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()

22:01:47 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.

22:01:48 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))

22:01:49 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:93

22:01:50 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

22:01:53 INFO      Mean acceptance fraction: 0.7218                                            emcee_sampler.py:157

22:01:54 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	1.013 -0.015 +0.017	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.005) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.762638
total	-6.762638

Values of statistical measures:

	statistical measures
AIC	18.231158
BIC	19.516740
DIC	17.308502
PDIC	1.885384

[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()

22:01:55 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
 *****************************************************
 ln(ev)=  -15.960051662356523      +/-  0.14355960779352378
  analysing data from chains/fit-.txt
 Total Likelihood Evaluations:         6168
 Sampling finished. Exiting MultiNest

22:01:56 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	1.013 -0.017 +0.015	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.004 +/- 0.005) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.762828
total	-6.762828

Values of statistical measures:

	statistical measures
AIC	18.231538
BIC	19.517120
DIC	17.477867
PDIC	1.976215
log(Z)	-6.931362

         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

4243it [00:04, 888.77it/s, +400 | bound: 8 | nc: 1 | ncall: 19785 | eff(%): 23.952 | loglstar:   -inf < -6.750 <    inf | logz: -16.308 +/-  0.146 | dlogz:  0.001 >  0.409]

22:02:01 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	1.013 -0.017 +0.015	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.005) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.763276
total	-6.763276

Values of statistical measures:

	statistical measures
AIC	18.232434
BIC	19.518016
DIC	17.488523
PDIC	1.981774
log(Z)	-7.082663

[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()

22:02:02 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:202

7289it [00:07, 1877.45it/s, batch: 0 | bound: 12 | nc: 1 | ncall: 26603 | eff(%): 27.216 | loglstar:   -inf < -6.754 <    inf | logz: -16.064 +/-  0.129 | dlogz:  0.004 >  0.010]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8038it [00:09, 916.76it/s, batch: 1 | bound: 3 | nc: 1 | ncall: 27702 | eff(%): 28.486 | loglstar: -8.573 < -7.297 < -7.291 | logz: -16.061 +/-  0.133 | stop:  1.439]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8864it [00:10, 919.54it/s, batch: 2 | bound: 2 | nc: 1 | ncall: 28588 | eff(%): 30.663 | loglstar: -9.097 < -7.839 < -8.573 | logz: -16.052 +/-  0.109 | stop:  1.130]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9159it [00:10, 851.68it/s, batch: 2 | bound: 2 | nc: 1 | ncall: 28908 | eff(%): 31.683 | loglstar: -9.097 < -6.753 < -8.573 | logz: -16.052 +/-  0.109 | stop:  0.938]

22:02:13 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	1.013 +/- 0.016	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.005) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.762426
total	-6.762426

Values of statistical measures:

	statistical measures
AIC	18.230734
BIC	19.516316
DIC	17.443250
PDIC	1.958775
log(Z)	-6.965615

[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()

22:02:14 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 [00:10<00:00, 57.10it/s]

22:02:26 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: 19
Scale Factor: 1.063141
Mean Integrated Autocorrelation Time: 3.22
Effective Sample Size: 3887.79
Number of Log Probability Evaluations: 65764
Effective Samples per Log Probability Evaluation: 0.059117

None

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.013 +/- 0.016	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.005) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.762503
total	-6.762503

Values of statistical measures:

	statistical measures
AIC	18.230888
BIC	19.516470
DIC	17.537700
PDIC	2.006148

[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()

22:02:27 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: 6481
[ultranest]   logZ = -16.2 +- 0.1157
[ultranest] Effective samples strategy satisfied (ESS = 969.3, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.47+-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:0.12 tail:0.41 total:0.42 required:<0.50
[ultranest] done iterating.

22:02:35 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	1.012 +/- 0.017	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.005) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-6.763056
total	-6.763056

Values of statistical measures:

	statistical measures
AIC	18.231994
BIC	19.517576
DIC	17.515318
PDIC	1.994916
log(Z)	-7.032023

[10]:

../_images/notebooks_sampler_docs_16_12.png

../_images/notebooks_sampler_docs_16_13.png

../_images/notebooks_sampler_docs_16_14.png