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()

22:17:12 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))

22:17:14 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:93

22:17:15 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

22:17:18 INFO      Mean acceptance fraction: 0.7206999999999999                                emcee_sampler.py:157

22:17:19 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	1.006 -0.013 +0.012	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.88 +/- 0.07) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.924711
total	-10.924711

Values of statistical measures:

	statistical measures
AIC	26.555304
BIC	27.840886
DIC	25.729262
PDIC	1.938592

[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()

22:17:20 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)=  -20.219864601307840      +/-  0.14375594527396279
 Total Likelihood Evaluations:         5347
 Sampling finished. Exiting MultiNest

22:17:21 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	1.006 -0.013 +0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.88 -0.07 +0.06) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.924957
total	-10.924957

Values of statistical measures:

	statistical measures
AIC	26.555797
BIC	27.841379
DIC	26.013032
PDIC	2.079833
log(Z)	-8.781376

         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

4099it [00:04, 880.07it/s, +400 | bound: 8 | nc: 1 | ncall: 19640 | eff(%): 23.384 | loglstar:   -inf < -10.919 <    inf | logz: -20.115 +/-  0.143 | dlogz:  0.001 >  0.409]

22:17:26 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	1.006 -0.012 +0.013	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.88 -0.06 +0.07) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.925352
total	-10.925352

Values of statistical measures:

	statistical measures
AIC	26.556587
BIC	27.842170
DIC	25.773575
PDIC	1.962053
log(Z)	-8.735692

[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()

22:17:27 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:202

7162it [00:07, 1772.40it/s, batch: 0 | bound: 12 | nc: 1 | ncall: 25789 | eff(%): 27.559 | loglstar:   -inf < -10.922 <    inf | logz: -20.019 +/-  0.127 | dlogz:  0.004 >  0.010]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8223it [00:08, 1248.81it/s, batch: 1 | bound: 3 | nc: 1 | ncall: 27270 | eff(%): 29.758 | loglstar: -12.773 < -11.260 < -11.400 | logz: -20.015 +/-  0.131 | stop:  1.516]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8858it [00:10, 663.61it/s, batch: 2 | bound: 2 | nc: 1 | ncall: 27940 | eff(%): 31.310 | loglstar: -13.203 < -12.125 < -12.769 | logz: -20.028 +/-  0.107 | stop:  1.071]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9554it [00:11, 700.79it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 28708 | eff(%): 32.987 | loglstar: -13.630 < -11.977 < -13.194 | logz: -20.034 +/-  0.100 | stop:  1.002]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9779it [00:11, 816.36it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 28963 | eff(%): 33.764 | loglstar: -13.630 < -10.919 < -13.194 | logz: -20.034 +/-  0.100 | stop:  0.897]

22:17:40 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	1.006 +/- 0.013	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.88 +/- 0.07) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.924779
total	-10.924779

Values of statistical measures:

	statistical measures
AIC	26.555440
BIC	27.841022
DIC	25.839795
PDIC	1.995218
log(Z)	-8.702847

[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()

         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, 60.35it/s]

22:17:51 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: 25
Scale Factor: 1.142734
Mean Integrated Autocorrelation Time: 3.13
Effective Sample Size: 3988.22
Number of Log Probability Evaluations: 65897
Effective Samples per Log Probability Evaluation: 0.060522

None

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.007 -0.013 +0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.88 +/- 0.07) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.924719
total	-10.924719

Values of statistical measures:

	statistical measures
AIC	26.555321
BIC	27.840903
DIC	26.042977
PDIC	2.096679

[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()

22:17:52 INFO      sampler set to ultranest                                                bayesian_analysis.py:202

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

[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 6121
[ultranest]   logZ = -20.23 +- 0.1392
[ultranest] Effective samples strategy satisfied (ESS = 971.5, 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=0.43, need <0.5)
[ultranest]   logZ error budget: single: 0.14 bs:0.14 tail:0.41 total:0.43 required:<0.50
[ultranest] done iterating.

22:17:59 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	1.007 -0.014 +0.012	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.88 -0.06 +0.07) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.925473
total	-10.925473

Values of statistical measures:

	statistical measures
AIC	26.556829
BIC	27.842411
DIC	25.916953
PDIC	2.032554
log(Z)	-8.790191

[10]:

../_images/notebooks_sampler_docs_16_12.png

../_images/notebooks_sampler_docs_16_13.png

../_images/notebooks_sampler_docs_16_14.png