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()

00:06:52 WARNING   The naima package is not available. Models that depend on it will not be         functions.py:47
                  available

         WARNING   The GSL library or the pygsl wrapper cannot be loaded. Models that depend on it  functions.py:68
                  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))

00:06:54 INFO      Using Gaussian statistic (equivalent to chi^2) with the provided errors.            XYLike.py:93

         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()

00:06:55 INFO      sampler set to emcee                                                    bayesian_analysis.py:202

00:06:57 INFO      Mean acceptance fraction: 0.7088                                            emcee_sampler.py:157

00:06:58 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	(9.93 +/- 0.13) x 10^-1	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.959 -0.034 +0.031) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-8.747515
total	-8.747515

Values of statistical measures:

	statistical measures
AIC	22.200913
BIC	23.486495
DIC	21.720799
PDIC	2.112838

[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()

00:06:59 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
 *****************************************************

 MultiNest Warning!
 Parameter            1  of mode            5  is converging towards the edge of the prior.
  analysing data from chains/fit-.txt ln(ev)=  -18.851350892197942      +/-  0.15093621567797696
 Total Likelihood Evaluations:         5623
 Sampling finished. Exiting MultiNest

         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	(9.94 -0.12 +0.10) x 10^-1	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.956 -0.031 +0.030) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-8.751447
total	-8.751447

Values of statistical measures:

	statistical measures
AIC	22.208776
BIC	23.494359
DIC	21.289796
PDIC	1.896275
log(Z)	-8.187038

         INFO      deleting the chain directory chains                                     multinest_sampler.py:255

WARNING:root:Too few points to create valid contours

[6]:

../_images/notebooks_sampler_docs_9_12.png

../_images/notebooks_sampler_docs_9_13.png

../_images/notebooks_sampler_docs_9_14.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()

00:07:00 INFO      sampler set to dynesty_nested                                           bayesian_analysis.py:202

4464it [00:04, 995.55it/s, +400 | bound: 9 | nc: 1 | ncall: 20360 | eff(%): 24.369 | loglstar:   -inf < -8.754 <    inf | logz: -18.863 +/-  0.151 | dlogz:  0.001 >  0.409]

00:07:05 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	(9.94 +/- 0.12) x 10^-1	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.958 -0.032 +0.031) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-8.747634
total	-8.747634

Values of statistical measures:

	statistical measures
AIC	22.201151
BIC	23.486733
DIC	21.437391
PDIC	1.970579
log(Z)	-8.192202

[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()

00:07:06 INFO      sampler set to dynesty_dynamic                                          bayesian_analysis.py:202

7398it [00:07, 1358.23it/s, batch: 0 | bound: 13 | nc: 4 | ncall: 27530 | eff(%): 26.393 | loglstar:   -inf < -8.765 <    inf | logz: -18.939 +/-  0.135 | dlogz:  0.010 >  0.010]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

8705it [00:08, 1169.33it/s, batch: 1 | bound: 3 | nc: 1 | ncall: 29275 | eff(%): 29.304 | loglstar: -10.646 < -9.149 < -9.213 | logz: -18.929 +/-  0.139 | stop:  1.410]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9558it [00:10, 1006.93it/s, batch: 2 | bound: 2 | nc: 1 | ncall: 30159 | eff(%): 31.499 | loglstar: -11.128 < -9.472 < -10.639 | logz: -18.930 +/-  0.113 | stop:  1.199]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

9783it [00:11, 544.93it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 30395 | eff(%): 31.646 | loglstar: -11.558 < -11.341 < -11.127 | logz: -18.931 +/-  0.106 | stop:  1.022]
WARNING DeprecationWarning: This an old stopping function that will be removed in future releases

10322it [00:11, 869.09it/s, batch: 3 | bound: 2 | nc: 1 | ncall: 30959 | eff(%): 33.341 | loglstar: -11.558 < -8.779 < -11.127 | logz: -18.931 +/-  0.106 | stop:  0.850]

00:07:18 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	(9.93 +/- 0.12) x 10^-1	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.958 +/- 0.032) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-8.747237
total	-8.747237

Values of statistical measures:

	statistical measures
AIC	22.200356
BIC	23.485938
DIC	21.549585
PDIC	2.027583
log(Z)	-8.220186

[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()

00:07:19 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 [00:09<00:00, 66.27it/s]

00:07:29 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: 35
Scale Factor: 1.164591
Mean Integrated Autocorrelation Time: 2.99
Effective Sample Size: 4180.1
Number of Log Probability Evaluations: 65810
Effective Samples per Log Probability Evaluation: 0.063518

None

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
demo.spectrum.main.Sin.K	(9.93 +/- 0.12) x 10^-1	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.958 +/- 0.031) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-8.747222
total	-8.747222

Values of statistical measures:

	statistical measures
AIC	22.200326
BIC	23.485908
DIC	21.447816
PDIC	1.976284

[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()

00:07:30 INFO      sampler set to ultranest                                                bayesian_analysis.py:202

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

[ultranest] Explored until L=-9
[ultranest] Likelihood function evaluations: 8559
[ultranest]   logZ = -18.66 +- 0.1348
[ultranest] Effective samples strategy satisfied (ESS = 977.9, 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=0.43, need <0.5)
[ultranest]   logZ error budget: single: 0.15 bs:0.13 tail:0.41 total:0.43 required:<0.50
[ultranest] done iterating.

00:07:38 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	(9.94 +/- 0.12) x 10^-1	1 / (cm2 keV s)
demo.spectrum.main.Sin.f	(9.960 -0.033 +0.029) x 10^-2	rad / keV

Values of -log(posterior) at the minimum:

	-log(posterior)
demo	-8.750222
total	-8.750222

Values of statistical measures:

	statistical measures
AIC	22.206326
BIC	23.491908
DIC	21.483093
PDIC	1.992386
log(Z)	-8.100086

[10]:

../_images/notebooks_sampler_docs_16_12.png

../_images/notebooks_sampler_docs_16_13.png

../_images/notebooks_sampler_docs_16_14.png