Bayesian Sampler Examples

Examples of running each sampler avaiable in 3ML.

Before, that, let’s discuss setting up configuration default sampler with default parameters. We can set in our configuration a default algorithm and default setup parameters for the samplers. This can ease fitting when we are doing exploratory data analysis.

With any of the samplers, you can pass keywords to access their setups. Read each pacakges documentation for more details.

[1]:

from threeML import *
from threeML.plugins.XYLike import XYLike

import numpy as np
import dynesty
from jupyterthemes import jtplot

%matplotlib inline
jtplot.style(context="talk", fscale=1, ticks=True, grid=False)
silence_warnings()
set_threeML_style()

05:15:25 WARNING   The naima package is not available. Models that depend on it will not be         functions.py:48
                  available

         WARNING   The GSL library or the pygsl wrapper cannot be loaded. Models that depend on it  functions.py:69
                  will not be available.

         WARNING   The ebltable package is not available. Models that depend on it will not be     absorption.py:33
                  available

[2]:

threeML_config.bayesian.default_sampler

[2]:

<Sampler.emcee: 'emcee'>

[3]:

threeML_config.bayesian.emcee_setup

[3]:

{'n_burnin': None, 'n_iterations': 500, 'n_walkers': 50, 'seed': 5123}

If you simply run bayes_analysis.sample() the default sampler and its default parameters will be used.

Let’s make some data to fit.

[4]:

sin = Sin(K=1, f=0.1)
sin.phi.fix = True
sin.K.prior = Log_uniform_prior(lower_bound=0.5, upper_bound=1.5)
sin.f.prior = Uniform_prior(lower_bound=0, upper_bound=0.5)

model = Model(PointSource("demo", 0, 0, spectral_shape=sin))

x = np.linspace(-2 * np.pi, 4 * np.pi, 20)
yerr = np.random.uniform(0.01, 0.2, 20)


xyl = XYLike.from_function("demo", sin, x, yerr)
xyl.plot()

bayes_analysis = BayesianAnalysis(model, DataList(xyl))

05:15:27 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:93

05:15:28 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:93

../_images/notebooks_sampler_docs_5_2.png

emcee

[5]:

bayes_analysis.set_sampler("emcee")
bayes_analysis.sampler.setup(n_walkers=20, n_iterations=500)
bayes_analysis.sample()

xyl.plot()
bayes_analysis.results.corner_plot()

         INFO      sampler set to emcee                                                    bayesian_analysis.py:202

05:15:31 INFO      Mean acceptance fraction: 0.7034999999999999                                emcee_sampler.py:157

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.010 -0.017 +0.015	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(10.00 -0.035 +0.04) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.831485
total	-6.831485

Values of statistical measures:

	statistical measures
AIC	18.368852
BIC	19.654435
DIC	17.847605
PDIC	2.088062

[5]:

../_images/notebooks_sampler_docs_7_12.png

../_images/notebooks_sampler_docs_7_13.png

../_images/notebooks_sampler_docs_7_14.png

multinest

[6]:

bayes_analysis.set_sampler("multinest")
bayes_analysis.sampler.setup(n_live_points=400, resume=False, auto_clean=True)
bayes_analysis.sample()

xyl.plot()
bayes_analysis.results.corner_plot()

05:15:32 INFO      sampler set to multinest                                                bayesian_analysis.py:202

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  400
 dimensionality =    2
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -16.574413947909981      +/-  0.14780047452922601
 Total Likelihood Evaluations:         5895
 Sampling finished. Exiting MultiNest

05:15:33 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.011 -0.017 +0.015	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(10.00 +/- 0.04) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.833564
total	-6.833564

Values of statistical measures:

	statistical measures
AIC	18.373010
BIC	19.658592
DIC	17.793983
PDIC	2.064988
log(Z)	-7.198177

         INFO      deleting the chain directory chains                                     multinest_sampler.py:255

[6]:

../_images/notebooks_sampler_docs_9_11.png

../_images/notebooks_sampler_docs_9_12.png

../_images/notebooks_sampler_docs_9_13.png

dynesty

[7]:

bayes_analysis.set_sampler("dynesty_nested")
bayes_analysis.sampler.setup(n_live_points=400)
bayes_analysis.sample()

xyl.plot()
bayes_analysis.results.corner_plot()

         INFO      sampler set to dynesty_nested                                           bayesian_analysis.py:202

4229it [00:04, 960.52it/s, +400 | bound: 9 | nc: 1 | ncall: 19355 | eff(%): 24.421 | loglstar:   -inf < -6.822 <    inf | logz: -16.342 +/-  0.146 | dlogz:  0.001 >  0.409]

05:15:38 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.010 -0.015 +0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(10.00 +/- 0.04) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.832491
total	-6.832491

Values of statistical measures:

	statistical measures
AIC	18.370865
BIC	19.656447
DIC	17.627936
PDIC	1.981979
log(Z)	-7.097053

[7]:

../_images/notebooks_sampler_docs_11_10.png

../_images/notebooks_sampler_docs_11_11.png

../_images/notebooks_sampler_docs_11_12.png

[8]:

bayes_analysis.set_sampler("dynesty_dynamic")
bayes_analysis.sampler.setup(
    stop_function=dynesty.utils.old_stopping_function, n_effective=None
)
bayes_analysis.sample()

xyl.plot()
bayes_analysis.results.corner_plot()

05:15:39 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:202

7347it [00:07, 1434.18it/s, batch: 0 | bound: 13 | nc: 1 | ncall: 27301 | eff(%): 26.559 | loglstar:   -inf < -6.829 <    inf | logz: -16.621 +/-  0.133 | dlogz:  0.007 >  0.010]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8614it [00:08, 1539.79it/s, batch: 1 | bound: 3 | nc: 1 | ncall: 28992 | eff(%): 29.490 | loglstar: -8.545 < -6.993 < -7.275 | logz: -16.614 +/-  0.137 | stop:  1.391]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9299it [00:09, 1074.66it/s, batch: 2 | bound: 2 | nc: 1 | ncall: 29737 | eff(%): 31.109 | loglstar: -9.042 < -7.252 < -8.543 | logz: -16.602 +/-  0.112 | stop:  1.054]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9551it [00:10, 631.33it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 30022 | eff(%): 31.322 | loglstar: -9.435 < -8.804 < -9.035 | logz: -16.601 +/-  0.104 | stop:  1.069]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

10006it [00:11, 902.01it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 30493 | eff(%): 32.814 | loglstar: -9.435 < -6.823 < -9.035 | logz: -16.601 +/-  0.104 | stop:  0.808]

05:15:50 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.010 +/- 0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(10.00 +/- 0.04) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.831431
total	-6.831431

Values of statistical measures:

	statistical measures
AIC	18.368745
BIC	19.654327
DIC	17.453666
PDIC	1.895196
log(Z)	-7.207368

[8]:

../_images/notebooks_sampler_docs_12_10.png

../_images/notebooks_sampler_docs_12_11.png

../_images/notebooks_sampler_docs_12_12.png

zeus

[9]:

bayes_analysis.set_sampler("zeus")
bayes_analysis.sampler.setup(n_walkers=20, n_iterations=500)
bayes_analysis.sample()

xyl.plot()
bayes_analysis.results.corner_plot()

05:15:51 INFO      sampler set to zeus                                                     bayesian_analysis.py:202

WARNING:root:The sampler class has been deprecated. Please use the new EnsembleSampler class.
The run method has been deprecated and it will be removed. Please use the new run_mcmc method.
Initialising ensemble of 20 walkers...
Sampling progress : 100%|██████████| 625/625 [00:08<00:00, 73.48it/s]

05:16:00 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Summary
-------
Number of Generations: 625
Number of Parameters: 2
Number of Walkers: 20
Number of Tuning Generations: 28
Scale Factor: 1.405118
Mean Integrated Autocorrelation Time: 2.9
Effective Sample Size: 4313.65
Number of Log Probability Evaluations: 64983
Effective Samples per Log Probability Evaluation: 0.066381

None

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.010 -0.016 +0.015	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(10.00 +/- 0.04) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.831516
total	-6.831516

Values of statistical measures:

	statistical measures
AIC	18.368914
BIC	19.654496
DIC	17.711939
PDIC	2.023822

[9]:

../_images/notebooks_sampler_docs_14_12.png

../_images/notebooks_sampler_docs_14_13.png

../_images/notebooks_sampler_docs_14_14.png

ultranest

[10]:

bayes_analysis.set_sampler("ultranest")
bayes_analysis.sampler.setup(
    min_num_live_points=400, frac_remain=0.5, use_mlfriends=False
)
bayes_analysis.sample()

xyl.plot()
bayes_analysis.results.corner_plot()

05:16:01 INFO      sampler set to ultranest                                                bayesian_analysis.py:202

[ultranest] Sampling 400 live points from prior ...

[ultranest] Explored until L=-7
[ultranest] Likelihood function evaluations: 7647
[ultranest]   logZ = -16.71 +- 0.107
[ultranest] Effective samples strategy satisfied (ESS = 984.9, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.42, need <0.5)
[ultranest]   logZ error budget: single: 0.15 bs:0.11 tail:0.41 total:0.42 required:<0.50
[ultranest] done iterating.

05:16:08 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.009 -0.014 +0.015	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(10.00 +/- 0.04) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.838926
total	-6.838926

Values of statistical measures:

	statistical measures
AIC	18.383735
BIC	19.669317
DIC	17.704202
PDIC	2.019729
log(Z)	-7.258329

[10]:

../_images/notebooks_sampler_docs_16_12.png

../_images/notebooks_sampler_docs_16_13.png

../_images/notebooks_sampler_docs_16_14.png