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 *

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()

Welcome to JupyROOT 6.22/02

13:44:56 WARNING   The naima package is not available. Models that depend on it will not be         functions.py:51
                  available

         WARNING   The GSL library or the pygsl wrapper cannot be loaded. Models that depend on it  functions.py:72
                  will not be available.

         WARNING   The ebltable package is not available. Models that depend on it will not be     absorption.py:37
                  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))

13:44:59 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:62

13:45:00 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:62

../_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()

13:45:01 INFO      sampler set to emcee                                                    bayesian_analysis.py:233

13:45:05 INFO      Mean acceptance fraction: 0.7174                                            emcee_sampler.py:157

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	(1.004 +/- 0.007) x 10^-1	rad / keV


Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.507075
total	-10.507075


Values of statistical measures:

	statistical measures
AIC	25.720033
BIC	27.005615
DIC	24.908258
PDIC	1.942690

[5]:

../_images/notebooks_sampler_docs_7_10.png

../_images/notebooks_sampler_docs_7_11.png

../_images/notebooks_sampler_docs_7_12.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()

13:45:08 INFO      sampler set to multinest                                                bayesian_analysis.py:233

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.012 -0.013 +0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 -0.008 +0.007) x 10^-1	rad / keV


Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.513789
total	-10.513789


Values of statistical measures:

	statistical measures
AIC	25.733460
BIC	27.019042
DIC	25.097662
PDIC	2.039027
log(Z)	-8.462505

13:45:09 INFO      deleting the chain directory chains                                     multinest_sampler.py:261

WARNING:root:Too few points to create valid contours

[6]:

../_images/notebooks_sampler_docs_9_9.png

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

 no. of live points =  400
 dimensionality =    2
 *****************************************************
 ln(ev)=  -19.485638820181762      +/-  0.14121093259648074
 Total Likelihood Evaluations:         5089
 Sampling finished. Exiting MultiNest

../_images/notebooks_sampler_docs_9_11.png

../_images/notebooks_sampler_docs_9_12.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()

13:45:10 INFO      sampler set to dynesty_nested                                           bayesian_analysis.py:233

3971it [00:10, 382.54it/s, +400 | bound: 8 | nc: 1 | ncall: 18560 | eff(%): 24.069 | loglstar:   -inf < -10.498 <    inf | logz: -19.376 +/-  0.141 | dlogz:  0.001 >  0.409]

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.011 +/- 0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.004 +/- 0.008) x 10^-1	rad / keV


Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.509168
total	-10.509168


Values of statistical measures:

	statistical measures
AIC	25.724219
BIC	27.009801
DIC	25.166964
PDIC	2.076518
log(Z)	-8.414852

[7]:

../_images/notebooks_sampler_docs_11_8.png

../_images/notebooks_sampler_docs_11_9.png

../_images/notebooks_sampler_docs_11_10.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()

13:45:22 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:233

7229it [00:17, 1226.11it/s, batch: 0 | bound: 13 | nc: 1 | ncall: 26526 | eff(%): 27.223 | loglstar:   -inf < -10.497 <    inf | logz: -19.388 +/-  0.126 | dlogz:  0.001 >  0.010]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8344it [00:19, 1075.87it/s, batch: 1 | bound: 3 | nc: 1 | ncall: 28077 | eff(%): 29.692 | loglstar: -12.329 < -10.533 < -11.013 | logz: -19.388 +/-  0.130 | stop:  1.584]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8958it [00:21, 594.44it/s, batch: 2 | bound: 2 | nc: 1 | ncall: 28729 | eff(%): 31.156 | loglstar: -12.757 < -10.557 < -12.318 | logz: -19.380 +/-  0.106 | stop:  1.128]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9180it [00:22, 329.13it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 28976 | eff(%): 31.202 | loglstar: -13.169 < -12.298 < -12.752 | logz: -19.402 +/-  0.099 | stop:  1.005]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9570it [00:23, 408.19it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 29421 | eff(%): 32.528 | loglstar: -13.169 < -10.500 < -12.752 | logz: -19.402 +/-  0.099 | stop:  0.984]

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.011 -0.013 +0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.008) x 10^-1	rad / keV


Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.506859
total	-10.506859


Values of statistical measures:

	statistical measures
AIC	25.719601
BIC	27.005183
DIC	24.927015
PDIC	1.955806
log(Z)	-8.433732

[8]:

../_images/notebooks_sampler_docs_12_8.png

../_images/notebooks_sampler_docs_12_9.png

../_images/notebooks_sampler_docs_12_10.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()

13:45:48 INFO      sampler set to zeus                                                     bayesian_analysis.py:233

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 [00:15<00:00, 39.75it/s]
Summary
-------
Number of Generations: 625
Number of Parameters: 2
Number of Walkers: 20
Number of Tuning Generations: 35
Scale Factor: 1.339496
Mean Integrated Autocorrelation Time: 3.38
Effective Sample Size: 3701.44
Number of Log Probability Evaluations: 65256
Effective Samples per Log Probability Evaluation: 0.056722

None
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.011 +/- 0.014	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.008) x 10^-1	rad / keV


Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.506904
total	-10.506904


Values of statistical measures:

	statistical measures
AIC	25.719690
BIC	27.005272
DIC	25.095113
PDIC	2.040639

[9]:

../_images/notebooks_sampler_docs_14_8.png

../_images/notebooks_sampler_docs_14_9.png

../_images/notebooks_sampler_docs_14_10.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()

13:46:05 INFO      sampler set to ultranest                                                bayesian_analysis.py:233

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

[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 8677
[ultranest]   logZ = -19.37 +- 0.09592
[ultranest] Effective samples strategy satisfied (ESS = 974.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.05 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:0.10 tail:0.41 total:0.42 required:<0.50
[ultranest] done iterating.
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	1.011 -0.014 +0.013	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(1.005 +/- 0.008) x 10^-1	rad / keV


Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-10.507074
total	-10.507074


Values of statistical measures:

	statistical measures
AIC	25.720031
BIC	27.005613
DIC	25.100989
PDIC	2.042336
log(Z)	-8.409488

[10]:

../_images/notebooks_sampler_docs_16_9.png

../_images/notebooks_sampler_docs_16_10.png

../_images/notebooks_sampler_docs_16_11.png