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PyMultiNest usage example
• for the Bayesian Inference in Subatomic Physics - A Marcus Wallenberg Symposium
https://www.chalmers.se/en/conference/Bayesian%20Inference%20in%20Subatomic%20
Physics/Pages/default.aspx
• by Johannes Buchner https://astrost.at/istics/, Max Planck Institute for extraterrestrial
Physics
• PyMultiNest GitHub repository with installation instructions
https://github.com/JohannesBuchner/PyMultiNest/
This example fits a broken powerlaw model.
1. generate data to work with.
2. define the model prior and likelihood functions, run with PyMultiNest
3. look at the posterior distribution
4. define an alternative model, look at the Bayes factor
5. measure how often such an extreme Bayes factor could arise from the alternative
(simpler) model by chance. (Get frequentist properties of Bayesian inference)
In [1]:
import numpy
import matplotlib.pyplot as plt
import json
import time
import scipy.stats
%matplotlib inline
prefix = 'hump_'
Generate data $$y \sim Poisson(N \times (x/x_\mathrm{brk})^k)$$
where k = 0 below x_brk, -1 above
In [2]:
numpy.random.seed(1)
x = numpy.linspace(1, 10, 40)
klo = 0
khi = -1
N = 10
xbrk = 4
ypred = N * numpy.where(x < xbrk, (x / xbrk)**klo, (x / xbrk)**khi)
y = numpy.random.poisson(ypred)
plt.plot(x, y, 'x ')
plt.plot(x, ypred)
#plt.show()
Out[2]: [<matplotlib.lines.Line2D at 0x7f3dcd6e4278>]
Specify model
In [3]:
parameters = ['N', 'xbrk', 'klo', 'khi']
ndim = len(parameters)
def prior_transform(cube):
params = cube.copy()
# N from 1e-2 to 1e2
params[0] = 10**(cube[0]*4 - 2)
# break energy 0...10
params[1] = cube[1]*10
# slopes: -2 to +2
params[2] = cube[2]*4 - 2
params[3] = cube[3]*4 - 2
return params
def model(params):
N, xbrk, klo, khi = params
return N * numpy.where(x < xbrk, (x / xbrk)**klo, (x / xbrk)**khi)
def loglike(params):
ypred = model(params)
loglike = scipy.stats.poisson.logpmf(y, ypred).sum()
if not numpy.isfinite(loglike):
loglike = -1e300
return loglike
Run analysis
In [4]:
from pymultinest.solve import solve
results = solve(LogLikelihood=loglike, Prior=prior_transform, n_dims=ndim,
n_live_points=1000,
outputfiles_basename=prefix, verbose=True, resume=True)
json.dump(parameters, open(prefix + 'params.json', 'w'))
print(results)
analysing data from hump_.txt
{'logZ': -102.74988182314499, 'logZerr': 0.07983781047465993, 'samples': array([[27
.54771054, 0.53811791, -1.36357521, -0.45436048],
[ 4.28315879, 6.39788903, -0.89087396, 0.84383966],
[ 6.59094115, 8.77070279, -0.27721136, 1.15676164],
...,
[ 8.81956935, 4.15645429, -0.56252224, -0.45793781],
[ 6.93849229, 5.16302085, -0.61798903, -0.38831301],
[ 7.74762965, 4.66945465, -0.67677528, -0.51356818]])}
Plot output
In [5]:
import corner
corner.corner(results['samples'], labels=parameters)
plt.show()
WARNING:root:Too few points to create valid contours
In [6]:
ypred = N * numpy.where(x < xbrk, (x / xbrk)**klo, (x / xbrk)**khi)
y = numpy.random.poisson(ypred)
plt.plot(x, y, 'x ')
plt.plot(x, ypred)
samples = results['samples']
for posterior_sample in samples[numpy.random.choice(len(samples), size=40)]:
plt.plot(x, model(posterior_sample), color='k', alpha=0.2)
Alternative model
In [7]:
def prior_transform2(cube):
params = cube * 100
# N from 1e-2 to 1e2
params[0] = 10**(cube[0]*4 - 2)
# slopes: -2 to +2
params[1] = cube[1]*4 - 2
return params
def model2(params):
N, k = params
return N * x**k
def loglike2(params):
ypred = model2(params)
loglike = scipy.stats.poisson.logpmf(y, ypred).sum()
if not numpy.isfinite(loglike):
loglike = -1e300
return loglike
prefix2 = 'hump_nobrk_'
from pymultinest.solve import solve
results2 = solve(LogLikelihood=loglike2, Prior=prior_transform2, n_dims=2,
n_live_points=1000,
outputfiles_basename=prefix2, verbose=True, resume=True)
print(results2)
analysing data from hump_nobrk_.txt
{'logZ': -109.09711107480186, 'logZerr': 0.07834950931779051, 'samples': array([[ 8
.8813058 , -0.07790949],
[19.42081685, -0.52613514],
[16.86204359, -0.66979213],
...,
[12.96384037, -0.3779294 ],
[13.26293915, -0.42036467],
[12.72889868, -0.37648294]])}
Look at Bayes factor
In [12]:
BF = results['logZ'] - results2['logZ']
print("Bayes factor: %.2f" % BF)
Bayes factor: 6.35
• This seems to be quite high. Lets do some simulations.
• In particular, parametric bootstrap:
o Monte Carlo simulate data at the posterior samples under the simpler model
o Fit them with the simpler and more complex models
o Get Bayes factor
• Look at Bayes factor distribution:
o How often would the true simpler model be mistaken for the more complex
model
o at some Bayes factor threshold
o --> means calibrating decision threshold (or equivalently, the Prior Odds ratio)
In [9]:
BFs = []
samples2 = results2['samples']
for posterior_sample in samples2[numpy.random.choice(len(samples2), size=400)]:
y = numpy.random.poisson(model2(posterior_sample))
results_base = solve(LogLikelihood=loglike2, Prior=prior_transform2, n_dims=2)
results_complex = solve(LogLikelihood=loglike, Prior=prior_transform, n_dims=4)
BFi = results_complex['logZ'] - results_base['logZ']
print(BFi)
BFs.append(BF)
analysing data from /tmp/tmpzp1ydre4pymultinest/.txt
analysing data from /tmp/tmpphxbzgq6pymultinest/.txt
0.09457069180707833
analysing data from /tmp/tmp8se3rcotpymultinest/.txt
analysing data from /tmp/tmpdffa0eqwpymultinest/.txt
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analysing data from /tmp/tmprx591j9ppymultinest/.txt
analysing data from /tmp/tmpylam5_pwpymultinest/.txt
0.23465283399885095
analysing data from /tmp/tmp9_n03vsspymultinest/.txt
analysing data from /tmp/tmpwrm7n7m0pymultinest/.txt
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analysing data from /tmp/tmpuz28h_ampymultinest/.txt
analysing data from /tmp/tmpm6u0dybipymultinest/.txt
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analysing data from /tmp/tmptygfo0i2pymultinest/.txt
analysing data from /tmp/tmppyuttacppymultinest/.txt
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analysing data from /tmp/tmpbfmhti5gpymultinest/.txt
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analysing data from /tmp/tmpt0the4tbpymultinest/.txt
analysing data from /tmp/tmpsfz0x2bxpymultinest/.txt
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analysing data from /tmp/tmpco3cvsh2pymultinest/.txt
analysing data from /tmp/tmpzh81fdngpymultinest/.txt
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analysing data from /tmp/tmp6h0dlzt_pymultinest/.txt
analysing data from /tmp/tmp0rb07nmwpymultinest/.txt
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analysing data from /tmp/tmpnno_dafcpymultinest/.txt
analysing data from /tmp/tmpnsimqn9zpymultinest/.txt
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analysing data from /tmp/tmpopy_3afdpymultinest/.txt
analysing data from /tmp/tmp20irgitspymultinest/.txt
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analysing data from /tmp/tmpg87dix4ipymultinest/.txt
analysing data from /tmp/tmpiupzivwipymultinest/.txt
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analysing data from /tmp/tmpny46l98wpymultinest/.txt
analysing data from /tmp/tmp63vtsa59pymultinest/.txt
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analysing data from /tmp/tmpx1a_a0qspymultinest/.txt
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-0.01496768478490651
analysing data from /tmp/tmp1vyfuf5wpymultinest/.txt
analysing data from /tmp/tmp9hntfirypymultinest/.txt
-0.2327765187435773
analysing data from /tmp/tmpfkmm8s6npymultinest/.txt
analysing data from /tmp/tmprad4nx62pymultinest/.txt
0.37166729253287656
analysing data from /tmp/tmpx57bk7jwpymultinest/.txt
analysing data from /tmp/tmp1cc6cgl5pymultinest/.txt
-0.04564368052602674
analysing data from /tmp/tmpuof1w_6upymultinest/.txt
analysing data from /tmp/tmpp_tn9azmpymultinest/.txt
0.2160371766246385
analysing data from /tmp/tmp9nd_6bpqpymultinest/.txt
analysing data from /tmp/tmpuj3_azhgpymultinest/.txt
0.042352272613854325
analysing data from /tmp/tmpkn5qj55ppymultinest/.txt
analysing data from /tmp/tmp2fed0kk9pymultinest/.txt
0.4263402403122427
analysing data from /tmp/tmp7fc8cdnupymultinest/.txt
analysing data from /tmp/tmpoace8jegpymultinest/.txt
-0.23212590957251678
analysing data from /tmp/tmpw8fpc8bhpymultinest/.txt
analysing data from /tmp/tmphfeq36y4pymultinest/.txt
1.2243239753746025
analysing data from /tmp/tmpgr20ga4opymultinest/.txt
analysing data from /tmp/tmpfxi_tzzopymultinest/.txt
0.6489790404213096
analysing data from /tmp/tmpc04r6kj9pymultinest/.txt
analysing data from /tmp/tmp6e2ghp8opymultinest/.txt
-0.2534543215826943
analysing data from /tmp/tmpcuylklclpymultinest/.txt
analysing data from /tmp/tmpzbs36pplpymultinest/.txt
-0.22922098279349257
analysing data from /tmp/tmpynlh0mzlpymultinest/.txt
analysing data from /tmp/tmptus68cywpymultinest/.txt
0.24629991933684892
analysing data from /tmp/tmp4bi0t_30pymultinest/.txt
analysing data from /tmp/tmpe25y58xppymultinest/.txt
0.1423353751523848
analysing data from /tmp/tmp9_gtj5c1pymultinest/.txt
analysing data from /tmp/tmpyfgkv4yrpymultinest/.txt
-0.020152790258535447
analysing data from /tmp/tmpe80fjrehpymultinest/.txt
analysing data from /tmp/tmprmg7v42_pymultinest/.txt
0.06294590848276016
Out[9]:
(array([142., 194., 39., 16., 4., 3., 0., 1., 0., 1.]),
array([-0.61964866, -0.11861287, 0.38242292, 0.88345871, 1.38449451,
1.8855303 , 2.38656609, 2.88760188, 3.38863767, 3.88967347,
4.39070926]),
<a list of 10 Patch objects>)
In [25]:
plt.clf()
plt.hist(BFs, histtype='step', label='BFs from null model')
lo, hi = plt.ylim()
plt.vlines(BF, lo, hi, label='actual BF')
plt.legend(loc='best')
Out[25]: <matplotlib.legend.Legend at 0x7f3deb93e4a8>
In [ ]:
