let us generate a few samples:

sampler.sample(strategy=BAT.MCMCSampling(nsteps=10_000, nchains=2));

Some interface to plotting tools are available

• The Great Triangular Confusion (GTC) plot:

sampler.gtc(figureSize=8, customLabelFont={'size':14}, customTickFont={'size':10});

findfont: Font family ['Arial'] not found. Falling back to DejaVu Sans.
findfont: Font family ['Arial'] not found. Falling back to DejaVu Sans.

• The corner plot:

sampler.corner(color='green');

HMC with Gradients

This at the moment only works with preiors defined as flat vectors!

llh = lambda x : -0.5 * np.dot(x, x)
grad = lambda x : -x
sampler = BAT_sampler(llh=llh, prior_specs=[Distributions.Uniform(-3, 3),], grad=grad, )

# Or alternatively:
# llh_and_grad = lambda x: (-0.5 * np.dot(x, x), -x)
# sampler = BAT_sampler(llh=llh, prior_specs=[Distributions.Uniform(-3, 3),], llh_and_grad=llh_and_grad)

sampler.sample(strategy=BAT.MCMCSampling(mcalg=BAT.HamiltonianMC()));
sampler.corner();