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

from packaging.version import Version
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()

17:47:45 WARNING   The naima package is not available. Models that depend on it will not be         functions.py:43
                  available

         WARNING   The GSL library or the pygsl wrapper cannot be loaded. Models that depend on it  functions.py:65
                  will not be available.

17:47:46 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))

17:47:48 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:90

         INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:90

../_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:186

17:47:50 INFO      Mean acceptance fraction: 0.6436999999999999                                emcee_sampler.py:145

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.007 -0.026 +0.017	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.0004 -0.0021 +0.0027) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-11.181299
total	-11.181299

Values of statistical measures:

	statistical measures
AIC	27.068481
BIC	28.354063
DIC	91.851159
PDIC	0.389275

[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()

17:47:51 INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 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)=  -21.214746953152403      +/-  0.15045954709225123
 Total Likelihood Evaluations:         5723
 Sampling finished. Exiting MultiNest

17:47:52 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.007 -0.021 +0.020	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.0003 -0.0022 +0.0021) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-11.181953
total	-11.181953

Values of statistical measures:

	statistical measures
AIC	27.069788
BIC	28.355370
DIC	26.298412
PDIC	1.966182
log(Z)	-9.213448

         INFO      deleting the chain directory chains                                     multinest_sampler.py:234

[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(nlive=400)
bayes_analysis.sample()

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

         INFO      sampler set to dynesty_nested                                           bayesian_analysis.py:186

4418it [00:05, 861.25it/s, +400 | bound: 13 | nc: 1 | ncall: 19518 | eff(%): 25.201 | loglstar:   -inf < -11.175 <    inf | logz: -21.165 +/-  0.150 | dlogz:  0.001 >  0.409]

17:47:57 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.007 -0.021 +0.019	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.0003 -0.0018 +0.0024) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-11.18151
total	-11.18151

Values of statistical measures:

	statistical measures
AIC	27.068902
BIC	28.354484
DIC	26.183630
PDIC	1.906263
log(Z)	-9.191686

[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()

if Version(dynesty.__version__) >= Version("3.0.0"):
    bayes_analysis.sample(n_effective=None)
else:
    bayes_analysis.sample(
        stop_function=dynesty.utils.old_stopping_function, n_effective=None
    )

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

17:47:58 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:186

16304it [00:18, 899.73it/s, batch: 8 | bound: 4 | nc: 1 | ncall: 37916 | eff(%): 42.944 | loglstar: -15.916 < -11.174 < -11.707 | logz: -21.071 +/-  0.077 | stop:  0.912]

17:48:16 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.007 +/- 0.021	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.0004 -0.0022 +0.0023) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-11.181315
total	-11.181315

Values of statistical measures:

	statistical measures
AIC	27.068513
BIC	28.354095
DIC	26.333027
PDIC	1.985095
log(Z)	-9.149370

[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()

17:48:17 INFO      sampler set to zeus                                                     bayesian_analysis.py:186

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 [00:05<00:00, 117.78it/s]

17:48:22 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Summary
-------
Number of Generations: 625
Number of Parameters: 2
Number of Walkers: 20
Number of Tuning Generations: 39
Scale Factor: 1.02588
Mean Integrated Autocorrelation Time: 2.77
Effective Sample Size: 4515.08
Number of Log Probability Evaluations: 66639
Effective Samples per Log Probability Evaluation: 0.067754

None

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.007 -0.021 +0.020	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.0004 -0.0022 +0.0023) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-11.181236
total	-11.181236

Values of statistical measures:

	statistical measures
AIC	27.068355
BIC	28.353937
DIC	26.405246
PDIC	2.020593

[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()

17:48:23 INFO      sampler set to ultranest                                                bayesian_analysis.py:186

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

[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 8503
[ultranest]   logZ = -20.9 +- 0.09528
[ultranest] Effective samples strategy satisfied (ESS = 986.4, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.08 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:0.10 tail:0.41 total:0.42 required:<0.50
[ultranest] done iterating.

17:48:27 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.008 -0.023 +0.019	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.0005 -0.0021 +0.0022) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-11.183516
total	-11.183516

Values of statistical measures:

	statistical measures
AIC	27.072915
BIC	28.358497
DIC	26.300029
PDIC	1.967939
log(Z)	-9.065638

[10]:

../_images/notebooks_sampler_docs_16_12.png

../_images/notebooks_sampler_docs_16_13.png

../_images/notebooks_sampler_docs_16_14.png