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()

00:48:37 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.

         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))

00:48:39 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:90

00:48:40 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

00:48:42 INFO      Mean acceptance fraction: 0.7119                                            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	(9.87 -0.17 +0.16) x 10^-1	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.018 +/- 0.008) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-14.695701
total	-14.695701

Values of statistical measures:

	statistical measures
AIC	34.097285
BIC	35.382867
DIC	33.265319
PDIC	1.937014

[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()

         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)=  -23.572032466369969      +/-  0.14018996012076002
 Total Likelihood Evaluations:         5670
 Sampling finished. Exiting MultiNest

00:48:43 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	(9.87 -0.16 +0.15) x 10^-1	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.018 +/- 0.008) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-14.695964
total	-14.695964

Values of statistical measures:

	statistical measures
AIC	34.097809
BIC	35.383392
DIC	33.382374
PDIC	1.993116
log(Z)	-10.237204

         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()

00:48:44 INFO      sampler set to dynesty_nested                                           bayesian_analysis.py:186

3965it [00:04, 882.82it/s, +400 | bound: 11 | nc: 1 | ncall: 18599 | eff(%): 23.985 | loglstar:   -inf < -14.708 <    inf | logz: -23.572 +/-  0.140 | dlogz:  0.001 >  0.409]

00:48:48 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	(9.87 -0.16 +0.17) x 10^-1	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.018 +/- 0.008) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-14.695796
total	-14.695796

Values of statistical measures:

	statistical measures
AIC	34.097474
BIC	35.383056
DIC	33.506475
PDIC	2.054773
log(Z)	-10.237314

[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()

00:48:49 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:186

15974it [00:17, 918.87it/s, batch: 8 | bound: 6 | nc: 1 | ncall: 36869 | eff(%): 43.274 | loglstar: -19.626 < -14.708 < -14.873 | logz: -23.527 +/-  0.074 | stop:  0.913]

00:49:06 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	(9.87 +/- 0.16) x 10^-1	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.018 -0.008 +0.009) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-14.695608
total	-14.695608

Values of statistical measures:

	statistical measures
AIC	34.097097
BIC	35.382680
DIC	33.378286
PDIC	1.991413
log(Z)	-10.219700

[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()

00:49:07 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, 120.38it/s]

00:49:12 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: 47
Scale Factor: 1.16969
Mean Integrated Autocorrelation Time: 2.85
Effective Sample Size: 4383.66
Number of Log Probability Evaluations: 65319
Effective Samples per Log Probability Evaluation: 0.067112

None

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	(9.87 -0.17 +0.16) x 10^-1	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.018 -0.008 +0.009) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-14.695601
total	-14.695601

Values of statistical measures:

	statistical measures
AIC	34.097084
BIC	35.382666
DIC	33.527317
PDIC	2.065139

[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()

00:49:13 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: 6987
[ultranest]   logZ = -23.4 +- 0.1056
[ultranest] Effective samples strategy satisfied (ESS = 971.2, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.42, need <0.5)
[ultranest]   logZ error budget: single: 0.14 bs:0.11 tail:0.41 total:0.42 required:<0.50
[ultranest] done iterating.

00:49:17 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	(9.88 -0.16 +0.15) x 10^-1	1 / (keV s cm2)
demo.spectrum.main.Sin.f	(1.017 -0.007 +0.008) x 10^-1	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-14.697308
total	-14.697308

Values of statistical measures:

	statistical measures
AIC	34.100499
BIC	35.386081
DIC	33.163963
PDIC	1.885623
log(Z)	-10.152962

[10]:

../_images/notebooks_sampler_docs_16_12.png

../_images/notebooks_sampler_docs_16_13.png

../_images/notebooks_sampler_docs_16_14.png