.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "tutorials/api/nested_sampling_Crab.py"
.. LINE NUMBERS ARE GIVEN BELOW.
.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_tutorials_api_nested_sampling_Crab.py>`
to download the full example code. or to run this example in your browser via Binder
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_tutorials_api_nested_sampling_Crab.py:
Bayesian analysis with nested sampling
======================================
A demonstration of a Bayesian analysis using the nested sampling technique.
.. GENERATED FROM PYTHON SOURCE LINES 11-85
Context
-------
1. Bayesian analysis
~~~~~~~~~~~~~~~~~~~~
Bayesian inference uses prior knowledge, in the form of a prior
distribution, in order to estimate posterior probabilities which we
traditionally visualise in the form of corner plots. These distributions
contain more information than a maximum likelihood fit as they reveal not
only the “best model” but provide a more accurate representation of errors and
correlation between parameters. In particular, non-Gaussian degeneracies are
complex to estimate with a maximum likelihood approach.
2. Limitations of the Markov Chain Monte Carlo approach
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A well-known approach to estimate this posterior distribution is the
Markov Chain Monte Carlo (MCMC). This uses an ensemble of walkers to
produce a chain of samples that after a convergence period will reach a
stationary state. *Once convergence* is reached, the successive elements
of the chain are samples of the target posterior distribution. However,
the weakness of the MCMC approach lies in the "*Once convergence*" part.
If the walkers are started far from the best likelihood region, the convergence time can be
long or never reached if the walkers fall in a local minima. The choice
of the initialisation point can become critical for complex models with
a high number of dimensions and the ability of these walkers to escape a
local minimum or to accurately describe a complex likelihood space is
not guaranteed.
3. Nested sampling approach
~~~~~~~~~~~~~~~~~~~~~~~~~~~
To overcome these issues, the nested sampling (NS) algorithm has
gained traction in physics and astronomy. It is a Monte Carlo
algorithm for computing an integral of the likelihood function over
the prior model parameter space introduced in
`Skilling, 2004 <https://ui.adsabs.harvard.edu/abs/2004AIPC..735..395S>`__.
The method performs this integral by evolving a collection of points
through the parameter space (see recent reviews from `Ashton et al.,
2022 <https://ui.adsabs.harvard.edu/abs/2022NRvMP...2...39A>`__, and
`Buchner, 2023 <http://arxiv.org/abs/2101.09675>`__). Without going
into too many details, one important specificity of the NS method is
that it starts from the entire parameter space and evolves a
collection of live points to map all minima (including multiple modes
if any), whereas Markov Chain Monte Carlo methods require an
initialisation point and the walkers will explore the local
likelihood. The ability of these walkers to escape a local minimum or
to accurately describe a complex likelihood space is not guaranteed.
This is a fundamental difference with MCMC or Minuit which will
only ever probe the vicinity along their minimisation paths and do not
have an overview of the global likelihood landscape. The analysis
using the NS framework is more CPU time consuming than a standard
classical fit, but it provides the full posterior distribution for all
parameters, which is out of reach with traditional fitting techniques
(N*(N-1)/2 contour plots to generate). In addition, it is more robust
to the choice of initialisation, requires less human intervention and
is therefore readily integrated in pipeline analysis. In Gammapy, we
used the NS implementation of the UltraNest package
(see `here <https://johannesbuchner.github.io/UltraNest/>`__ for more information), one of the
leading package in Astronomy (already used in Cosmology and in
X-rays).
For a nice visualisation of the NS method see here : `sampling
visualisation <https://johannesbuchner.github.io/UltraNest/method.html#visualisation>`__.
And for a tutorial of UltraNest applied to X-ray fitting with concrete examples and questions see : `BXA
Tutorial <https://peterboorman.com/tutorial_bxa.html>`__.
**Note: please cite UltraNest if used for a paper**
If you are using the "UltraNest" library for a paper, please follow its citation scheme:
`Cite UltraNest <https://johannesbuchner.github.io/UltraNest/issues.html#how-should-i-cite-ultranest>`__.
.. GENERATED FROM PYTHON SOURCE LINES 87-94
Proposed approach
-----------------
In this example, we will perform a Bayesian analysis with multiple 1D
spectra of the Crab nebula data and investigate their posterior
distributions.
.. GENERATED FROM PYTHON SOURCE LINES 97-102
Setup
-----
As usual, we’ll start with some setup …
.. GENERATED FROM PYTHON SOURCE LINES 102-117
.. code-block:: Python
import matplotlib.pyplot as plt
import numpy as np
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
.. GENERATED FROM PYTHON SOURCE LINES 118-124
Loading the spectral datasets
-----------------------------
Here we will load a few Crab 1D spectral data for which we will do a
fit.
.. GENERATED FROM PYTHON SOURCE LINES 124-133
.. code-block:: Python
path = "$GAMMAPY_DATA/joint-crab/spectra/hess/"
datasets = Datasets()
for id in ["23526", "23559", "23592"]:
dataset = SpectrumDatasetOnOff.read(f"{path}pha_obs{id}.fits")
datasets.append(dataset)
.. GENERATED FROM PYTHON SOURCE LINES 134-142
Model definition
----------------
Now we want to define the spectral model that will be fitted to the
data.
The Crab spectra will be fitted here with a simple powerlaw for
simplicity.
.. GENERATED FROM PYTHON SOURCE LINES 142-146
.. code-block:: Python
model = SkyModel.create(spectral_model="pl", name="crab")
.. GENERATED FROM PYTHON SOURCE LINES 147-161
.. WARNING:: Priors definition:
Unlike a traditional fit where priors on the
parameters are optional, here it is inherent to the Bayesian approach and
are therefore mandatory.
In this case we will set (min,max) prior that will define the
boundaries in which the sampling will be performed.
Note that it is usually recommended to use a `~gammapy.modeling.models.LogUniformPrior` for
the parameters that have a large amplitude range like the
`amplitude` parameter.
A `~gammapy.modeling.models.UniformPrior` means that the samples will be drawn with uniform
probability between the (min,max) values in the linear or log space
in the case of a `~gammapy.modeling.models.LogUniformPrior`.
.. GENERATED FROM PYTHON SOURCE LINES 162-169
.. code-block:: Python
model.spectral_model.amplitude.prior = LogUniformPrior(min=1e-12, max=1e-10)
model.spectral_model.index.prior = UniformPrior(min=1, max=5)
datasets.models = [model]
print(datasets.models)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.000 +/- 0.00
amplitude : 1.00e-12 +/- 0.0e+00 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
.. GENERATED FROM PYTHON SOURCE LINES 170-200
Defining the sampler and options
--------------------------------
As for the `~gammapy.modeling.Fit` object, the `~gammapy.modeling.Sampler` object can receive
different backend (although just one is available for now).
The `~gammapy.modeling.Sampler` comes with “reasonable” default parameters, but you can
change them via the `sampler_opts` dictionnary.
Here is a short description of the most relevant parameters that you
could change :
- `live_points`: minimum number of live points throughout the run.
More points allow to discover multiple peaks if existing, but is
slower. To test the Prior boundaries and for debugging, a lower
number (~100) can be used before a production run with more points
(~400 or more).
- `frac_remain`: the cut-off condition for the integration, set by the maximum
allowed fraction of posterior mass left in the live points vs the dead points. High
values (e.g., 0.5) are faster and can be used if the posterior
distribution is a relatively simple shape. A low value (1e-1, 1e-2)
is optimal for finding peaks, but slower.
- `log_dir`: directory where the output files will be stored.
If set to None, no files will be written. If set to a string, a directory
will be created containing the ongoing status of the run and final results.
For time consuming analysis, it is highly recommended to use that option to monitor
the run and restart it in case of a crash (with `resume=True`).
**Important note:** unlike the MCMC method, you don’t need to define the
number of steps for which the sampler will run. The algorithm will
automatically stop once a convergence criteria has been reached.
.. GENERATED FROM PYTHON SOURCE LINES 200-210
.. code-block:: Python
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
.. GENERATED FROM PYTHON SOURCE LINES 211-214
Next we can run the sampler on a given dataset.
No options are accepted in the run method.
.. GENERATED FROM PYTHON SOURCE LINES 214-218
.. code-block:: Python
result_joint = sampler.run(datasets)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.98) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************** ****** ***** ***********| +1.0e-10
Z=-inf(0.00%) | Like=-4842.51..-64.69 [-4842.5105..-355.2563] | it/evals=0/301 eff=0.0000% N=300
Z=-589.1(0.00%) | Like=-553.66..-64.69 [-4842.5105..-355.2563] | it/evals=21/322 eff=95.4545% N=300
Z=-552.2(0.00%) | Like=-545.22..-60.14 [-4842.5105..-355.2563] | it/evals=30/332 eff=93.7500% N=300
Z=-529.8(0.00%) | Like=-524.52..-60.14 [-4842.5105..-355.2563] | it/evals=49/354 eff=90.7407% N=300
Z=-512.7(0.00%) | Like=-506.60..-60.14 [-4842.5105..-355.2563] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.07) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************** ****** ***** ****** ***| +1.0e-10
Z=-503.4(0.00%) | Like=-498.00..-60.14 [-4842.5105..-355.2563] | it/evals=67/376 eff=88.1579% N=300
Z=-473.7(0.00%) | Like=-466.19..-59.69 [-4842.5105..-355.2563] | it/evals=88/398 eff=89.7959% N=300
Z=-469.8(0.00%) | Like=-463.29..-59.69 [-4842.5105..-355.2563] | it/evals=90/400 eff=90.0000% N=300
Z=-452.6(0.00%) | Like=-446.85..-59.69 [-4842.5105..-355.2563] | it/evals=109/424 eff=87.9032% N=300
Z=-440.7(0.00%) | Like=-435.30..-59.69 [-4842.5105..-355.2563] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.20) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************ ******* * *| +1.0e-10
Z=-431.3(0.00%) | Like=-423.86..-59.69 [-4842.5105..-355.2563] | it/evals=134/455 eff=86.4516% N=300
Z=-418.8(0.00%) | Like=-410.69..-59.69 [-4842.5105..-355.2563] | it/evals=150/472 eff=87.2093% N=300
Z=-401.0(0.00%) | Like=-393.78..-59.69 [-4842.5105..-355.2563] | it/evals=167/496 eff=85.2041% N=300
Z=-390.5(0.00%) | Like=-383.62..-59.69 [-4842.5105..-355.2563] | it/evals=180/512 eff=84.9057% N=300
Z=-369.6(0.00%) | Like=-361.14..-59.69 [-4842.5105..-355.2563] | it/evals=198/534 eff=84.6154% N=300
Mono-modal Volume: ~exp(-4.26) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ********************************************* *| +1.0e-10
Z=-365.7(0.00%) | Like=-358.08..-59.69 [-4842.5105..-355.2563] | it/evals=201/539 eff=84.1004% N=300
Z=-350.8(0.00%) | Like=-342.05..-59.69 [-354.1947..-200.5357] | it/evals=210/551 eff=83.6653% N=300
Z=-337.0(0.00%) | Like=-331.57..-59.69 [-354.1947..-200.5357] | it/evals=227/574 eff=82.8467% N=300
Z=-327.2(0.00%) | Like=-320.66..-59.69 [-354.1947..-200.5357] | it/evals=240/590 eff=82.7586% N=300
Z=-312.0(0.00%) | Like=-304.71..-59.69 [-354.1947..-200.5357] | it/evals=255/612 eff=81.7308% N=300
Mono-modal Volume: ~exp(-4.43) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************************** *| +1.0e-10
Z=-299.1(0.00%) | Like=-290.64..-59.69 [-354.1947..-200.5357] | it/evals=268/633 eff=80.4805% N=300
Z=-296.1(0.00%) | Like=-289.08..-59.69 [-354.1947..-200.5357] | it/evals=270/635 eff=80.5970% N=300
Z=-280.4(0.00%) | Like=-271.38..-59.69 [-354.1947..-200.5357] | it/evals=287/657 eff=80.3922% N=300
Z=-268.5(0.00%) | Like=-263.09..-59.69 [-354.1947..-200.5357] | it/evals=300/673 eff=80.4290% N=300
Z=-257.9(0.00%) | Like=-249.91..-59.69 [-354.1947..-200.5357] | it/evals=319/695 eff=80.7595% N=300
Z=-247.3(0.00%) | Like=-240.63..-59.69 [-354.1947..-200.5357] | it/evals=330/708 eff=80.8824% N=300
Mono-modal Volume: ~exp(-5.11) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ******************************************* *| +1.0e-10
Z=-245.0(0.00%) | Like=-239.45..-59.69 [-354.1947..-200.5357] | it/evals=335/714 eff=80.9179% N=300
Z=-231.3(0.00%) | Like=-224.69..-59.65 [-354.1947..-200.5357] | it/evals=354/735 eff=81.3793% N=300
Z=-225.6(0.00%) | Like=-219.37..-59.65 [-354.1947..-200.5357] | it/evals=360/742 eff=81.4480% N=300
Z=-221.3(0.00%) | Like=-215.19..-59.28 [-354.1947..-200.5357] | it/evals=368/764 eff=79.3103% N=300
Z=-211.5(0.00%) | Like=-205.61..-59.28 [-354.1947..-200.5357] | it/evals=387/785 eff=79.7938% N=300
Z=-210.6(0.00%) | Like=-204.05..-59.28 [-354.1947..-200.5357] | it/evals=390/792 eff=79.2683% N=300
Mono-modal Volume: ~exp(-5.11) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-205.9(0.00%) | Like=-199.71..-59.28 [-200.4350..-138.3325] | it/evals=403/815 eff=78.2524% N=300
Z=-198.9(0.00%) | Like=-192.93..-59.28 [-200.4350..-138.3325] | it/evals=420/835 eff=78.5047% N=300
Z=-193.4(0.00%) | Like=-187.07..-59.28 [-200.4350..-138.3325] | it/evals=438/857 eff=78.6355% N=300
Z=-189.3(0.00%) | Like=-183.50..-59.28 [-200.4350..-138.3325] | it/evals=450/878 eff=77.8547% N=300
Z=-184.7(0.00%) | Like=-178.53..-59.28 [-200.4350..-138.3325] | it/evals=466/900 eff=77.6667% N=300
Mono-modal Volume: ~exp(-5.11) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| ******************************** **********| +1.0e-10
Z=-181.0(0.00%) | Like=-173.72..-59.28 [-200.4350..-138.3325] | it/evals=480/919 eff=77.5444% N=300
Z=-174.5(0.00%) | Like=-168.46..-59.28 [-200.4350..-138.3325] | it/evals=494/941 eff=77.0671% N=300
Z=-172.5(0.00%) | Like=-167.09..-59.28 [-200.4350..-138.3325] | it/evals=506/963 eff=76.3198% N=300
Z=-171.9(0.00%) | Like=-166.28..-59.28 [-200.4350..-138.3325] | it/evals=510/970 eff=76.1194% N=300
Z=-167.0(0.00%) | Like=-160.45..-59.28 [-200.4350..-138.3325] | it/evals=526/992 eff=76.0116% N=300
Mono-modal Volume: ~exp(-5.29) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| ******************************* **********| +1.0e-10
Z=-164.5(0.00%) | Like=-158.47..-59.28 [-200.4350..-138.3325] | it/evals=536/1008 eff=75.7062% N=300
Z=-163.6(0.00%) | Like=-157.31..-59.28 [-200.4350..-138.3325] | it/evals=540/1012 eff=75.8427% N=300
Z=-159.5(0.00%) | Like=-152.18..-59.28 [-200.4350..-138.3325] | it/evals=553/1034 eff=75.3406% N=300
Z=-152.4(0.00%) | Like=-145.65..-59.28 [-200.4350..-138.3325] | it/evals=569/1056 eff=75.2646% N=300
Z=-152.0(0.00%) | Like=-145.62..-59.28 [-200.4350..-138.3325] | it/evals=570/1058 eff=75.1979% N=300
Z=-148.5(0.00%) | Like=-142.13..-59.28 [-200.4350..-138.3325] | it/evals=584/1080 eff=74.8718% N=300
Z=-145.1(0.00%) | Like=-138.99..-59.28 [-200.4350..-138.3325] | it/evals=600/1102 eff=74.8130% N=300
Mono-modal Volume: ~exp(-6.14) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| ************************************* | +1.0e-10
Z=-144.5(0.00%) | Like=-138.35..-59.28 [-200.4350..-138.3325] | it/evals=603/1105 eff=74.9068% N=300
Z=-141.0(0.00%) | Like=-134.18..-59.28 [-138.3150..-103.3803] | it/evals=621/1127 eff=75.0907% N=300
Z=-138.5(0.00%) | Like=-131.60..-59.28 [-138.3150..-103.3803] | it/evals=630/1138 eff=75.1790% N=300
Z=-135.9(0.00%) | Like=-129.68..-59.28 [-138.3150..-103.3803] | it/evals=648/1160 eff=75.3488% N=300
Z=-133.9(0.00%) | Like=-127.45..-59.28 [-138.3150..-103.3803] | it/evals=660/1178 eff=75.1708% N=300
Mono-modal Volume: ~exp(-6.33) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 ************************* +4.0 | +5.0
amplitude: +1.0e-12| ********************************** | +1.0e-10
Z=-131.9(0.00%) | Like=-125.39..-59.28 [-138.3150..-103.3803] | it/evals=670/1195 eff=74.8603% N=300
Z=-127.9(0.00%) | Like=-120.79..-59.28 [-138.3150..-103.3803] | it/evals=686/1217 eff=74.8092% N=300
Z=-126.8(0.00%) | Like=-120.51..-59.28 [-138.3150..-103.3803] | it/evals=690/1222 eff=74.8373% N=300
Z=-124.8(0.00%) | Like=-118.88..-59.28 [-138.3150..-103.3803] | it/evals=703/1245 eff=74.3915% N=300
Z=-121.1(0.00%) | Like=-114.97..-59.28 [-138.3150..-103.3803] | it/evals=720/1267 eff=74.4571% N=300
Mono-modal Volume: ~exp(-6.42) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.7 | +5.0
amplitude: +1.0e-12| ******************************** | +1.0e-10
Z=-119.0(0.00%) | Like=-112.73..-59.28 [-138.3150..-103.3803] | it/evals=737/1288 eff=74.5951% N=300
Z=-116.2(0.00%) | Like=-109.46..-59.28 [-138.3150..-103.3803] | it/evals=750/1308 eff=74.4048% N=300
Z=-113.4(0.00%) | Like=-107.16..-59.09 [-138.3150..-103.3803] | it/evals=769/1330 eff=74.6602% N=300
Z=-111.9(0.00%) | Like=-105.73..-59.09 [-138.3150..-103.3803] | it/evals=780/1341 eff=74.9280% N=300
Z=-109.4(0.00%) | Like=-102.69..-59.09 [-103.2797..-81.5359] | it/evals=799/1362 eff=75.2354% N=300
Mono-modal Volume: ~exp(-6.74) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| **************************** +7.6e-11 | +1.0e-10
Z=-108.5(0.00%) | Like=-102.05..-59.01 [-103.2797..-81.5359] | it/evals=804/1369 eff=75.2105% N=300
Z=-107.8(0.00%) | Like=-101.08..-59.01 [-103.2797..-81.5359] | it/evals=810/1375 eff=75.3488% N=300
Z=-105.9(0.00%) | Like=-99.60..-59.01 [-103.2797..-81.5359] | it/evals=825/1398 eff=75.1366% N=300
Z=-104.3(0.00%) | Like=-98.04..-59.01 [-103.2797..-81.5359] | it/evals=840/1417 eff=75.2014% N=300
Z=-102.2(0.00%) | Like=-95.77..-59.01 [-103.2797..-81.5359] | it/evals=858/1439 eff=75.3292% N=300
Z=-100.8(0.00%) | Like=-94.21..-59.01 [-103.2797..-81.5359] | it/evals=870/1461 eff=74.9354% N=300
Mono-modal Volume: ~exp(-6.74) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ************************* +7.3e-11 | +1.0e-10
Z=-98.4(0.00%) | Like=-91.91..-59.01 [-103.2797..-81.5359] | it/evals=887/1481 eff=75.1058% N=300
Z=-96.8(0.00%) | Like=-90.12..-59.01 [-103.2797..-81.5359] | it/evals=900/1499 eff=75.0626% N=300
Z=-94.7(0.00%) | Like=-87.96..-59.01 [-103.2797..-81.5359] | it/evals=914/1521 eff=74.8567% N=300
Z=-93.1(0.00%) | Like=-86.68..-59.01 [-103.2797..-81.5359] | it/evals=930/1542 eff=74.8792% N=300
Mono-modal Volume: ~exp(-7.09) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************** +7.0e-11 | +1.0e-10
Z=-92.5(0.00%) | Like=-85.93..-59.01 [-103.2797..-81.5359] | it/evals=938/1554 eff=74.8006% N=300
Z=-90.9(0.00%) | Like=-84.72..-59.01 [-103.2797..-81.5359] | it/evals=957/1575 eff=75.0588% N=300
Z=-90.7(0.00%) | Like=-84.21..-59.01 [-103.2797..-81.5359] | it/evals=960/1579 eff=75.0586% N=300
Z=-89.2(0.00%) | Like=-82.50..-59.01 [-103.2797..-81.5359] | it/evals=979/1600 eff=75.3077% N=300
Z=-88.2(0.00%) | Like=-81.67..-59.01 [-103.2797..-81.5359] | it/evals=990/1612 eff=75.4573% N=300
Mono-modal Volume: ~exp(-7.25) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******************** +6.7e-11 | +1.0e-10
Z=-87.1(0.00%) | Like=-80.49..-59.01 [-81.5161..-69.7248] | it/evals=1005/1628 eff=75.6777% N=300
Z=-86.1(0.00%) | Like=-79.64..-59.01 [-81.5161..-69.7248] | it/evals=1020/1644 eff=75.8929% N=300
Z=-85.0(0.00%) | Like=-78.67..-59.01 [-81.5161..-69.7248] | it/evals=1036/1667 eff=75.7864% N=300
Z=-84.1(0.00%) | Like=-77.60..-59.01 [-81.5161..-69.7248] | it/evals=1050/1684 eff=75.8671% N=300
Z=-83.2(0.00%) | Like=-76.47..-59.01 [-81.5161..-69.7248] | it/evals=1064/1706 eff=75.6757% N=300
Mono-modal Volume: ~exp(-7.55) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.4e-11 | +1.0e-10
Z=-82.5(0.00%) | Like=-75.89..-59.01 [-81.5161..-69.7248] | it/evals=1072/1727 eff=75.1226% N=300
Z=-82.0(0.00%) | Like=-75.32..-59.01 [-81.5161..-69.7248] | it/evals=1080/1738 eff=75.1043% N=300
Z=-81.0(0.00%) | Like=-74.70..-59.01 [-81.5161..-69.7248] | it/evals=1098/1759 eff=75.2570% N=300
Z=-80.4(0.00%) | Like=-73.70..-59.01 [-81.5161..-69.7248] | it/evals=1110/1773 eff=75.3564% N=300
Z=-79.7(0.00%) | Like=-73.02..-59.01 [-81.5161..-69.7248] | it/evals=1122/1795 eff=75.0502% N=300
Mono-modal Volume: ~exp(-7.58) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.0e-11 **************** +6.1e-11 | +1.0e-10
Z=-78.8(0.00%) | Like=-72.32..-59.01 [-81.5161..-69.7248] | it/evals=1139/1815 eff=75.1815% N=300
Z=-78.7(0.00%) | Like=-72.32..-59.01 [-81.5161..-69.7248] | it/evals=1140/1816 eff=75.1979% N=300
Z=-78.0(0.00%) | Like=-71.73..-59.01 [-81.5161..-69.7248] | it/evals=1157/1838 eff=75.2276% N=300
Z=-77.6(0.00%) | Like=-71.32..-58.89 [-81.5161..-69.7248] | it/evals=1170/1856 eff=75.1928% N=300
Z=-77.1(0.00%) | Like=-71.02..-58.89 [-81.5161..-69.7248] | it/evals=1186/1879 eff=75.1108% N=300
Z=-76.7(0.00%) | Like=-70.40..-58.89 [-81.5161..-69.7248] | it/evals=1200/1897 eff=75.1409% N=300
Mono-modal Volume: ~exp(-7.84) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************** +6.0e-11 | +1.0e-10
Z=-76.5(0.00%) | Like=-70.02..-58.89 [-81.5161..-69.7248] | it/evals=1206/1910 eff=74.9068% N=300
Z=-75.9(0.00%) | Like=-69.45..-58.89 [-69.7163..-65.7294] | it/evals=1223/1931 eff=74.9847% N=300
Z=-75.7(0.00%) | Like=-69.35..-58.89 [-69.7163..-65.7294] | it/evals=1230/1941 eff=74.9543% N=300
Z=-75.2(0.00%) | Like=-68.95..-58.89 [-69.7163..-65.7294] | it/evals=1245/1963 eff=74.8647% N=300
Z=-74.8(0.01%) | Like=-68.45..-58.89 [-69.7163..-65.7294] | it/evals=1260/1983 eff=74.8663% N=300
Mono-modal Volume: ~exp(-7.84) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.8e-11 | +1.0e-10
Z=-74.3(0.01%) | Like=-67.72..-58.89 [-69.7163..-65.7294] | it/evals=1277/2004 eff=74.9413% N=300
Z=-73.9(0.02%) | Like=-67.39..-58.89 [-69.7163..-65.7294] | it/evals=1290/2025 eff=74.7826% N=300
Z=-73.4(0.03%) | Like=-66.83..-58.89 [-69.7163..-65.7294] | it/evals=1305/2048 eff=74.6568% N=300
Z=-73.0(0.04%) | Like=-66.35..-58.89 [-69.7163..-65.7294] | it/evals=1320/2071 eff=74.5342% N=300
Z=-72.6(0.06%) | Like=-66.04..-58.89 [-69.7163..-65.7294] | it/evals=1334/2093 eff=74.4004% N=300
Mono-modal Volume: ~exp(-7.88) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************ +5.6e-11 | +1.0e-10
Z=-72.4(0.07%) | Like=-65.80..-58.89 [-69.7163..-65.7294] | it/evals=1340/2104 eff=74.2794% N=300
Z=-72.1(0.10%) | Like=-65.47..-58.89 [-65.7037..-65.2088] | it/evals=1350/2114 eff=74.4212% N=300
Z=-71.7(0.15%) | Like=-65.08..-58.89 [-65.1194..-65.0808] | it/evals=1366/2135 eff=74.4414% N=300
Z=-71.3(0.21%) | Like=-64.73..-58.80 [-64.7496..-64.7292] | it/evals=1379/2157 eff=74.2596% N=300
Z=-71.3(0.22%) | Like=-64.70..-58.80 [-64.7045..-64.7019]*| it/evals=1380/2158 eff=74.2734% N=300
Z=-70.9(0.32%) | Like=-64.36..-58.80 [-64.3574..-64.3482]*| it/evals=1397/2180 eff=74.3085% N=300
Mono-modal Volume: ~exp(-8.54) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +2.9 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********** +5.5e-11 | +1.0e-10
Z=-70.7(0.41%) | Like=-64.20..-58.80 [-64.2016..-64.1922]*| it/evals=1407/2194 eff=74.2872% N=300
Z=-70.6(0.43%) | Like=-64.16..-58.80 [-64.1604..-64.1534]*| it/evals=1410/2197 eff=74.3279% N=300
Z=-70.3(0.58%) | Like=-63.92..-58.80 [-63.9230..-63.9157]*| it/evals=1426/2219 eff=74.3095% N=300
Z=-70.1(0.74%) | Like=-63.62..-58.80 [-63.6409..-63.6233] | it/evals=1440/2236 eff=74.3802% N=300
Z=-69.8(1.02%) | Like=-63.33..-58.80 [-63.3424..-63.3251] | it/evals=1458/2258 eff=74.4637% N=300
Z=-69.6(1.21%) | Like=-63.08..-58.77 [-63.0769..-63.0671]*| it/evals=1470/2279 eff=74.2799% N=300
Mono-modal Volume: ~exp(-8.88) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.4 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ********* +5.4e-11 | +1.0e-10
Z=-69.5(1.28%) | Like=-62.97..-58.77 [-62.9696..-62.9658]*| it/evals=1474/2285 eff=74.2569% N=300
Z=-69.3(1.71%) | Like=-62.82..-58.77 [-62.8166..-62.8151]*| it/evals=1491/2306 eff=74.3270% N=300
Z=-69.1(1.89%) | Like=-62.71..-58.77 [-62.7532..-62.7148] | it/evals=1500/2317 eff=74.3679% N=300
Z=-69.0(2.29%) | Like=-62.53..-58.77 [-62.5690..-62.5299] | it/evals=1514/2340 eff=74.2157% N=300
Z=-68.8(2.68%) | Like=-62.35..-58.76 [-62.3667..-62.3527] | it/evals=1527/2362 eff=74.0543% N=300
Z=-68.7(2.78%) | Like=-62.33..-58.76 [-62.3405..-62.3297] | it/evals=1530/2366 eff=74.0561% N=300
Mono-modal Volume: ~exp(-9.29) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******** +5.3e-11 | +1.0e-10
Z=-68.6(3.15%) | Like=-62.22..-58.76 [-62.2355..-62.2199] | it/evals=1541/2381 eff=74.0509% N=300
Z=-68.4(3.90%) | Like=-61.96..-58.76 [-61.9593..-61.9587]*| it/evals=1560/2402 eff=74.2150% N=300
Z=-68.2(4.54%) | Like=-61.84..-58.76 [-61.8635..-61.8425] | it/evals=1574/2424 eff=74.1055% N=300
Z=-68.1(5.38%) | Like=-61.73..-58.76 [-61.7375..-61.7261] | it/evals=1590/2446 eff=74.0913% N=300
Z=-67.9(6.37%) | Like=-61.58..-58.75 [-61.6099..-61.5791] | it/evals=1607/2467 eff=74.1578% N=300
Mono-modal Volume: ~exp(-9.43) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******** +5.2e-11 | +1.0e-10
Z=-67.9(6.38%) | Like=-61.58..-58.75 [-61.5762..-61.5745]*| it/evals=1608/2468 eff=74.1697% N=300
Z=-67.8(7.05%) | Like=-61.44..-58.75 [-61.4585..-61.4413] | it/evals=1620/2481 eff=74.2779% N=300
Z=-67.6(8.53%) | Like=-61.24..-58.75 [-61.2448..-61.2328] | it/evals=1641/2502 eff=74.5232% N=300
Z=-67.6(9.26%) | Like=-61.18..-58.75 [-61.1947..-61.1823] | it/evals=1650/2512 eff=74.5931% N=300
Z=-67.4(10.39%) | Like=-61.01..-58.75 [-61.0279..-61.0083] | it/evals=1667/2534 eff=74.6195% N=300
Mono-modal Volume: ~exp(-9.47) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ****** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.4(10.96%) | Like=-60.94..-58.75 [-60.9416..-60.9412]*| it/evals=1675/2545 eff=74.6102% N=300
Z=-67.3(11.38%) | Like=-60.93..-58.75 [-60.9253..-60.9209]*| it/evals=1680/2550 eff=74.6667% N=300
Z=-67.2(12.76%) | Like=-60.77..-58.75 [-60.8034..-60.7690] | it/evals=1698/2572 eff=74.7359% N=300
Z=-67.1(13.89%) | Like=-60.71..-58.75 [-60.7073..-60.7054]*| it/evals=1710/2587 eff=74.7704% N=300
Z=-67.0(15.10%) | Like=-60.64..-58.75 [-60.6408..-60.6386]*| it/evals=1722/2609 eff=74.5777% N=300
Z=-66.9(16.91%) | Like=-60.54..-58.75 [-60.5419..-60.5181] | it/evals=1740/2630 eff=74.6781% N=300
Mono-modal Volume: ~exp(-9.75) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.9(17.15%) | Like=-60.51..-58.75 [-60.5105..-60.5042]*| it/evals=1742/2632 eff=74.6998% N=300
Z=-66.8(19.02%) | Like=-60.41..-58.75 [-60.4239..-60.4088] | it/evals=1760/2653 eff=74.7981% N=300
Z=-66.8(19.97%) | Like=-60.34..-58.75 [-60.3436..-60.3388]*| it/evals=1770/2663 eff=74.9048% N=300
Z=-66.7(21.76%) | Like=-60.26..-58.75 [-60.2646..-60.2595]*| it/evals=1787/2684 eff=74.9581% N=300
Z=-66.6(23.16%) | Like=-60.21..-58.75 [-60.2149..-60.2146]*| it/evals=1800/2703 eff=74.9064% N=300
Mono-modal Volume: ~exp(-10.33) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.6(24.11%) | Like=-60.16..-58.75 [-60.1678..-60.1566] | it/evals=1809/2716 eff=74.8758% N=300
Z=-66.5(25.83%) | Like=-60.06..-58.75 [-60.0857..-60.0600] | it/evals=1824/2739 eff=74.7847% N=300
Z=-66.5(26.60%) | Like=-60.04..-58.75 [-60.0387..-60.0331]*| it/evals=1830/2746 eff=74.8160% N=300
Z=-66.4(28.88%) | Like=-59.97..-58.75 [-59.9653..-59.9577]*| it/evals=1848/2768 eff=74.8784% N=300
Z=-66.3(30.25%) | Like=-59.91..-58.75 [-59.9095..-59.9086]*| it/evals=1860/2784 eff=74.8792% N=300
Z=-66.3(31.98%) | Like=-59.86..-58.75 [-59.8643..-59.8621]*| it/evals=1874/2806 eff=74.7805% N=300
Mono-modal Volume: ~exp(-10.47) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.3(32.27%) | Like=-59.85..-58.75 [-59.8542..-59.8509]*| it/evals=1876/2808 eff=74.8006% N=300
Z=-66.2(33.95%) | Like=-59.82..-58.75 [-59.8210..-59.8198]*| it/evals=1890/2826 eff=74.8219% N=300
Z=-66.1(36.19%) | Like=-59.76..-58.75 [-59.7646..-59.7549]*| it/evals=1910/2847 eff=74.9902% N=300
Z=-66.1(37.39%) | Like=-59.71..-58.75 [-59.7125..-59.7061]*| it/evals=1920/2859 eff=75.0293% N=300
Z=-66.1(39.56%) | Like=-59.67..-58.75 [-59.6746..-59.6733]*| it/evals=1937/2883 eff=74.9903% N=300
Mono-modal Volume: ~exp(-10.47) Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ***** +4.9e-11 | +1.0e-10
Z=-66.0(41.09%) | Like=-59.64..-58.75 [-59.6434..-59.6422]*| it/evals=1950/2899 eff=75.0289% N=300
Z=-66.0(42.65%) | Like=-59.61..-58.75 [-59.6073..-59.6066]*| it/evals=1963/2921 eff=74.8951% N=300
Z=-65.9(44.44%) | Like=-59.56..-58.75 [-59.5646..-59.5638]*| it/evals=1978/2943 eff=74.8392% N=300
Z=-65.9(44.69%) | Like=-59.56..-58.75 [-59.5638..-59.5466] | it/evals=1980/2945 eff=74.8582% N=300
Z=-65.9(46.57%) | Like=-59.49..-58.75 [-59.4906..-59.4902]*| it/evals=1995/2968 eff=74.7751% N=300
Z=-65.9(48.08%) | Like=-59.47..-58.75 [-59.4708..-59.4702]*| it/evals=2008/2990 eff=74.6468% N=300
Mono-modal Volume: ~exp(-10.64) * Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.9(48.36%) | Like=-59.47..-58.75 [-59.4669..-59.4644]*| it/evals=2010/2992 eff=74.6657% N=300
Z=-65.8(50.14%) | Like=-59.44..-58.75 [-59.4402..-59.4365]*| it/evals=2025/3014 eff=74.6131% N=300
Z=-65.8(51.83%) | Like=-59.38..-58.75 [-59.3775..-59.3710]*| it/evals=2040/3035 eff=74.5887% N=300
Z=-65.8(53.49%) | Like=-59.34..-58.75 [-59.3386..-59.3378]*| it/evals=2055/3057 eff=74.5375% N=300
Z=-65.7(55.19%) | Like=-59.31..-58.75 [-59.3141..-59.3114]*| it/evals=2070/3082 eff=74.4069% N=300
Mono-modal Volume: ~exp(-11.27) * Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.7(55.88%) | Like=-59.30..-58.75 [-59.2981..-59.2957]*| it/evals=2077/3092 eff=74.3911% N=300
Z=-65.7(57.96%) | Like=-59.27..-58.75 [-59.2735..-59.2710]*| it/evals=2095/3113 eff=74.4756% N=300
Z=-65.7(58.47%) | Like=-59.26..-58.75 [-59.2636..-59.2626]*| it/evals=2100/3118 eff=74.5209% N=300
Z=-65.6(60.22%) | Like=-59.24..-58.75 [-59.2436..-59.2432]*| it/evals=2117/3139 eff=74.5685% N=300
Z=-65.6(61.55%) | Like=-59.22..-58.75 [-59.2207..-59.2203]*| it/evals=2130/3157 eff=74.5537% N=300
Mono-modal Volume: ~exp(-11.27) Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.6(63.01%) | Like=-59.21..-58.75 [-59.2058..-59.2056]*| it/evals=2145/3177 eff=74.5568% N=300
Z=-65.6(64.42%) | Like=-59.18..-58.75 [-59.1772..-59.1760]*| it/evals=2160/3195 eff=74.6114% N=300
Z=-65.5(65.98%) | Like=-59.15..-58.75 [-59.1544..-59.1528]*| it/evals=2176/3216 eff=74.6228% N=300
Z=-65.5(67.20%) | Like=-59.14..-58.75 [-59.1391..-59.1378]*| it/evals=2190/3237 eff=74.5659% N=300
Z=-65.5(68.51%) | Like=-59.12..-58.75 [-59.1186..-59.1171]*| it/evals=2205/3259 eff=74.5184% N=300
Mono-modal Volume: ~exp(-11.49) * Expected Volume: exp(-7.37) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.5(68.99%) | Like=-59.11..-58.75 [-59.1149..-59.1149]*| it/evals=2211/3270 eff=74.4444% N=300
Z=-65.5(69.77%) | Like=-59.10..-58.75 [-59.1011..-59.0976]*| it/evals=2220/3282 eff=74.4467% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3285
[ultranest] logZ = -65.11 +- 0.1001
[ultranest] Effective samples strategy satisfied (ESS = 993.1, 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=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.13 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.123 +- 0.336
single instance: logZ = -65.123 +- 0.134
bootstrapped : logZ = -65.107 +- 0.210
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.414 │ ▁▁▁▁▁▂▂▂▃▃▄▄▆▅▅▇▆▆▇▅▅▄▄▄▄▃▂▁▁▁▁▁▁▁▁▁▁ │2.974 2.671 +- 0.086
amplitude : 0.0000000000329│ ▁ ▁ ▁▁▁▁▂▃▃▂▄▅▆▇▇▆▇▇▆▇▅▅▄▄▂▂▁▁▁▁▁▁ ▁▁ │0.0000000000559 0.0000000000444 +- 0.0000000000030
.. GENERATED FROM PYTHON SOURCE LINES 219-270
Understanding the outputs
-------------------------
In the Jupyter notebook, you should be able to see an interactive
visualisation of how the parameter space shrinks which starts from the
(min,max) shrinks down towards the optimal parameters.
The output above is filled with interesting information. Here we
provide a short description of the most relevant information provided
above.
For more detailed information see the `UltraNest
docs <https://johannesbuchner.github.io/UltraNest/issues.html#what-does-the-status-line-mean>`__.
**During the sampling**
`Z=-68.8(0.53%) | Like=-63.96..-58.75 [-63.9570..-63.9539]*| it/evals=640/1068 eff=73.7327% N=300`
Some important information here is:
- Progress (0.53%): the completed fraction of the integral. This is not a time progress bar.
Stays at zero for a good fraction of the run.
- Efficiency (eff value) of the sampling: this indicates out of the proposed new points,
how many were accepted. If your efficiency is too small (<<1%), maybe
you should revise your priors (e.g use a LogUniform prior for the
normalisation).
**Final outputs**
The final lines indicate that all three “convergence” strategies are
satisfied (samples, posterior uncertainty, and evidence uncertainty).
`logZ = -65.104 +- 0.292`
The main goal of the Nested sampling algorithm is to estimate Z (the
Bayesian evidence) which is given above together with an uncertainty.
In a similar way to deltaLogLike and deltaAIC, deltaLogZ values can be
used for model comparison.
For more information see : `on the use of the evidence for model comparison
<https://ned.ipac.caltech.edu/level5/Sept13/Trotta/Trotta4.html>`__.
An interesting comparison of the efficiency and false discovery rate of
model selection with deltaLogLike and deltaLogZ is given in Appendix C of
`Buchner et al., 2014 <https://ui.adsabs.harvard.edu/abs/2014A%2526A...564A.125B%252F/>`__.
**Results stored on disk**
if `log_dir` is set to a name where the results will be stored, then
a directory is created containing many useful results and plots.
A description of these outputs is given in the `Ultranest
docs <https://johannesbuchner.github.io/UltraNest/performance.html#output-files>`__.
.. GENERATED FROM PYTHON SOURCE LINES 273-276
Results
-------
.. GENERATED FROM PYTHON SOURCE LINES 279-294
Within a Bayesian analysis, the concept of best-fit has to be viewed
differently from what is done in a gradient descent fit.
The output of the Bayesian analysis is the posterior distribution and
there is no “best-fit” output.
One has to define, based on the posteriors, what we want to consider
as “best-fit” and several options are possible:
- the mean of the distribution
- the median
- the lowest likelihood value
By default the `~gammapy.modeling.models.DatasetModels` will be updated with the `mean` of
the posterior distributions.
.. GENERATED FROM PYTHON SOURCE LINES 294-298
.. code-block:: Python
print(result_joint.models)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.671 +/- 0.09
amplitude : 4.44e-11 +/- 3.0e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
.. GENERATED FROM PYTHON SOURCE LINES 299-303
The `~gammapy.modeling.Sampler` class returns a very rich dictionnary.
The most “standard” information about the posterior distributions can
be found in :
.. GENERATED FROM PYTHON SOURCE LINES 303-307
.. code-block:: Python
print(result_joint.sampler_results["posterior"])
.. rst-class:: sphx-glr-script-out
.. code-block:: none
{'mean': [2.671437963604333, 4.4379999340797087e-11], 'stdev': [0.08632788455158054, 3.0230326072490516e-12], 'median': [2.6701716063531142, 4.4274342516343e-11], 'errlo': [2.582916276593124, 4.137212345350785e-11], 'errup': [2.7590040954310164, 4.741342138489821e-11], 'information_gain_bits': [2.7137192768396257, 3.1240113731335293]}
.. GENERATED FROM PYTHON SOURCE LINES 308-319
Besides mean, errors, etc, an interesting value is the
`information gain` which estimates how much the posterior
distribution has shrinked with respect to the prior (i.e. how much
we’ve learned). A value < 1 means that the parameter is poorly
constrained within the prior range (we haven't learned much with respect to our prior assumption).
For a physical example see this
`example <https://arxiv.org/abs/2205.00009>`__.
The `~gammapy.modeling.SamplerResult` dictionary contains also other interesting
information :
.. GENERATED FROM PYTHON SOURCE LINES 319-323
.. code-block:: Python
print(result_joint.sampler_results.keys())
.. rst-class:: sphx-glr-script-out
.. code-block:: none
dict_keys(['niter', 'logz', 'logzerr', 'logz_bs', 'logz_single', 'logzerr_tail', 'logzerr_bs', 'ess', 'H', 'Herr', 'posterior', 'weighted_samples', 'samples', 'maximum_likelihood', 'ncall', 'paramnames', 'logzerr_single', 'insertion_order_MWW_test'])
.. GENERATED FROM PYTHON SOURCE LINES 324-327
Of particular interest, the samples used in the process to approximate
the posterior distribution can be accessed via :
.. GENERATED FROM PYTHON SOURCE LINES 327-337
.. code-block:: Python
for i, n in enumerate(model.parameters.free_parameters.names):
s = result_joint.samples[:, i]
fig, ax = plt.subplots()
ax.hist(s, bins=30)
ax.axvline(np.mean(s), ls="--", color="red")
ax.set_xlabel(n)
plt.show()
.. rst-class:: sphx-glr-horizontal
*
.. image-sg:: /tutorials/api/images/sphx_glr_nested_sampling_Crab_001.png
:alt: nested sampling Crab
:srcset: /tutorials/api/images/sphx_glr_nested_sampling_Crab_001.png
:class: sphx-glr-multi-img
*
.. image-sg:: /tutorials/api/images/sphx_glr_nested_sampling_Crab_002.png
:alt: nested sampling Crab
:srcset: /tutorials/api/images/sphx_glr_nested_sampling_Crab_002.png
:class: sphx-glr-multi-img
.. GENERATED FROM PYTHON SOURCE LINES 338-346
While the above plots are interesting, the real strength of the Bayesian
analysis is to visualise all parameters correlations which is usually
done using “corner plots”.
Ultranest corner plot function is a wrapper around the `corner
<https://corner.readthedocs.io/en/latest/api>`__ package.
See the above link for optional keywords.
Other packages exist for corner plots, like
`chainconsumer <https://samreay.github.io/ChainConsumer>`__ which is discussed later in this tutorial.
.. GENERATED FROM PYTHON SOURCE LINES 346-361
.. code-block:: Python
from ultranest.plot import cornerplot
cornerplot(
result_joint.sampler_results,
plot_datapoints=True,
plot_density=True,
bins=20,
title_fmt=".2e",
smooth=False,
)
plt.show()
.. image-sg:: /tutorials/api/images/sphx_glr_nested_sampling_Crab_003.png
:alt: index = ${2.67e+00}_{-8.71e-02}^{+8.88e-02}$, amplitude = ${4.43e-11}_{-2.90e-12}^{+3.13e-12}$
:srcset: /tutorials/api/images/sphx_glr_nested_sampling_Crab_003.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 363-369
Individual run analysis
-----------------------
Now we’ll analyse several Crab runs individually so that we can compare
them.
.. GENERATED FROM PYTHON SOURCE LINES 369-375
.. code-block:: Python
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
.. rst-class:: sphx-glr-script-out
.. code-block:: none
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.18) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************* ******** ******** ********| +1.0e-10
Z=-inf(0.00%) | Like=-1231.24..-21.10 [-1231.2416..-109.7661] | it/evals=0/301 eff=0.0000% N=300
Z=-175.3(0.00%) | Like=-170.50..-21.10 [-1231.2416..-109.7661] | it/evals=30/335 eff=85.7143% N=300
Z=-165.2(0.00%) | Like=-160.54..-21.10 [-1231.2416..-109.7661] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.18) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*************************** ** *** **** ***** *| +1.0e-10
Z=-155.8(0.00%) | Like=-151.14..-21.10 [-1231.2416..-109.7661] | it/evals=90/400 eff=90.0000% N=300
Z=-144.5(0.00%) | Like=-139.33..-21.10 [-1231.2416..-109.7661] | it/evals=120/438 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.48) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|*************************** ** *** ** * ***** *| +1.0e-10
Z=-140.0(0.00%) | Like=-134.86..-21.10 [-1231.2416..-109.7661] | it/evals=134/453 eff=87.5817% N=300
Z=-133.6(0.00%) | Like=-128.29..-21.10 [-1231.2416..-109.7661] | it/evals=150/472 eff=87.2093% N=300
Z=-122.6(0.00%) | Like=-117.20..-21.10 [-1231.2416..-109.7661] | it/evals=180/508 eff=86.5385% N=300
Mono-modal Volume: ~exp(-4.48) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ***************************** *** ** * ***** *| +1.0e-10
Z=-113.4(0.00%) | Like=-107.69..-21.10 [-109.3278..-69.7998] | it/evals=202/545 eff=82.4490% N=300
Z=-110.1(0.00%) | Like=-104.37..-21.10 [-109.3278..-69.7998] | it/evals=210/556 eff=82.0312% N=300
Z=-102.2(0.00%) | Like=-97.07..-21.10 [-109.3278..-69.7998] | it/evals=240/596 eff=81.0811% N=300
Z=-95.1(0.00%) | Like=-89.85..-21.10 [-109.3278..-69.7998] | it/evals=266/637 eff=78.9318% N=300
Mono-modal Volume: ~exp(-4.79) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************** **** ***** | +1.0e-10
Z=-94.6(0.00%) | Like=-89.05..-21.10 [-109.3278..-69.7998] | it/evals=268/639 eff=79.0560% N=300
Z=-94.1(0.00%) | Like=-88.85..-21.10 [-109.3278..-69.7998] | it/evals=270/642 eff=78.9474% N=300
Z=-87.1(0.00%) | Like=-81.86..-21.10 [-109.3278..-69.7998] | it/evals=300/678 eff=79.3651% N=300
Z=-81.8(0.00%) | Like=-77.09..-21.10 [-109.3278..-69.7998] | it/evals=330/718 eff=78.9474% N=300
Mono-modal Volume: ~exp(-4.79) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ******************************* **** ** | +1.0e-10
Z=-79.1(0.00%) | Like=-74.39..-21.10 [-109.3278..-69.7998] | it/evals=350/761 eff=75.9219% N=300
Z=-78.1(0.00%) | Like=-73.41..-21.10 [-109.3278..-69.7998] | it/evals=360/772 eff=76.2712% N=300
Z=-76.3(0.00%) | Like=-71.72..-21.10 [-109.3278..-69.7998] | it/evals=382/813 eff=74.4639% N=300
Z=-75.6(0.00%) | Like=-71.09..-21.10 [-109.3278..-69.7998] | it/evals=390/823 eff=74.5698% N=300
Mono-modal Volume: ~exp(-4.79) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** * | +5.0
amplitude: +1.0e-12| ******************************* ****** | +1.0e-10
Z=-73.1(0.00%) | Like=-67.86..-21.10 [-69.2927..-47.6463] | it/evals=414/861 eff=73.7968% N=300
Z=-72.3(0.00%) | Like=-67.34..-21.10 [-69.2927..-47.6463] | it/evals=420/870 eff=73.6842% N=300
Z=-70.1(0.00%) | Like=-65.01..-20.88 [-69.2927..-47.6463] | it/evals=439/912 eff=71.7320% N=300
Z=-68.9(0.00%) | Like=-63.83..-20.88 [-69.2927..-47.6463] | it/evals=450/935 eff=70.8661% N=300
Mono-modal Volume: ~exp(-5.12) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ****************************** *** +7.9e-11| +1.0e-10
Z=-67.0(0.00%) | Like=-61.99..-20.88 [-69.2927..-47.6463] | it/evals=469/966 eff=70.4204% N=300
Z=-66.0(0.00%) | Like=-61.01..-20.88 [-69.2927..-47.6463] | it/evals=480/980 eff=70.5882% N=300
Z=-63.5(0.00%) | Like=-58.25..-20.88 [-69.2927..-47.6463] | it/evals=505/1022 eff=69.9446% N=300
Z=-63.0(0.00%) | Like=-58.12..-20.88 [-69.2927..-47.6463] | it/evals=510/1031 eff=69.7674% N=300
Mono-modal Volume: ~exp(-5.87) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| **************************** +4.0 | +5.0
amplitude: +1.0e-12| ******************************** +7.5e-11 | +1.0e-10
Z=-61.1(0.00%) | Like=-56.11..-20.54 [-69.2927..-47.6463] | it/evals=536/1069 eff=69.7009% N=300
Z=-60.7(0.00%) | Like=-55.35..-20.54 [-69.2927..-47.6463] | it/evals=540/1075 eff=69.6774% N=300
Z=-57.6(0.00%) | Like=-52.27..-20.54 [-69.2927..-47.6463] | it/evals=567/1115 eff=69.5706% N=300
Z=-57.3(0.00%) | Like=-52.14..-20.54 [-69.2927..-47.6463] | it/evals=570/1120 eff=69.5122% N=300
Z=-55.5(0.00%) | Like=-50.41..-20.54 [-69.2927..-47.6463] | it/evals=594/1160 eff=69.0698% N=300
Z=-55.0(0.00%) | Like=-49.91..-20.54 [-69.2927..-47.6463] | it/evals=600/1168 eff=69.1244% N=300
Mono-modal Volume: ~exp(-6.12) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +3.8 | +5.0
amplitude: +1.0e-12| ************************** * +7.1e-11 | +1.0e-10
Z=-54.8(0.00%) | Like=-49.69..-20.54 [-69.2927..-47.6463] | it/evals=603/1174 eff=68.9931% N=300
Z=-52.9(0.00%) | Like=-47.74..-20.54 [-69.2927..-47.6463] | it/evals=630/1208 eff=69.3833% N=300
Z=-51.1(0.00%) | Like=-45.84..-20.54 [-47.6004..-34.7508] | it/evals=660/1246 eff=69.7674% N=300
Mono-modal Volume: ~exp(-6.12) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.7 | +5.0
amplitude: +1.0e-12| ************************* +6.4e-11 | +1.0e-10
Z=-49.0(0.00%) | Like=-43.63..-20.47 [-47.6004..-34.7508] | it/evals=687/1284 eff=69.8171% N=300
Z=-48.8(0.00%) | Like=-43.58..-20.47 [-47.6004..-34.7508] | it/evals=690/1287 eff=69.9088% N=300
Z=-47.5(0.00%) | Like=-42.26..-20.47 [-47.6004..-34.7508] | it/evals=713/1328 eff=69.3580% N=300
Z=-47.0(0.00%) | Like=-41.78..-20.47 [-47.6004..-34.7508] | it/evals=720/1336 eff=69.4981% N=300
Mono-modal Volume: ~exp(-6.57) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************* +3.5 | +5.0
amplitude: +1.0e-12| ************************ +6.1e-11 | +1.0e-10
Z=-45.9(0.00%) | Like=-40.43..-20.47 [-47.6004..-34.7508] | it/evals=737/1359 eff=69.5940% N=300
Z=-45.0(0.00%) | Like=-39.47..-20.47 [-47.6004..-34.7508] | it/evals=750/1375 eff=69.7674% N=300
Z=-42.9(0.00%) | Like=-37.36..-20.47 [-47.6004..-34.7508] | it/evals=778/1414 eff=69.8384% N=300
Z=-42.7(0.00%) | Like=-37.23..-20.47 [-47.6004..-34.7508] | it/evals=780/1416 eff=69.8925% N=300
Mono-modal Volume: ~exp(-6.73) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************* +5.9e-11 | +1.0e-10
Z=-41.4(0.00%) | Like=-36.01..-20.47 [-47.6004..-34.7508] | it/evals=804/1448 eff=70.0348% N=300
Z=-41.1(0.00%) | Like=-35.73..-20.47 [-47.6004..-34.7508] | it/evals=810/1454 eff=70.1906% N=300
Z=-39.7(0.00%) | Like=-34.41..-20.47 [-34.7450..-27.8260] | it/evals=840/1493 eff=70.4107% N=300
Z=-38.4(0.00%) | Like=-32.95..-20.47 [-34.7450..-27.8260] | it/evals=868/1533 eff=70.3974% N=300
Z=-38.4(0.00%) | Like=-32.89..-20.47 [-34.7450..-27.8260] | it/evals=870/1535 eff=70.4453% N=300
Mono-modal Volume: ~exp(-7.04) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ******************* +5.6e-11 | +1.0e-10
Z=-38.3(0.00%) | Like=-32.82..-20.47 [-34.7450..-27.8260] | it/evals=871/1536 eff=70.4693% N=300
Z=-37.1(0.00%) | Like=-31.71..-20.47 [-34.7450..-27.8260] | it/evals=900/1576 eff=70.5329% N=300
Z=-36.2(0.00%) | Like=-30.89..-20.47 [-34.7450..-27.8260] | it/evals=927/1617 eff=70.3872% N=300
Z=-36.1(0.00%) | Like=-30.82..-20.47 [-34.7450..-27.8260] | it/evals=930/1623 eff=70.2948% N=300
Mono-modal Volume: ~exp(-7.22) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 *************** +3.2 | +5.0
amplitude: +1.0e-12| ***************** +5.4e-11 | +1.0e-10
Z=-35.9(0.00%) | Like=-30.50..-20.47 [-34.7450..-27.8260] | it/evals=938/1637 eff=70.1571% N=300
Z=-35.2(0.01%) | Like=-29.69..-20.47 [-34.7450..-27.8260] | it/evals=960/1662 eff=70.4846% N=300
Z=-34.3(0.02%) | Like=-28.83..-20.47 [-34.7450..-27.8260] | it/evals=987/1701 eff=70.4497% N=300
Z=-34.2(0.02%) | Like=-28.77..-20.47 [-34.7450..-27.8260] | it/evals=990/1708 eff=70.3125% N=300
Mono-modal Volume: ~exp(-7.22) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| **************** +5.1e-11 | +1.0e-10
Z=-33.6(0.04%) | Like=-28.14..-20.47 [-34.7450..-27.8260] | it/evals=1014/1744 eff=70.2216% N=300
Z=-33.4(0.05%) | Like=-28.05..-20.47 [-34.7450..-27.8260] | it/evals=1020/1757 eff=70.0069% N=300
Z=-32.7(0.10%) | Like=-27.43..-20.47 [-27.8217..-26.9374] | it/evals=1050/1797 eff=70.1403% N=300
Mono-modal Volume: ~exp(-7.37) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************ +3.1 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-32.3(0.16%) | Like=-27.07..-20.47 [-27.8217..-26.9374] | it/evals=1072/1832 eff=69.9739% N=300
Z=-32.2(0.18%) | Like=-26.85..-20.47 [-26.9209..-26.7991] | it/evals=1080/1843 eff=69.9935% N=300
Z=-31.7(0.31%) | Like=-26.24..-20.47 [-26.2745..-26.2446] | it/evals=1107/1883 eff=69.9305% N=300
Z=-31.6(0.33%) | Like=-26.15..-20.47 [-26.1500..-26.1348] | it/evals=1110/1886 eff=69.9874% N=300
Z=-31.1(0.53%) | Like=-25.71..-20.47 [-25.7096..-25.6921] | it/evals=1137/1925 eff=69.9692% N=300
Mono-modal Volume: ~exp(-7.37) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 *********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-31.1(0.55%) | Like=-25.69..-20.47 [-25.6864..-25.6837]*| it/evals=1140/1929 eff=69.9816% N=300
Z=-30.7(0.82%) | Like=-25.26..-20.47 [-25.2750..-25.2647] | it/evals=1164/1969 eff=69.7424% N=300
Z=-30.6(0.88%) | Like=-25.18..-20.47 [-25.2018..-25.1821] | it/evals=1170/1980 eff=69.6429% N=300
Z=-30.2(1.25%) | Like=-24.81..-20.47 [-24.8100..-24.7928] | it/evals=1195/2023 eff=69.3558% N=300
Z=-30.2(1.32%) | Like=-24.73..-20.47 [-24.7417..-24.7254] | it/evals=1200/2032 eff=69.2841% N=300
Mono-modal Volume: ~exp(-8.25) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +2.9 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********** +4.6e-11 | +1.0e-10
Z=-30.1(1.44%) | Like=-24.70..-20.47 [-24.7084..-24.6974] | it/evals=1206/2041 eff=69.2705% N=300
Z=-29.8(1.98%) | Like=-24.36..-20.47 [-24.4037..-24.3579] | it/evals=1230/2066 eff=69.6489% N=300
Z=-29.4(2.76%) | Like=-24.02..-20.47 [-24.0171..-24.0116]*| it/evals=1260/2105 eff=69.8061% N=300
Mono-modal Volume: ~exp(-8.25) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-29.2(3.67%) | Like=-23.74..-20.47 [-23.7351..-23.7178] | it/evals=1286/2142 eff=69.8154% N=300
Z=-29.1(3.84%) | Like=-23.69..-20.47 [-23.7042..-23.6934] | it/evals=1290/2146 eff=69.8808% N=300
Z=-28.9(4.84%) | Like=-23.41..-20.47 [-23.4064..-23.4061]*| it/evals=1314/2186 eff=69.6713% N=300
Z=-28.8(5.15%) | Like=-23.37..-20.47 [-23.3684..-23.3473] | it/evals=1320/2193 eff=69.7306% N=300
Mono-modal Volume: ~exp(-8.30) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.6(6.19%) | Like=-23.11..-20.47 [-23.1251..-23.1086] | it/evals=1340/2221 eff=69.7553% N=300
Z=-28.5(6.87%) | Like=-23.02..-20.47 [-23.0249..-23.0222]*| it/evals=1350/2233 eff=69.8396% N=300
Z=-28.3(9.10%) | Like=-22.80..-20.47 [-22.8011..-22.7963]*| it/evals=1380/2271 eff=70.0152% N=300
Mono-modal Volume: ~exp(-8.30) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.2e-11 | +1.0e-10
Z=-28.1(11.06%) | Like=-22.64..-20.47 [-22.6375..-22.6305]*| it/evals=1407/2309 eff=70.0348% N=300
Z=-28.1(11.31%) | Like=-22.63..-20.47 [-22.6282..-22.6237]*| it/evals=1410/2313 eff=70.0447% N=300
Z=-27.9(13.64%) | Like=-22.37..-20.47 [-22.3749..-22.3726]*| it/evals=1438/2355 eff=69.9757% N=300
Z=-27.9(13.86%) | Like=-22.36..-20.47 [-22.3630..-22.3600]*| it/evals=1440/2358 eff=69.9708% N=300
Z=-27.7(16.46%) | Like=-22.18..-20.47 [-22.1841..-22.1765]*| it/evals=1466/2397 eff=69.9094% N=300
Z=-27.7(16.89%) | Like=-22.16..-20.47 [-22.1612..-22.1318] | it/evals=1470/2405 eff=69.8337% N=300
Mono-modal Volume: ~exp(-8.47) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.1e-11 | +1.0e-10
Z=-27.6(17.38%) | Like=-22.10..-20.47 [-22.1014..-22.0852] | it/evals=1474/2409 eff=69.8909% N=300
Z=-27.5(20.22%) | Like=-21.94..-20.47 [-21.9372..-21.9233] | it/evals=1500/2444 eff=69.9627% N=300
Z=-27.3(23.37%) | Like=-21.81..-20.47 [-21.8090..-21.8052]*| it/evals=1526/2483 eff=69.9038% N=300
Z=-27.3(23.77%) | Like=-21.80..-20.46 [-21.8019..-21.8005]*| it/evals=1530/2490 eff=69.8630% N=300
Mono-modal Volume: ~exp(-9.08) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******* +4.1e-11 | +1.0e-10
Z=-27.3(25.11%) | Like=-21.75..-20.46 [-21.7496..-21.7478]*| it/evals=1541/2504 eff=69.9183% N=300
Z=-27.2(27.46%) | Like=-21.65..-20.46 [-21.6641..-21.6508] | it/evals=1560/2526 eff=70.0809% N=300
Z=-27.1(30.72%) | Like=-21.50..-20.46 [-21.5131..-21.4988] | it/evals=1585/2566 eff=69.9470% N=300
Z=-27.0(31.35%) | Like=-21.49..-20.46 [-21.4854..-21.4850]*| it/evals=1590/2571 eff=70.0132% N=300
Mono-modal Volume: ~exp(-9.75) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.0(33.81%) | Like=-21.43..-20.46 [-21.4347..-21.4327]*| it/evals=1608/2597 eff=70.0044% N=300
Z=-26.9(35.44%) | Like=-21.39..-20.46 [-21.3941..-21.3926]*| it/evals=1620/2610 eff=70.1299% N=300
Z=-26.8(39.03%) | Like=-21.30..-20.46 [-21.3033..-21.3025]*| it/evals=1649/2648 eff=70.2300% N=300
Z=-26.8(39.13%) | Like=-21.30..-20.46 [-21.3025..-21.2990]*| it/evals=1650/2650 eff=70.2128% N=300
Mono-modal Volume: ~exp(-9.87) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.7(42.38%) | Like=-21.22..-20.46 [-21.2216..-21.2189]*| it/evals=1675/2679 eff=70.4077% N=300
Z=-26.7(42.96%) | Like=-21.20..-20.46 [-21.2033..-21.2026]*| it/evals=1680/2685 eff=70.4403% N=300
Z=-26.6(46.56%) | Like=-21.15..-20.46 [-21.1480..-21.1472]*| it/evals=1710/2720 eff=70.6612% N=300
Z=-26.6(50.00%) | Like=-21.10..-20.46 [-21.1046..-21.1038]*| it/evals=1738/2761 eff=70.6217% N=300
Z=-26.6(50.24%) | Like=-21.10..-20.46 [-21.1022..-21.0995]*| it/evals=1740/2764 eff=70.6169% N=300
Mono-modal Volume: ~exp(-9.91) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***** +3.9e-11 | +1.0e-10
Z=-26.6(50.47%) | Like=-21.10..-20.46 [-21.0992..-21.0977]*| it/evals=1742/2766 eff=70.6407% N=300
Z=-26.5(53.57%) | Like=-21.03..-20.46 [-21.0345..-21.0334]*| it/evals=1770/2806 eff=70.6305% N=300
Z=-26.4(56.86%) | Like=-20.97..-20.46 [-20.9726..-20.9721]*| it/evals=1799/2847 eff=70.6321% N=300
Z=-26.4(56.98%) | Like=-20.97..-20.46 [-20.9721..-20.9715]*| it/evals=1800/2849 eff=70.6159% N=300
Mono-modal Volume: ~exp(-9.91) Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 **** +3.8e-11 | +1.0e-10
Z=-26.4(59.48%) | Like=-20.93..-20.46 [-20.9324..-20.9324]*| it/evals=1824/2886 eff=70.5336% N=300
Z=-26.4(60.11%) | Like=-20.93..-20.46 [-20.9281..-20.9244]*| it/evals=1830/2897 eff=70.4659% N=300
Z=-26.3(62.43%) | Like=-20.90..-20.46 [-20.8972..-20.8951]*| it/evals=1854/2937 eff=70.3072% N=300
Z=-26.3(63.00%) | Like=-20.89..-20.46 [-20.8889..-20.8885]*| it/evals=1860/2945 eff=70.3214% N=300
Mono-modal Volume: ~exp(-9.99) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.3(64.62%) | Like=-20.86..-20.46 [-20.8566..-20.8532]*| it/evals=1876/2969 eff=70.2885% N=300
Z=-26.3(65.90%) | Like=-20.84..-20.46 [-20.8385..-20.8370]*| it/evals=1890/2988 eff=70.3125% N=300
Z=-26.2(68.24%) | Like=-20.81..-20.46 [-20.8062..-20.8055]*| it/evals=1916/3027 eff=70.2604% N=300
Z=-26.2(68.60%) | Like=-20.80..-20.46 [-20.8000..-20.7999]*| it/evals=1920/3037 eff=70.1498% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3061
[ultranest] logZ = -25.87 +- 0.1122
[ultranest] Effective samples strategy satisfied (ESS = 973.5, 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=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.11 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.863 +- 0.352
single instance: logZ = -25.863 +- 0.122
bootstrapped : logZ = -25.874 +- 0.234
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.12 │ ▁▁▁▁▁▁▁▂▂▃▄▅▅▆▇▇▆▇▆▅▄▃▃▂▁▁▁▁▁▁▁ ▁ ▁ │3.17 2.57 +- 0.12
amplitude : 0.0000000000216│ ▁ ▁▁▁▁▁▂▂▃▄▅▄▆▆▆▇▇▆▅▅▅▃▃▂▂▂▁▁▁▁▁▁▁ ▁▁ │0.0000000000487 0.0000000000341 +- 0.0000000000037
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.02) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************* ************** ********* * **** * | +1.0e-10
Z=-inf(0.00%) | Like=-878.02..-19.96 [-878.0224..-135.7551] | it/evals=0/301 eff=0.0000% N=300
Z=-224.0(0.00%) | Like=-219.31..-19.96 [-878.0224..-135.7551] | it/evals=30/333 eff=90.9091% N=300
Z=-211.7(0.00%) | Like=-207.47..-19.53 [-878.0224..-135.7551] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.02) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************* ************************** **** * | +1.0e-10
Z=-199.1(0.00%) | Like=-193.83..-19.53 [-878.0224..-135.7551] | it/evals=90/400 eff=90.0000% N=300
Z=-184.8(0.00%) | Like=-178.59..-19.53 [-878.0224..-135.7551] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.31) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************* ************** *********** **** * | +1.0e-10
Z=-175.1(0.00%) | Like=-170.20..-19.53 [-878.0224..-135.7551] | it/evals=134/451 eff=88.7417% N=300
Z=-166.9(0.00%) | Like=-162.18..-19.53 [-878.0224..-135.7551] | it/evals=150/469 eff=88.7574% N=300
Z=-153.9(0.00%) | Like=-148.35..-19.53 [-878.0224..-135.7551] | it/evals=180/506 eff=87.3786% N=300
Mono-modal Volume: ~exp(-4.64) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ************************************* * **** * | +1.0e-10
Z=-144.2(0.00%) | Like=-138.51..-19.53 [-878.0224..-135.7551] | it/evals=201/531 eff=87.0130% N=300
Z=-141.7(0.00%) | Like=-136.96..-19.53 [-878.0224..-135.7551] | it/evals=210/542 eff=86.7769% N=300
Z=-131.5(0.00%) | Like=-126.20..-19.53 [-134.9490..-66.3680] | it/evals=240/580 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.75) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ************************************ ******** | +1.0e-10
Z=-120.6(0.00%) | Like=-114.54..-19.17 [-134.9490..-66.3680] | it/evals=268/613 eff=85.6230% N=300
Z=-119.7(0.00%) | Like=-114.19..-19.17 [-134.9490..-66.3680] | it/evals=270/615 eff=85.7143% N=300
Z=-109.5(0.00%) | Like=-103.81..-19.17 [-134.9490..-66.3680] | it/evals=300/652 eff=85.2273% N=300
Z=-100.5(0.00%) | Like=-94.49..-19.17 [-134.9490..-66.3680] | it/evals=330/691 eff=84.3990% N=300
Mono-modal Volume: ~exp(-5.03) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ******************************************* | +1.0e-10
Z=-99.2(0.00%) | Like=-93.43..-19.17 [-134.9490..-66.3680] | it/evals=335/696 eff=84.5960% N=300
Z=-89.4(0.00%) | Like=-83.87..-19.17 [-134.9490..-66.3680] | it/evals=360/727 eff=84.3091% N=300
Z=-82.5(0.00%) | Like=-76.29..-19.17 [-134.9490..-66.3680] | it/evals=390/765 eff=83.8710% N=300
Mono-modal Volume: ~exp(-5.27) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ***************************************** | +1.0e-10
Z=-78.9(0.00%) | Like=-73.33..-19.17 [-134.9490..-66.3680] | it/evals=402/783 eff=83.2298% N=300
Z=-76.3(0.00%) | Like=-70.75..-19.17 [-134.9490..-66.3680] | it/evals=420/802 eff=83.6653% N=300
Z=-68.5(0.00%) | Like=-62.61..-19.17 [-66.1106..-40.7452] | it/evals=450/840 eff=83.3333% N=300
Mono-modal Volume: ~exp(-5.67) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ***************************************| +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-64.5(0.00%) | Like=-58.76..-19.17 [-66.1106..-40.7452] | it/evals=469/868 eff=82.5704% N=300
Z=-62.5(0.00%) | Like=-57.01..-19.17 [-66.1106..-40.7452] | it/evals=480/883 eff=82.3328% N=300
Z=-58.0(0.00%) | Like=-52.68..-19.17 [-66.1106..-40.7452] | it/evals=510/921 eff=82.1256% N=300
Z=-55.6(0.00%) | Like=-50.50..-19.17 [-66.1106..-40.7452] | it/evals=535/961 eff=80.9380% N=300
Mono-modal Volume: ~exp(-5.96) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-55.5(0.00%) | Like=-50.33..-19.17 [-66.1106..-40.7452] | it/evals=536/962 eff=80.9668% N=300
Z=-55.1(0.00%) | Like=-49.75..-19.17 [-66.1106..-40.7452] | it/evals=540/966 eff=81.0811% N=300
Z=-52.0(0.00%) | Like=-46.39..-19.17 [-66.1106..-40.7452] | it/evals=570/1000 eff=81.4286% N=300
Z=-49.3(0.00%) | Like=-43.82..-19.17 [-66.1106..-40.7452] | it/evals=599/1039 eff=81.0555% N=300
Z=-49.2(0.00%) | Like=-43.81..-19.17 [-66.1106..-40.7452] | it/evals=600/1040 eff=81.0811% N=300
Mono-modal Volume: ~exp(-5.96) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ***************************** | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-47.1(0.00%) | Like=-41.92..-19.17 [-66.1106..-40.7452] | it/evals=624/1078 eff=80.2057% N=300
Z=-46.7(0.00%) | Like=-41.55..-19.17 [-66.1106..-40.7452] | it/evals=630/1084 eff=80.3571% N=300
Z=-45.1(0.00%) | Like=-40.11..-19.17 [-40.7271..-29.9790] | it/evals=656/1124 eff=79.6117% N=300
Z=-44.9(0.00%) | Like=-39.84..-19.17 [-40.7271..-29.9790] | it/evals=660/1131 eff=79.4224% N=300
Mono-modal Volume: ~exp(-6.41) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 *************************** | +5.0
amplitude: +1.0e-12| +2.6e-11 ************************************| +1.0e-10
Z=-44.3(0.00%) | Like=-39.09..-19.17 [-40.7271..-29.9790] | it/evals=670/1143 eff=79.4781% N=300
Z=-43.3(0.00%) | Like=-38.15..-19.17 [-40.7271..-29.9790] | it/evals=690/1166 eff=79.6767% N=300
Z=-41.6(0.00%) | Like=-36.27..-19.17 [-40.7271..-29.9790] | it/evals=720/1205 eff=79.5580% N=300
Mono-modal Volume: ~exp(-6.41) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ************************ +4.0 | +5.0
amplitude: +1.0e-12| +2.9e-11 ******************************** | +1.0e-10
Z=-40.1(0.00%) | Like=-34.72..-19.17 [-40.7271..-29.9790] | it/evals=749/1241 eff=79.5962% N=300
Z=-40.0(0.00%) | Like=-34.60..-19.17 [-40.7271..-29.9790] | it/evals=750/1243 eff=79.5334% N=300
Z=-38.4(0.00%) | Like=-33.21..-19.17 [-40.7271..-29.9790] | it/evals=778/1283 eff=79.1455% N=300
Z=-38.4(0.00%) | Like=-33.18..-19.17 [-40.7271..-29.9790] | it/evals=780/1285 eff=79.1878% N=300
Mono-modal Volume: ~exp(-6.81) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ********************** +3.9 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************************** | +1.0e-10
Z=-37.4(0.00%) | Like=-32.33..-19.17 [-40.7271..-29.9790] | it/evals=804/1315 eff=79.2118% N=300
Z=-37.2(0.00%) | Like=-32.11..-19.17 [-40.7271..-29.9790] | it/evals=810/1324 eff=79.1016% N=300
Z=-36.1(0.00%) | Like=-30.84..-19.17 [-40.7271..-29.9790] | it/evals=840/1358 eff=79.3951% N=300
Z=-35.0(0.00%) | Like=-29.75..-19.17 [-29.9629..-26.0645] | it/evals=869/1397 eff=79.2160% N=300
Z=-35.0(0.00%) | Like=-29.71..-19.17 [-29.9629..-26.0645] | it/evals=870/1399 eff=79.1629% N=300
Mono-modal Volume: ~exp(-6.90) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ******************* +3.7 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************************** | +1.0e-10
Z=-34.9(0.00%) | Like=-29.71..-19.17 [-29.9629..-26.0645] | it/evals=871/1401 eff=79.1099% N=300
Z=-34.0(0.01%) | Like=-28.66..-19.17 [-29.9629..-26.0645] | it/evals=900/1437 eff=79.1557% N=300
Z=-33.2(0.02%) | Like=-28.04..-19.17 [-29.9629..-26.0645] | it/evals=928/1476 eff=78.9116% N=300
Z=-33.1(0.02%) | Like=-27.99..-19.17 [-29.9629..-26.0645] | it/evals=930/1479 eff=78.8804% N=300
Mono-modal Volume: ~exp(-6.90) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 ****************** +3.6 | +5.0
amplitude: +1.0e-12| +3.5e-11 ************************ | +1.0e-10
Z=-32.5(0.03%) | Like=-27.22..-19.17 [-29.9629..-26.0645] | it/evals=958/1514 eff=78.9127% N=300
Z=-32.4(0.03%) | Like=-27.16..-19.17 [-29.9629..-26.0645] | it/evals=960/1518 eff=78.8177% N=300
Z=-31.6(0.07%) | Like=-26.27..-19.17 [-29.9629..-26.0645] | it/evals=990/1557 eff=78.7589% N=300
Mono-modal Volume: ~exp(-7.63) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********************* +7.9e-11| +1.0e-10
Z=-31.2(0.11%) | Like=-25.74..-19.16 [-26.0579..-25.5823] | it/evals=1005/1573 eff=78.9474% N=300
Z=-30.8(0.16%) | Like=-25.50..-19.16 [-25.4986..-25.4925]*| it/evals=1020/1590 eff=79.0698% N=300
Z=-30.2(0.29%) | Like=-24.94..-19.16 [-24.9402..-24.9299] | it/evals=1050/1624 eff=79.3051% N=300
Mono-modal Volume: ~exp(-7.84) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.6e-11 | +1.0e-10
Z=-29.8(0.43%) | Like=-24.58..-19.16 [-24.6027..-24.5849] | it/evals=1072/1651 eff=79.3486% N=300
Z=-29.7(0.50%) | Like=-24.47..-19.16 [-24.4739..-24.4568] | it/evals=1080/1663 eff=79.2370% N=300
Z=-29.3(0.78%) | Like=-23.97..-19.16 [-23.9949..-23.9653] | it/evals=1106/1702 eff=78.8873% N=300
Z=-29.2(0.84%) | Like=-23.89..-19.16 [-23.8937..-23.8543] | it/evals=1110/1706 eff=78.9474% N=300
Mono-modal Volume: ~exp(-7.84) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****************** +7.4e-11 | +1.0e-10
Z=-28.7(1.29%) | Like=-23.55..-19.16 [-23.5585..-23.5471] | it/evals=1139/1742 eff=78.9875% N=300
Z=-28.7(1.32%) | Like=-23.54..-19.16 [-23.5381..-23.5250] | it/evals=1140/1743 eff=79.0021% N=300
Z=-28.4(1.89%) | Like=-23.13..-19.16 [-23.1326..-23.1292]*| it/evals=1168/1782 eff=78.8124% N=300
Z=-28.3(1.93%) | Like=-23.13..-19.16 [-23.1262..-23.1079] | it/evals=1170/1785 eff=78.7879% N=300
Z=-28.0(2.57%) | Like=-22.84..-19.16 [-22.8384..-22.8379]*| it/evals=1195/1825 eff=78.3607% N=300
Z=-28.0(2.72%) | Like=-22.76..-19.16 [-22.7606..-22.7566]*| it/evals=1200/1830 eff=78.4314% N=300
Mono-modal Volume: ~exp(-7.84) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 *************** +7.0e-11 | +1.0e-10
Z=-27.7(3.60%) | Like=-22.37..-19.16 [-22.3703..-22.3364] | it/evals=1226/1865 eff=78.3387% N=300
Z=-27.7(3.78%) | Like=-22.28..-19.16 [-22.2850..-22.2716] | it/evals=1230/1871 eff=78.2941% N=300
Z=-27.3(5.21%) | Like=-21.96..-19.16 [-21.9581..-21.9577]*| it/evals=1257/1910 eff=78.0745% N=300
Z=-27.3(5.41%) | Like=-21.88..-19.16 [-21.8843..-21.8653] | it/evals=1260/1914 eff=78.0669% N=300
Mono-modal Volume: ~exp(-8.26) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +6.9e-11 | +1.0e-10
Z=-27.2(6.31%) | Like=-21.75..-19.16 [-21.7497..-21.7480]*| it/evals=1273/1930 eff=78.0982% N=300
Z=-27.0(7.38%) | Like=-21.63..-19.16 [-21.6253..-21.6199]*| it/evals=1290/1953 eff=78.0399% N=300
Z=-26.7(9.38%) | Like=-21.39..-19.16 [-21.3900..-21.3851]*| it/evals=1316/1992 eff=77.7778% N=300
Z=-26.7(9.68%) | Like=-21.36..-19.16 [-21.3641..-21.3558]*| it/evals=1320/2000 eff=77.6471% N=300
Mono-modal Volume: ~exp(-8.61) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.5(11.36%) | Like=-21.23..-19.16 [-21.2265..-21.2233]*| it/evals=1340/2028 eff=77.5463% N=300
Z=-26.5(12.23%) | Like=-21.17..-19.16 [-21.1716..-21.1660]*| it/evals=1350/2042 eff=77.4971% N=300
Z=-26.3(14.61%) | Like=-20.97..-19.16 [-20.9719..-20.9678]*| it/evals=1376/2081 eff=77.2600% N=300
Z=-26.3(14.97%) | Like=-20.94..-19.16 [-20.9358..-20.9344]*| it/evals=1380/2086 eff=77.2676% N=300
Z=-26.1(17.65%) | Like=-20.78..-19.16 [-20.7781..-20.7659] | it/evals=1405/2125 eff=76.9863% N=300
Mono-modal Volume: ~exp(-9.15) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.6 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.6e-11 | +1.0e-10
Z=-26.1(17.89%) | Like=-20.76..-19.16 [-20.7622..-20.7518] | it/evals=1407/2127 eff=77.0115% N=300
Z=-26.1(18.25%) | Like=-20.75..-19.16 [-20.7485..-20.7481]*| it/evals=1410/2130 eff=77.0492% N=300
Z=-25.9(21.68%) | Like=-20.56..-19.16 [-20.5572..-20.5560]*| it/evals=1440/2169 eff=77.0465% N=300
Z=-25.8(24.67%) | Like=-20.45..-19.16 [-20.4511..-20.4440]*| it/evals=1464/2209 eff=76.6894% N=300
Z=-25.7(25.33%) | Like=-20.43..-19.16 [-20.4339..-20.4297]*| it/evals=1470/2216 eff=76.7223% N=300
Mono-modal Volume: ~exp(-9.15) Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 ********* +6.5e-11 | +1.0e-10
Z=-25.6(28.23%) | Like=-20.34..-19.16 [-20.3448..-20.3410]*| it/evals=1494/2251 eff=76.5761% N=300
Z=-25.6(28.92%) | Like=-20.33..-19.16 [-20.3255..-20.3246]*| it/evals=1500/2257 eff=76.6479% N=300
Z=-25.5(32.23%) | Like=-20.22..-19.16 [-20.2189..-20.2165]*| it/evals=1526/2296 eff=76.4529% N=300
Z=-25.5(32.74%) | Like=-20.20..-19.16 [-20.2027..-20.1993]*| it/evals=1530/2301 eff=76.4618% N=300
Mono-modal Volume: ~exp(-9.16) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.4(34.09%) | Like=-20.16..-19.16 [-20.1644..-20.1638]*| it/evals=1541/2324 eff=76.1364% N=300
Z=-25.4(36.39%) | Like=-20.10..-19.16 [-20.0987..-20.0986]*| it/evals=1560/2349 eff=76.1347% N=300
Z=-25.3(40.10%) | Like=-19.99..-19.16 [-19.9896..-19.9888]*| it/evals=1589/2387 eff=76.1380% N=300
Z=-25.3(40.25%) | Like=-19.99..-19.16 [-19.9888..-19.9886]*| it/evals=1590/2388 eff=76.1494% N=300
Mono-modal Volume: ~exp(-9.69) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ******** +6.3e-11 | +1.0e-10
Z=-25.2(42.46%) | Like=-19.95..-19.16 [-19.9497..-19.9492]*| it/evals=1608/2417 eff=75.9565% N=300
Z=-25.2(44.02%) | Like=-19.92..-19.16 [-19.9221..-19.9190]*| it/evals=1620/2434 eff=75.9138% N=300
Z=-25.1(47.55%) | Like=-19.85..-19.16 [-19.8528..-19.8511]*| it/evals=1648/2473 eff=75.8399% N=300
Z=-25.1(47.80%) | Like=-19.85..-19.16 [-19.8498..-19.8457]*| it/evals=1650/2476 eff=75.8272% N=300
Mono-modal Volume: ~exp(-9.69) Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-25.0(50.81%) | Like=-19.81..-19.16 [-19.8052..-19.8033]*| it/evals=1675/2512 eff=75.7233% N=300
Z=-25.0(51.40%) | Like=-19.80..-19.16 [-19.7958..-19.7850] | it/evals=1680/2518 eff=75.7439% N=300
Z=-25.0(54.57%) | Like=-19.72..-19.16 [-19.7229..-19.7225]*| it/evals=1707/2557 eff=75.6314% N=300
Z=-24.9(54.94%) | Like=-19.72..-19.16 [-19.7201..-19.7186]*| it/evals=1710/2561 eff=75.6303% N=300
Z=-24.9(57.77%) | Like=-19.69..-19.16 [-19.6872..-19.6863]*| it/evals=1736/2601 eff=75.4455% N=300
Z=-24.9(58.17%) | Like=-19.68..-19.16 [-19.6825..-19.6817]*| it/evals=1740/2606 eff=75.4553% N=300
Mono-modal Volume: ~exp(-9.94) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.9(58.36%) | Like=-19.68..-19.16 [-19.6802..-19.6790]*| it/evals=1742/2609 eff=75.4439% N=300
Z=-24.8(61.26%) | Like=-19.63..-19.16 [-19.6346..-19.6332]*| it/evals=1770/2641 eff=75.6087% N=300
Z=-24.8(64.19%) | Like=-19.59..-19.16 [-19.5938..-19.5884]*| it/evals=1800/2677 eff=75.7257% N=300
Mono-modal Volume: ~exp(-10.25) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.1e-11 | +1.0e-10
Z=-24.8(65.01%) | Like=-19.58..-19.16 [-19.5783..-19.5745]*| it/evals=1809/2690 eff=75.6904% N=300
Z=-24.7(66.94%) | Like=-19.55..-19.16 [-19.5470..-19.5470]*| it/evals=1830/2713 eff=75.8392% N=300
Z=-24.7(69.57%) | Like=-19.51..-19.16 [-19.5098..-19.5091]*| it/evals=1860/2749 eff=75.9494% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2755
[ultranest] logZ = -24.36 +- 0.07651
[ultranest] Effective samples strategy satisfied (ESS = 966.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.08 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -24.343 +- 0.283
single instance: logZ = -24.343 +- 0.119
bootstrapped : logZ = -24.356 +- 0.108
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.33 │ ▁▁▁▁▁▁▂▄▄▄▇▇▆▇▇▇▅▆▅▅▃▃▂▂▂▁▁▁▁▁▁▁▁ ▁ │3.57 2.83 +- 0.16
amplitude : 0.0000000000353│ ▁▁▁▁▁▁▂▂▃▄▄▆▆▇▇▅▆▆▅▄▃▃▂▂▂▁▁▁▁▁ ▁▁▁ ▁ │0.0000000000835 0.0000000000551 +- 0.0000000000057
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.10) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************** *** ********* ** **** * ***| +1.0e-10
Z=-inf(0.00%) | Like=-1195.15..-14.38 [-1195.1461..-100.2557] | it/evals=0/301 eff=0.0000% N=300
Z=-155.1(0.00%) | Like=-150.99..-13.67 [-1195.1461..-100.2557] | it/evals=30/331 eff=96.7742% N=300
Z=-146.8(0.00%) | Like=-142.12..-13.67 [-1195.1461..-100.2557] | it/evals=60/363 eff=95.2381% N=300
Mono-modal Volume: ~exp(-4.43) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************** *** ************ **** * ***| +1.0e-10
Z=-144.9(0.00%) | Like=-140.32..-13.67 [-1195.1461..-100.2557] | it/evals=67/372 eff=93.0556% N=300
Z=-138.5(0.00%) | Like=-133.83..-13.67 [-1195.1461..-100.2557] | it/evals=90/397 eff=92.7835% N=300
Z=-130.2(0.00%) | Like=-124.46..-13.67 [-1195.1461..-100.2557] | it/evals=120/432 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.43) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************ ***************** *****| +1.0e-10
Z=-121.0(0.00%) | Like=-115.73..-13.67 [-1195.1461..-100.2557] | it/evals=150/464 eff=91.4634% N=300
Z=-112.8(0.00%) | Like=-107.97..-13.67 [-1195.1461..-100.2557] | it/evals=180/497 eff=91.3706% N=300
Mono-modal Volume: ~exp(-4.93) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|************************ ***************** *****| +1.0e-10
Z=-106.9(0.00%) | Like=-101.97..-13.67 [-1195.1461..-100.2557] | it/evals=201/522 eff=90.5405% N=300
Z=-104.6(0.00%) | Like=-99.75..-13.67 [-100.2196..-48.4326] | it/evals=210/532 eff=90.5172% N=300
Z=-97.0(0.00%) | Like=-91.67..-13.67 [-100.2196..-48.4326] | it/evals=240/567 eff=89.8876% N=300
Mono-modal Volume: ~exp(-4.93) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ***************************************** *****| +1.0e-10
Z=-91.0(0.00%) | Like=-85.06..-13.67 [-100.2196..-48.4326] | it/evals=269/602 eff=89.0728% N=300
Z=-90.6(0.00%) | Like=-84.81..-13.67 [-100.2196..-48.4326] | it/evals=270/604 eff=88.8158% N=300
Z=-82.9(0.00%) | Like=-77.48..-13.67 [-100.2196..-48.4326] | it/evals=300/639 eff=88.4956% N=300
Z=-74.2(0.00%) | Like=-67.86..-13.67 [-100.2196..-48.4326] | it/evals=330/676 eff=87.7660% N=300
Mono-modal Volume: ~exp(-5.14) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| **************************************** *****| +1.0e-10
Z=-72.6(0.00%) | Like=-67.15..-13.67 [-100.2196..-48.4326] | it/evals=335/681 eff=87.9265% N=300
Z=-67.3(0.00%) | Like=-61.83..-13.67 [-100.2196..-48.4326] | it/evals=360/712 eff=87.3786% N=300
Z=-61.4(0.00%) | Like=-56.06..-13.67 [-100.2196..-48.4326] | it/evals=390/748 eff=87.0536% N=300
Mono-modal Volume: ~exp(-5.31) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-59.5(0.00%) | Like=-54.13..-13.67 [-100.2196..-48.4326] | it/evals=402/764 eff=86.6379% N=300
Z=-56.2(0.00%) | Like=-50.57..-13.46 [-100.2196..-48.4326] | it/evals=420/792 eff=85.3659% N=300
Z=-52.0(0.00%) | Like=-46.92..-13.46 [-48.4268..-31.6589] | it/evals=449/831 eff=84.5574% N=300
Z=-51.9(0.00%) | Like=-46.83..-13.46 [-48.4268..-31.6589] | it/evals=450/832 eff=84.5865% N=300
Mono-modal Volume: ~exp(-5.33) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-49.8(0.00%) | Like=-44.99..-13.46 [-48.4268..-31.6589] | it/evals=469/863 eff=83.3037% N=300
Z=-48.9(0.00%) | Like=-44.00..-13.46 [-48.4268..-31.6589] | it/evals=480/876 eff=83.3333% N=300
Z=-46.7(0.00%) | Like=-41.96..-13.40 [-48.4268..-31.6589] | it/evals=506/916 eff=82.1429% N=300
Z=-46.4(0.00%) | Like=-41.62..-13.40 [-48.4268..-31.6589] | it/evals=510/923 eff=81.8620% N=300
Mono-modal Volume: ~exp(-5.89) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-44.8(0.00%) | Like=-39.94..-13.40 [-48.4268..-31.6589] | it/evals=536/960 eff=81.2121% N=300
Z=-44.5(0.00%) | Like=-39.75..-13.40 [-48.4268..-31.6589] | it/evals=540/965 eff=81.2030% N=300
Z=-42.4(0.00%) | Like=-37.47..-13.40 [-48.4268..-31.6589] | it/evals=568/1004 eff=80.6818% N=300
Z=-42.3(0.00%) | Like=-37.41..-13.40 [-48.4268..-31.6589] | it/evals=570/1006 eff=80.7365% N=300
Z=-40.3(0.00%) | Like=-35.09..-13.40 [-48.4268..-31.6589] | it/evals=599/1047 eff=80.1874% N=300
Z=-40.2(0.00%) | Like=-34.95..-13.40 [-48.4268..-31.6589] | it/evals=600/1048 eff=80.2139% N=300
Mono-modal Volume: ~exp(-6.02) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-40.0(0.00%) | Like=-34.49..-13.40 [-48.4268..-31.6589] | it/evals=603/1053 eff=80.0797% N=300
Z=-38.1(0.00%) | Like=-33.18..-13.40 [-48.4268..-31.6589] | it/evals=630/1090 eff=79.7468% N=300
Z=-36.6(0.00%) | Like=-31.77..-13.40 [-48.4268..-31.6589] | it/evals=656/1130 eff=79.0361% N=300
Z=-36.4(0.00%) | Like=-31.64..-13.40 [-31.6407..-22.3764] | it/evals=660/1135 eff=79.0419% N=300
Mono-modal Volume: ~exp(-6.02) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 *************************** +4.1 | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-35.0(0.00%) | Like=-29.86..-13.40 [-31.6407..-22.3764] | it/evals=688/1171 eff=78.9897% N=300
Z=-34.9(0.00%) | Like=-29.81..-13.40 [-31.6407..-22.3764] | it/evals=690/1175 eff=78.8571% N=300
Z=-33.4(0.00%) | Like=-28.37..-13.40 [-31.6407..-22.3764] | it/evals=718/1215 eff=78.4699% N=300
Z=-33.3(0.00%) | Like=-28.20..-13.40 [-31.6407..-22.3764] | it/evals=720/1217 eff=78.5169% N=300
Mono-modal Volume: ~exp(-6.53) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 *********************** +3.9 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-32.5(0.00%) | Like=-27.48..-13.40 [-31.6407..-22.3764] | it/evals=737/1241 eff=78.3209% N=300
Z=-31.9(0.00%) | Like=-26.88..-13.40 [-31.6407..-22.3764] | it/evals=750/1258 eff=78.2881% N=300
Z=-30.7(0.00%) | Like=-25.57..-13.39 [-31.6407..-22.3764] | it/evals=778/1297 eff=78.0341% N=300
Z=-30.6(0.00%) | Like=-25.42..-13.39 [-31.6407..-22.3764] | it/evals=780/1299 eff=78.0781% N=300
Mono-modal Volume: ~exp(-6.53) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ********************* +3.7 | +5.0
amplitude: +1.0e-12| +2.5e-11 ********************************** **| +1.0e-10
Z=-29.5(0.00%) | Like=-24.53..-13.39 [-31.6407..-22.3764] | it/evals=810/1335 eff=78.2609% N=300
Z=-28.5(0.00%) | Like=-23.41..-13.39 [-31.6407..-22.3764] | it/evals=840/1374 eff=78.2123% N=300
Z=-27.6(0.01%) | Like=-22.51..-13.39 [-31.6407..-22.3764] | it/evals=867/1414 eff=77.8276% N=300
Z=-27.5(0.01%) | Like=-22.48..-13.39 [-31.6407..-22.3764] | it/evals=870/1419 eff=77.7480% N=300
Mono-modal Volume: ~exp(-6.67) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************* | +1.0e-10
Z=-27.5(0.01%) | Like=-22.47..-13.39 [-31.6407..-22.3764] | it/evals=871/1421 eff=77.6985% N=300
Z=-26.9(0.02%) | Like=-21.90..-13.39 [-22.3257..-20.1270] | it/evals=897/1462 eff=77.1945% N=300
Z=-26.8(0.03%) | Like=-21.81..-13.39 [-22.3257..-20.1270] | it/evals=900/1466 eff=77.1870% N=300
Z=-26.2(0.05%) | Like=-21.17..-13.39 [-22.3257..-20.1270] | it/evals=929/1505 eff=77.0954% N=300
Z=-26.1(0.05%) | Like=-21.17..-13.39 [-22.3257..-20.1270] | it/evals=930/1509 eff=76.9231% N=300
Mono-modal Volume: ~exp(-6.69) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************************** | +1.0e-10
Z=-26.0(0.06%) | Like=-20.98..-13.39 [-22.3257..-20.1270] | it/evals=938/1518 eff=77.0115% N=300
Z=-25.5(0.10%) | Like=-20.51..-13.39 [-22.3257..-20.1270] | it/evals=960/1548 eff=76.9231% N=300
Z=-25.0(0.17%) | Like=-20.01..-13.39 [-20.1218..-19.8519] | it/evals=986/1588 eff=76.5528% N=300
Z=-24.9(0.18%) | Like=-19.95..-13.39 [-20.1218..-19.8519] | it/evals=990/1593 eff=76.5661% N=300
Mono-modal Volume: ~exp(-7.26) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.0e-11 *************************** | +1.0e-10
Z=-24.7(0.24%) | Like=-19.55..-13.39 [-19.5519..-19.5413] | it/evals=1005/1618 eff=76.2519% N=300
Z=-24.4(0.33%) | Like=-19.18..-13.35 [-19.1823..-19.1337] | it/evals=1020/1636 eff=76.3473% N=300
Z=-23.8(0.56%) | Like=-18.73..-13.35 [-18.7336..-18.6791] | it/evals=1050/1675 eff=76.3636% N=300
Mono-modal Volume: ~exp(-7.30) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************ | +1.0e-10
Z=-23.5(0.80%) | Like=-18.31..-13.35 [-18.3085..-18.3032]*| it/evals=1072/1704 eff=76.3533% N=300
Z=-23.3(0.91%) | Like=-18.17..-13.35 [-18.1651..-18.1560]*| it/evals=1080/1715 eff=76.3251% N=300
Z=-22.9(1.46%) | Like=-17.68..-13.35 [-17.6765..-17.6611] | it/evals=1110/1750 eff=76.5517% N=300
Z=-22.5(2.13%) | Like=-17.32..-13.35 [-17.3226..-17.3039] | it/evals=1136/1789 eff=76.2928% N=300
Mono-modal Volume: ~exp(-7.74) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************** +7.8e-11| +1.0e-10
Z=-22.5(2.23%) | Like=-17.30..-13.35 [-17.3005..-17.2763] | it/evals=1139/1792 eff=76.3405% N=300
Z=-22.5(2.26%) | Like=-17.28..-13.35 [-17.3005..-17.2763] | it/evals=1140/1793 eff=76.3563% N=300
Z=-22.1(3.01%) | Like=-17.00..-13.31 [-17.0048..-16.9891] | it/evals=1167/1830 eff=76.2745% N=300
Z=-22.1(3.12%) | Like=-16.91..-13.31 [-16.9751..-16.9074] | it/evals=1170/1833 eff=76.3209% N=300
Z=-21.8(4.04%) | Like=-16.70..-13.31 [-16.7209..-16.7046] | it/evals=1192/1872 eff=75.8270% N=300
Z=-21.8(4.36%) | Like=-16.65..-13.31 [-16.6672..-16.6461] | it/evals=1200/1881 eff=75.9013% N=300
Mono-modal Volume: ~exp(-7.83) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.5e-11 | +1.0e-10
Z=-21.7(4.65%) | Like=-16.59..-13.31 [-16.5892..-16.5701] | it/evals=1206/1890 eff=75.8491% N=300
Z=-21.5(5.81%) | Like=-16.41..-13.31 [-16.4253..-16.4103] | it/evals=1230/1917 eff=76.0668% N=300
Z=-21.2(7.27%) | Like=-16.19..-13.31 [-16.1854..-16.1807]*| it/evals=1257/1956 eff=75.9058% N=300
Z=-21.2(7.41%) | Like=-16.18..-13.31 [-16.1790..-16.1693]*| it/evals=1260/1959 eff=75.9494% N=300
Mono-modal Volume: ~exp(-7.83) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.3e-11 | +1.0e-10
Z=-21.1(8.69%) | Like=-15.95..-13.31 [-15.9532..-15.9416] | it/evals=1282/1995 eff=75.6342% N=300
Z=-21.0(9.28%) | Like=-15.89..-13.31 [-15.9096..-15.8912] | it/evals=1290/2006 eff=75.6155% N=300
Z=-20.8(11.55%) | Like=-15.68..-13.31 [-15.6999..-15.6829] | it/evals=1320/2045 eff=75.6447% N=300
Mono-modal Volume: ~exp(-8.40) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 **************** +7.1e-11 | +1.0e-10
Z=-20.7(13.19%) | Like=-15.49..-13.31 [-15.4857..-15.4657] | it/evals=1340/2079 eff=75.3232% N=300
Z=-20.6(14.00%) | Like=-15.40..-13.31 [-15.3952..-15.3925]*| it/evals=1350/2089 eff=75.4612% N=300
Z=-20.4(16.72%) | Like=-15.24..-13.31 [-15.2374..-15.2352]*| it/evals=1378/2128 eff=75.3829% N=300
Z=-20.4(16.84%) | Like=-15.23..-13.31 [-15.2283..-15.2253]*| it/evals=1380/2133 eff=75.2864% N=300
Mono-modal Volume: ~exp(-8.40) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.0e-11 *************** +6.9e-11 | +1.0e-10
Z=-20.2(19.72%) | Like=-15.10..-13.31 [-15.0987..-15.0908]*| it/evals=1407/2170 eff=75.2406% N=300
Z=-20.2(19.93%) | Like=-15.08..-13.31 [-15.0793..-15.0784]*| it/evals=1410/2174 eff=75.2401% N=300
Z=-20.1(23.05%) | Like=-14.91..-13.31 [-14.9116..-14.9078]*| it/evals=1440/2212 eff=75.3138% N=300
Z=-19.9(26.12%) | Like=-14.79..-13.31 [-14.7945..-14.7914]*| it/evals=1470/2249 eff=75.4233% N=300
Mono-modal Volume: ~exp(-8.52) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************** +6.7e-11 | +1.0e-10
Z=-19.9(26.60%) | Like=-14.78..-13.31 [-14.7815..-14.7787]*| it/evals=1474/2253 eff=75.4736% N=300
Z=-19.8(29.45%) | Like=-14.66..-13.31 [-14.6705..-14.6569] | it/evals=1500/2285 eff=75.5668% N=300
Z=-19.7(32.66%) | Like=-14.54..-13.31 [-14.5394..-14.5393]*| it/evals=1528/2324 eff=75.4941% N=300
Z=-19.7(32.88%) | Like=-14.53..-13.31 [-14.5393..-14.5263] | it/evals=1530/2331 eff=75.3323% N=300
Mono-modal Volume: ~exp(-8.85) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ******* +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 ************* +6.6e-11 | +1.0e-10
Z=-19.7(34.22%) | Like=-14.50..-13.31 [-14.5009..-14.4998]*| it/evals=1541/2345 eff=75.3545% N=300
Z=-19.6(36.43%) | Like=-14.43..-13.31 [-14.4265..-14.4260]*| it/evals=1560/2366 eff=75.5082% N=300
Z=-19.5(39.76%) | Like=-14.33..-13.31 [-14.3341..-14.3329]*| it/evals=1589/2405 eff=75.4869% N=300
Z=-19.5(39.86%) | Like=-14.33..-13.31 [-14.3329..-14.3289]*| it/evals=1590/2406 eff=75.4986% N=300
Mono-modal Volume: ~exp(-9.01) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.4e-11 | +1.0e-10
Z=-19.5(42.01%) | Like=-14.29..-13.31 [-14.2880..-14.2870]*| it/evals=1608/2431 eff=75.4575% N=300
Z=-19.4(43.35%) | Like=-14.25..-13.31 [-14.2527..-14.2525]*| it/evals=1620/2447 eff=75.4541% N=300
Z=-19.4(46.65%) | Like=-14.19..-13.31 [-14.1904..-14.1828]*| it/evals=1650/2484 eff=75.5495% N=300
Mono-modal Volume: ~exp(-9.21) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.3e-11 | +1.0e-10
Z=-19.3(49.63%) | Like=-14.09..-13.31 [-14.0942..-14.0900]*| it/evals=1675/2521 eff=75.4165% N=300
Z=-19.3(50.19%) | Like=-14.09..-13.31 [-14.0883..-14.0865]*| it/evals=1680/2527 eff=75.4378% N=300
Z=-19.2(53.37%) | Like=-14.02..-13.30 [-14.0184..-14.0173]*| it/evals=1710/2565 eff=75.4967% N=300
Z=-19.2(56.21%) | Like=-13.96..-13.30 [-13.9623..-13.9613]*| it/evals=1737/2604 eff=75.3906% N=300
Z=-19.2(56.53%) | Like=-13.96..-13.30 [-13.9595..-13.9548]*| it/evals=1740/2607 eff=75.4226% N=300
Mono-modal Volume: ~exp(-9.75) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.2e-11 | +1.0e-10
Z=-19.2(56.70%) | Like=-13.95..-13.30 [-13.9538..-13.9538]*| it/evals=1742/2609 eff=75.4439% N=300
Z=-19.1(59.45%) | Like=-13.91..-13.30 [-13.9093..-13.9037]*| it/evals=1770/2642 eff=75.5764% N=300
Z=-19.1(62.37%) | Like=-13.85..-13.30 [-13.8536..-13.8525]*| it/evals=1800/2673 eff=75.8534% N=300
Mono-modal Volume: ~exp(-9.75) Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.1e-11 | +1.0e-10
Z=-19.0(64.69%) | Like=-13.81..-13.30 [-13.8121..-13.8118]*| it/evals=1825/2709 eff=75.7576% N=300
Z=-19.0(65.14%) | Like=-13.81..-13.30 [-13.8084..-13.8081]*| it/evals=1830/2716 eff=75.7450% N=300
Z=-19.0(67.52%) | Like=-13.76..-13.30 [-13.7619..-13.7597]*| it/evals=1856/2756 eff=75.5700% N=300
Z=-19.0(67.88%) | Like=-13.76..-13.30 [-13.7568..-13.7567]*| it/evals=1860/2764 eff=75.4870% N=300
Mono-modal Volume: ~exp(-9.75) Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.0e-11 | +1.0e-10
Z=-19.0(69.78%) | Like=-13.71..-13.30 [-13.7139..-13.7139]*| it/evals=1883/2800 eff=75.3200% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2802
[ultranest] logZ = -18.59 +- 0.07521
[ultranest] Effective samples strategy satisfied (ESS = 1025.4, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.11 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.08 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -18.593 +- 0.305
single instance: logZ = -18.593 +- 0.118
bootstrapped : logZ = -18.595 +- 0.155
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.11 │ ▁ ▁▁▁▁▂▃▃▄▅▆▆▇▇▇▆▆▅▄▅▃▂▂▁▁▁▁▁▁ ▁ │3.70 2.76 +- 0.18
amplitude : 0.0000000000260│ ▁▁ ▁▁▁▁▂▃▄▅▅▇▆▇▇▇▇▇▅▄▅▃▂▂▁▁▁▁▁▁▁▁▁▁ ▁ │0.0000000000893 0.0000000000536 +- 0.0000000000081
.. GENERATED FROM PYTHON SOURCE LINES 376-384
Comparing the posterior distribution of all runs
------------------------------------------------
For a comparison of different posterior distributions, we can use the
package chainconsumer.
As this is not a Gammapy dependency, you’ll need to install it.
More info here : https://samreay.github.io/ChainConsumer/
.. GENERATED FROM PYTHON SOURCE LINES 384-410
.. code-block:: Python
# Uncomment this if you have installed `chainconsumer`.
# from chainconsumer import Chain, ChainConfig, ChainConsumer, PlotConfig, Truth, make_sample
# from pandas import DataFrame
# c = ChainConsumer()
# def create_chain(result, name, color="k"):
# return Chain(
# samples=DataFrame(result, columns=["index", "amplitude"]),
# name=name,
# color=color,
# smooth=7,
# shade=False,
# linewidth=1.0,
# cmap="magma",
# show_contour_labels=True,
# kde= True
# )
# c.add_chain(create_chain(result_joint.samples, "joint"))
# c.add_chain(create_chain(result_0.samples, "run0", "g"))
# c.add_chain(create_chain(result_1.samples, "run1", "b"))
# c.add_chain(create_chain(result_2.samples, "run2", "y"))
# fig = c.plotter.plot()
# plt.show()
.. GENERATED FROM PYTHON SOURCE LINES 411-426
Corner plot comparison
----------------------
.. figure:: ../../_static/cornerplot-multiple-runs-Crab.png
:alt: Corner plot of Crab runs
Corner plot comparing the three Crab runs.
We can see the joint analysis allows to better constrain the
parameters than the individual runs (more observation time is of
course better).
One can note as well that one of the run has a notably different
amplitude (possibly due to calibrations or/and atmospheric issues).
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (0 minutes 40.686 seconds)
.. _sphx_glr_download_tutorials_api_nested_sampling_Crab.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: binder-badge
.. image:: images/binder_badge_logo.svg
:target: https://mybinder.org/v2/gh/gammapy/gammapy-webpage/main?urlpath=lab/tree/notebooks/dev/tutorials/api/nested_sampling_Crab.ipynb
:alt: Launch binder
:width: 150 px
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: nested_sampling_Crab.ipynb <nested_sampling_Crab.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: nested_sampling_Crab.py <nested_sampling_Crab.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: nested_sampling_Crab.zip <nested_sampling_Crab.zip>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_