Note
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Bayesian analysis with nested sampling#
A demonstration of a Bayesian analysis using the nested sampling technique.
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. The method performs this integral by evolving a collection of points through the parameter space (see recent reviews from Ashton et al., 2022, and Buchner, 2023). 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 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. And for a tutorial of UltraNest applied to X-ray fitting with concrete examples and questions see : BXA Tutorial.
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.
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.
Setup#
As usual, we’ll start with some setup …
import matplotlib.pyplot as plt
import numpy as np
import astropy.units as u
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
Loading the spectral datasets#
Here we will load a few Crab 1D spectral data for which we will do a fit.
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.
model = SkyModel.create(spectral_model="pl", name="crab")
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 LogUniformPrior for
the parameters that have a large amplitude range like the
amplitude parameter.
A 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 LogUniformPrior.
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)
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
Defining the sampler and options#
As for the Fit object, the Sampler object can receive
different backend (although just one is available for now).
The Sampler comes with “reasonable” default parameters, but you can
change them via the sampler_opts dictionary.
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 (withresume=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.
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
Next we can run the sampler on a given dataset. No options are accepted in the run method.
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.00) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************** ***************** ****** ** **| +1.0e-10
Z=-inf(0.00%) | Like=-4197.27..-62.36 [-4197.2706..-345.4305] | it/evals=0/301 eff=0.0000% N=300
Z=-586.2(0.00%) | Like=-572.51..-62.36 [-4197.2706..-345.4305] | it/evals=22/323 eff=95.6522% N=300
Z=-552.2(0.00%) | Like=-546.27..-62.36 [-4197.2706..-345.4305] | it/evals=30/331 eff=96.7742% N=300
Z=-533.5(0.00%) | Like=-528.42..-61.66 [-4197.2706..-345.4305] | it/evals=49/353 eff=92.4528% N=300
Z=-520.0(0.00%) | Like=-513.96..-61.66 [-4197.2706..-345.4305] | it/evals=60/365 eff=92.3077% N=300
Mono-modal Volume: ~exp(-4.00) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************************** ** * | +1.0e-10
Z=-491.6(0.00%) | Like=-482.89..-61.66 [-4197.2706..-345.4305] | it/evals=78/385 eff=91.7647% N=300
Z=-480.4(0.00%) | Like=-473.99..-61.66 [-4197.2706..-345.4305] | it/evals=90/400 eff=90.0000% N=300
Z=-462.7(0.00%) | Like=-456.39..-61.66 [-4197.2706..-345.4305] | it/evals=109/426 eff=86.5079% N=300
Z=-447.7(0.00%) | Like=-441.79..-61.66 [-4197.2706..-345.4305] | it/evals=120/438 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.61) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************************** *** * | +1.0e-10
Z=-424.4(0.00%) | Like=-418.73..-61.66 [-4197.2706..-345.4305] | it/evals=134/455 eff=86.4516% N=300
Z=-411.2(0.00%) | Like=-404.31..-61.66 [-4197.2706..-345.4305] | it/evals=150/472 eff=87.2093% N=300
Z=-390.2(0.00%) | Like=-383.93..-61.66 [-4197.2706..-345.4305] | it/evals=168/494 eff=86.5979% N=300
Z=-378.4(0.00%) | Like=-372.16..-61.66 [-4197.2706..-345.4305] | it/evals=180/510 eff=85.7143% N=300
Z=-351.8(0.00%) | Like=-345.57..-61.66 [-4197.2706..-345.4305] | it/evals=199/532 eff=85.7759% N=300
Mono-modal Volume: ~exp(-4.61) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| * ********************************************| +5.0
amplitude: +1.0e-12| **************************************** *** * | +1.0e-10
Z=-340.7(0.00%) | Like=-332.26..-61.66 [-345.2895..-190.4399] | it/evals=210/547 eff=85.0202% N=300
Z=-324.7(0.00%) | Like=-318.34..-61.66 [-345.2895..-190.4399] | it/evals=230/569 eff=85.5019% N=300
Z=-317.8(0.00%) | Like=-310.06..-61.66 [-345.2895..-190.4399] | it/evals=240/582 eff=85.1064% N=300
Z=-298.3(0.00%) | Like=-291.72..-60.25 [-345.2895..-190.4399] | it/evals=257/604 eff=84.5395% N=300
Mono-modal Volume: ~exp(-4.73) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| **************************************** ** * | +1.0e-10
Z=-287.8(0.00%) | Like=-281.40..-60.25 [-345.2895..-190.4399] | it/evals=268/618 eff=84.2767% N=300
Z=-286.7(0.00%) | Like=-278.73..-60.25 [-345.2895..-190.4399] | it/evals=270/620 eff=84.3750% N=300
Z=-273.2(0.00%) | Like=-265.99..-60.25 [-345.2895..-190.4399] | it/evals=287/642 eff=83.9181% N=300
Z=-260.9(0.00%) | Like=-254.76..-60.25 [-345.2895..-190.4399] | it/evals=300/664 eff=82.4176% N=300
Z=-245.4(0.00%) | Like=-239.20..-60.25 [-345.2895..-190.4399] | it/evals=316/686 eff=81.8653% N=300
Z=-234.4(0.00%) | Like=-227.82..-59.33 [-345.2895..-190.4399] | it/evals=330/701 eff=82.2943% N=300
Mono-modal Volume: ~exp(-4.73) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *************************************** ** **| +1.0e-10
Z=-226.5(0.00%) | Like=-219.67..-59.08 [-345.2895..-190.4399] | it/evals=345/721 eff=81.9477% N=300
Z=-216.4(0.00%) | Like=-210.50..-59.08 [-345.2895..-190.4399] | it/evals=360/740 eff=81.8182% N=300
Z=-208.7(0.00%) | Like=-202.06..-59.08 [-345.2895..-190.4399] | it/evals=375/763 eff=80.9935% N=300
Z=-201.3(0.00%) | Like=-194.65..-59.08 [-345.2895..-190.4399] | it/evals=390/787 eff=80.0821% N=300
Mono-modal Volume: ~exp(-5.23) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| ************************************* ** *| +1.0e-10
Z=-195.7(0.00%) | Like=-188.02..-59.08 [-190.0855..-126.7141] | it/evals=402/815 eff=78.0583% N=300
Z=-189.4(0.00%) | Like=-183.30..-59.08 [-190.0855..-126.7141] | it/evals=415/838 eff=77.1375% N=300
Z=-187.4(0.00%) | Like=-180.32..-59.08 [-190.0855..-126.7141] | it/evals=420/845 eff=77.0642% N=300
Z=-182.2(0.00%) | Like=-176.27..-59.08 [-190.0855..-126.7141] | it/evals=435/869 eff=76.4499% N=300
Z=-177.4(0.00%) | Like=-171.21..-59.08 [-190.0855..-126.7141] | it/evals=450/891 eff=76.1421% N=300
Z=-171.4(0.00%) | Like=-165.27..-59.08 [-190.0855..-126.7141] | it/evals=463/914 eff=75.4072% N=300
Mono-modal Volume: ~exp(-5.38) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| ************************************* ** *| +1.0e-10
Z=-169.4(0.00%) | Like=-163.32..-59.08 [-190.0855..-126.7141] | it/evals=469/922 eff=75.4019% N=300
Z=-167.2(0.00%) | Like=-161.80..-59.08 [-190.0855..-126.7141] | it/evals=480/939 eff=75.1174% N=300
Z=-163.9(0.00%) | Like=-157.35..-59.08 [-190.0855..-126.7141] | it/evals=495/962 eff=74.7734% N=300
Z=-158.7(0.00%) | Like=-152.49..-59.08 [-190.0855..-126.7141] | it/evals=510/981 eff=74.8899% N=300
Z=-153.2(0.00%) | Like=-147.28..-59.08 [-190.0855..-126.7141] | it/evals=523/1007 eff=73.9745% N=300
Z=-151.1(0.00%) | Like=-145.12..-59.08 [-190.0855..-126.7141] | it/evals=534/1030 eff=73.1507% N=300
Mono-modal Volume: ~exp(-5.79) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| ************************************ ** | +1.0e-10
Z=-150.7(0.00%) | Like=-144.82..-59.08 [-190.0855..-126.7141] | it/evals=536/1032 eff=73.2240% N=300
Z=-150.0(0.00%) | Like=-144.14..-59.08 [-190.0855..-126.7141] | it/evals=540/1036 eff=73.3696% N=300
Z=-143.9(0.00%) | Like=-137.93..-59.08 [-190.0855..-126.7141] | it/evals=558/1060 eff=73.4211% N=300
Z=-140.9(0.00%) | Like=-134.69..-59.08 [-190.0855..-126.7141] | it/evals=570/1075 eff=73.5484% N=300
Z=-138.1(0.00%) | Like=-132.12..-59.08 [-190.0855..-126.7141] | it/evals=584/1098 eff=73.1830% N=300
Z=-134.7(0.00%) | Like=-128.36..-59.08 [-190.0855..-126.7141] | it/evals=600/1118 eff=73.3496% N=300
Mono-modal Volume: ~exp(-6.05) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +4.0 | +5.0
amplitude: +1.0e-12| ********************************* * | +1.0e-10
Z=-134.1(0.00%) | Like=-127.38..-59.08 [-190.0855..-126.7141] | it/evals=603/1125 eff=73.0909% N=300
Z=-129.5(0.00%) | Like=-123.07..-59.08 [-126.3080..-90.4748] | it/evals=619/1147 eff=73.0815% N=300
Z=-126.9(0.00%) | Like=-120.57..-59.08 [-126.3080..-90.4748] | it/evals=630/1158 eff=73.4266% N=300
Z=-123.4(0.00%) | Like=-116.58..-59.08 [-126.3080..-90.4748] | it/evals=646/1180 eff=73.4091% N=300
Z=-120.7(0.00%) | Like=-114.58..-59.08 [-126.3080..-90.4748] | it/evals=660/1204 eff=73.0088% N=300
Mono-modal Volume: ~exp(-6.31) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 *********************** +3.8 | +5.0
amplitude: +1.0e-12| ******************************* | +1.0e-10
Z=-119.0(0.00%) | Like=-112.80..-59.08 [-126.3080..-90.4748] | it/evals=670/1219 eff=72.9053% N=300
Z=-116.5(0.00%) | Like=-110.64..-59.08 [-126.3080..-90.4748] | it/evals=685/1241 eff=72.7949% N=300
Z=-115.9(0.00%) | Like=-109.68..-59.08 [-126.3080..-90.4748] | it/evals=690/1249 eff=72.7081% N=300
Z=-113.8(0.00%) | Like=-108.03..-59.08 [-126.3080..-90.4748] | it/evals=706/1271 eff=72.7085% N=300
Z=-111.1(0.00%) | Like=-104.82..-58.88 [-126.3080..-90.4748] | it/evals=720/1289 eff=72.8008% N=300
Mono-modal Volume: ~exp(-6.52) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| **************************** +7.8e-11| +1.0e-10
Z=-109.0(0.00%) | Like=-102.94..-58.88 [-126.3080..-90.4748] | it/evals=737/1312 eff=72.8261% N=300
Z=-107.3(0.00%) | Like=-101.04..-58.88 [-126.3080..-90.4748] | it/evals=750/1328 eff=72.9572% N=300
Z=-104.9(0.00%) | Like=-98.30..-58.79 [-126.3080..-90.4748] | it/evals=763/1352 eff=72.5285% N=300
Z=-102.9(0.00%) | Like=-96.60..-58.79 [-126.3080..-90.4748] | it/evals=777/1375 eff=72.2791% N=300
Z=-102.5(0.00%) | Like=-96.36..-58.79 [-126.3080..-90.4748] | it/evals=780/1380 eff=72.2222% N=300
Z=-101.0(0.00%) | Like=-94.99..-58.79 [-126.3080..-90.4748] | it/evals=795/1403 eff=72.0762% N=300
Mono-modal Volume: ~exp(-6.63) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| +2.5e-11 ************************* +7.3e-11 | +1.0e-10
Z=-100.1(0.00%) | Like=-94.00..-58.79 [-126.3080..-90.4748] | it/evals=804/1415 eff=72.1076% N=300
Z=-99.4(0.00%) | Like=-93.08..-58.79 [-126.3080..-90.4748] | it/evals=810/1422 eff=72.1925% N=300
Z=-97.7(0.00%) | Like=-90.98..-58.79 [-126.3080..-90.4748] | it/evals=826/1444 eff=72.2028% N=300
Z=-96.2(0.00%) | Like=-89.73..-58.79 [-90.3816..-75.0647] | it/evals=840/1465 eff=72.1030% N=300
Z=-94.7(0.00%) | Like=-88.52..-58.79 [-90.3816..-75.0647] | it/evals=853/1489 eff=71.7410% N=300
Z=-93.3(0.00%) | Like=-87.01..-58.79 [-90.3816..-75.0647] | it/evals=870/1511 eff=71.8415% N=300
Mono-modal Volume: ~exp(-7.17) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************** +7.1e-11 | +1.0e-10
Z=-93.2(0.00%) | Like=-86.91..-58.79 [-90.3816..-75.0647] | it/evals=871/1512 eff=71.8647% N=300
Z=-91.5(0.00%) | Like=-85.25..-58.79 [-90.3816..-75.0647] | it/evals=888/1535 eff=71.9028% N=300
Z=-90.4(0.00%) | Like=-84.26..-58.79 [-90.3816..-75.0647] | it/evals=900/1552 eff=71.8850% N=300
Z=-89.3(0.00%) | Like=-83.23..-58.79 [-90.3816..-75.0647] | it/evals=917/1574 eff=71.9780% N=300
Z=-88.4(0.00%) | Like=-82.35..-58.79 [-90.3816..-75.0647] | it/evals=930/1590 eff=72.0930% N=300
Mono-modal Volume: ~exp(-7.17) Expected Volume: exp(-3.13) 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=-87.5(0.00%) | Like=-81.11..-58.79 [-90.3816..-75.0647] | it/evals=945/1611 eff=72.0824% N=300
Z=-86.5(0.00%) | Like=-80.33..-58.79 [-90.3816..-75.0647] | it/evals=959/1635 eff=71.8352% N=300
Z=-86.4(0.00%) | Like=-80.19..-58.79 [-90.3816..-75.0647] | it/evals=960/1637 eff=71.8025% N=300
Z=-85.2(0.00%) | Like=-79.08..-58.79 [-90.3816..-75.0647] | it/evals=979/1659 eff=72.0383% N=300
Z=-84.6(0.00%) | Like=-78.43..-58.78 [-90.3816..-75.0647] | it/evals=990/1679 eff=71.7912% N=300
Z=-83.7(0.00%) | Like=-77.51..-58.78 [-90.3816..-75.0647] | it/evals=1004/1702 eff=71.6120% N=300
Mono-modal Volume: ~exp(-7.17) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************* +6.5e-11 | +1.0e-10
Z=-82.9(0.00%) | Like=-76.65..-58.78 [-90.3816..-75.0647] | it/evals=1018/1722 eff=71.5893% N=300
Z=-82.8(0.00%) | Like=-76.61..-58.78 [-90.3816..-75.0647] | it/evals=1020/1726 eff=71.5288% N=300
Z=-82.0(0.00%) | Like=-75.85..-58.76 [-90.3816..-75.0647] | it/evals=1035/1750 eff=71.3793% N=300
Z=-81.2(0.00%) | Like=-74.83..-58.76 [-75.0423..-66.5935] | it/evals=1050/1774 eff=71.2347% N=300
Z=-80.4(0.00%) | Like=-74.26..-58.76 [-75.0423..-66.5935] | it/evals=1066/1796 eff=71.2567% N=300
Mono-modal Volume: ~exp(-7.61) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************** +6.3e-11 | +1.0e-10
Z=-80.2(0.00%) | Like=-74.00..-58.76 [-75.0423..-66.5935] | it/evals=1072/1805 eff=71.2292% N=300
Z=-79.8(0.00%) | Like=-73.62..-58.76 [-75.0423..-66.5935] | it/evals=1080/1815 eff=71.2871% N=300
Z=-79.1(0.00%) | Like=-72.91..-58.76 [-75.0423..-66.5935] | it/evals=1097/1839 eff=71.2801% N=300
Z=-78.5(0.00%) | Like=-72.08..-58.76 [-75.0423..-66.5935] | it/evals=1110/1856 eff=71.3368% N=300
Z=-77.6(0.00%) | Like=-71.31..-58.76 [-75.0423..-66.5935] | it/evals=1128/1878 eff=71.4829% N=300
Mono-modal Volume: ~exp(-7.66) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************** +6.1e-11 | +1.0e-10
Z=-77.1(0.00%) | Like=-70.74..-58.76 [-75.0423..-66.5935] | it/evals=1139/1896 eff=71.3659% N=300
Z=-77.1(0.00%) | Like=-70.67..-58.76 [-75.0423..-66.5935] | it/evals=1140/1898 eff=71.3392% N=300
Z=-76.4(0.00%) | Like=-70.26..-58.76 [-75.0423..-66.5935] | it/evals=1159/1920 eff=71.5432% N=300
Z=-76.0(0.00%) | Like=-69.62..-58.76 [-75.0423..-66.5935] | it/evals=1170/1935 eff=71.5596% N=300
Z=-75.3(0.00%) | Like=-68.67..-58.76 [-75.0423..-66.5935] | it/evals=1188/1957 eff=71.6958% N=300
Z=-74.8(0.01%) | Like=-68.42..-58.76 [-75.0423..-66.5935] | it/evals=1200/1973 eff=71.7274% N=300
Mono-modal Volume: ~exp(-7.78) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********* +3.1 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************* +5.9e-11 | +1.0e-10
Z=-74.6(0.01%) | Like=-68.25..-58.76 [-75.0423..-66.5935] | it/evals=1206/1981 eff=71.7430% N=300
Z=-74.0(0.01%) | Like=-67.78..-58.76 [-75.0423..-66.5935] | it/evals=1225/2003 eff=71.9319% N=300
Z=-73.9(0.01%) | Like=-67.61..-58.76 [-75.0423..-66.5935] | it/evals=1230/2013 eff=71.8039% N=300
Z=-73.4(0.02%) | Like=-67.08..-58.76 [-75.0423..-66.5935] | it/evals=1245/2036 eff=71.7166% N=300
Z=-73.0(0.03%) | Like=-66.67..-58.76 [-75.0423..-66.5935] | it/evals=1260/2058 eff=71.6724% N=300
Mono-modal Volume: ~exp(-8.57) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************* +5.7e-11 | +1.0e-10
Z=-72.6(0.05%) | Like=-66.29..-58.76 [-66.5912..-65.2296] | it/evals=1273/2076 eff=71.6779% N=300
Z=-72.2(0.07%) | Like=-65.80..-58.76 [-66.5912..-65.2296] | it/evals=1290/2094 eff=71.9064% N=300
Z=-71.7(0.13%) | Like=-65.35..-58.76 [-66.5912..-65.2296] | it/evals=1311/2116 eff=72.1916% N=300
Z=-71.4(0.16%) | Like=-65.04..-58.76 [-65.1371..-65.0367] | it/evals=1320/2126 eff=72.2892% N=300
Mono-modal Volume: ~exp(-8.57) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.5e-11 | +1.0e-10
Z=-71.0(0.26%) | Like=-64.63..-58.76 [-64.6274..-64.5989] | it/evals=1340/2147 eff=72.5501% N=300
Z=-70.8(0.32%) | Like=-64.43..-58.76 [-64.4281..-64.4161] | it/evals=1350/2168 eff=72.2698% N=300
Z=-70.5(0.44%) | Like=-64.26..-58.76 [-64.2577..-64.2461] | it/evals=1366/2190 eff=72.2751% N=300
Z=-70.2(0.56%) | Like=-64.10..-58.76 [-64.1028..-64.0600] | it/evals=1380/2209 eff=72.2892% N=300
Z=-70.1(0.67%) | Like=-63.92..-58.76 [-63.9249..-63.9246]*| it/evals=1391/2232 eff=71.9979% N=300
Z=-69.9(0.81%) | Like=-63.74..-58.76 [-63.7363..-63.6824] | it/evals=1406/2255 eff=71.9182% N=300
Mono-modal Volume: ~exp(-8.72) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********** +5.4e-11 | +1.0e-10
Z=-69.8(0.82%) | Like=-63.68..-58.76 [-63.7363..-63.6824] | it/evals=1407/2256 eff=71.9325% N=300
Z=-69.8(0.87%) | Like=-63.60..-58.76 [-63.6007..-63.5813] | it/evals=1410/2260 eff=71.9388% N=300
Z=-69.5(1.11%) | Like=-63.35..-58.76 [-63.3541..-63.3481]*| it/evals=1428/2283 eff=72.0121% N=300
Z=-69.4(1.30%) | Like=-63.21..-58.76 [-63.2126..-63.1910] | it/evals=1440/2300 eff=72.0000% N=300
Z=-69.2(1.59%) | Like=-63.02..-58.76 [-63.0316..-63.0190] | it/evals=1455/2322 eff=71.9585% N=300
Z=-69.0(1.94%) | Like=-62.85..-58.76 [-62.8645..-62.8543] | it/evals=1470/2341 eff=72.0235% N=300
Mono-modal Volume: ~exp(-8.98) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.4 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********* +5.3e-11 | +1.0e-10
Z=-69.0(2.05%) | Like=-62.77..-58.76 [-62.7723..-62.7600] | it/evals=1474/2345 eff=72.0782% N=300
Z=-68.8(2.44%) | Like=-62.60..-58.76 [-62.5953..-62.5745] | it/evals=1490/2367 eff=72.0851% N=300
Z=-68.7(2.70%) | Like=-62.45..-58.76 [-62.4473..-62.4289] | it/evals=1500/2380 eff=72.1154% N=300
Z=-68.5(3.22%) | Like=-62.26..-58.76 [-62.2550..-62.2435] | it/evals=1517/2402 eff=72.1694% N=300
Z=-68.3(3.73%) | Like=-62.06..-58.76 [-62.0838..-62.0558] | it/evals=1530/2420 eff=72.1698% N=300
Mono-modal Volume: ~exp(-9.22) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******** +5.2e-11 | +1.0e-10
Z=-68.2(4.19%) | Like=-61.96..-58.76 [-61.9583..-61.9381] | it/evals=1541/2434 eff=72.2118% N=300
Z=-68.1(4.96%) | Like=-61.77..-58.76 [-61.7687..-61.7530] | it/evals=1557/2456 eff=72.2171% N=300
Z=-68.0(5.10%) | Like=-61.73..-58.76 [-61.7332..-61.7332]*| it/evals=1560/2459 eff=72.2557% N=300
Z=-67.9(5.84%) | Like=-61.55..-58.76 [-61.5549..-61.5310] | it/evals=1574/2483 eff=72.1026% N=300
Z=-67.8(6.58%) | Like=-61.44..-58.76 [-61.4392..-61.4206] | it/evals=1587/2506 eff=71.9402% N=300
Z=-67.7(6.75%) | Like=-61.39..-58.76 [-61.4175..-61.3891] | it/evals=1590/2510 eff=71.9457% N=300
Z=-67.6(7.52%) | Like=-61.29..-58.76 [-61.2928..-61.2866]*| it/evals=1602/2533 eff=71.7421% N=300
Mono-modal Volume: ~exp(-9.53) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******** +5.1e-11 | +1.0e-10
Z=-67.6(7.96%) | Like=-61.23..-58.76 [-61.2331..-61.2300]*| it/evals=1608/2542 eff=71.7217% N=300
Z=-67.5(8.89%) | Like=-61.16..-58.76 [-61.1597..-61.1594]*| it/evals=1620/2556 eff=71.8085% N=300
Z=-67.3(10.00%) | Like=-61.04..-58.76 [-61.0432..-61.0415]*| it/evals=1638/2578 eff=71.9052% N=300
Z=-67.2(11.00%) | Like=-60.93..-58.76 [-60.9263..-60.9241]*| it/evals=1650/2594 eff=71.9268% N=300
Z=-67.1(12.14%) | Like=-60.86..-58.76 [-60.8613..-60.8429] | it/evals=1664/2617 eff=71.8170% N=300
Mono-modal Volume: ~exp(-9.53) 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=-67.0(13.46%) | Like=-60.71..-58.76 [-60.7238..-60.7122] | it/evals=1680/2637 eff=71.8870% N=300
Z=-66.9(14.94%) | Like=-60.63..-58.76 [-60.6282..-60.6222]*| it/evals=1695/2659 eff=71.8525% N=300
Z=-66.9(15.83%) | Like=-60.56..-58.76 [-60.5583..-60.5580]*| it/evals=1705/2683 eff=71.5485% N=300
Z=-66.8(16.37%) | Like=-60.54..-58.76 [-60.5393..-60.5260] | it/evals=1710/2690 eff=71.5481% N=300
Z=-66.7(17.88%) | Like=-60.42..-58.76 [-60.4189..-60.4142]*| it/evals=1725/2713 eff=71.4878% N=300
Z=-66.7(18.88%) | Like=-60.36..-58.76 [-60.3567..-60.3553]*| it/evals=1735/2736 eff=71.2233% N=300
Z=-66.7(19.43%) | Like=-60.34..-58.76 [-60.3363..-60.3346]*| it/evals=1740/2751 eff=70.9914% N=300
Mono-modal Volume: ~exp(-10.18) * 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=-66.6(19.62%) | Like=-60.33..-58.75 [-60.3330..-60.3287]*| it/evals=1742/2753 eff=71.0151% N=300
Z=-66.6(21.30%) | Like=-60.27..-58.75 [-60.2696..-60.2671]*| it/evals=1756/2776 eff=70.9208% N=300
Z=-66.5(22.78%) | Like=-60.23..-58.75 [-60.2262..-60.2245]*| it/evals=1770/2796 eff=70.9135% N=300
Z=-66.4(24.49%) | Like=-60.14..-58.75 [-60.1367..-60.1183] | it/evals=1785/2819 eff=70.8615% N=300
Z=-66.4(26.13%) | Like=-60.08..-58.75 [-60.0804..-60.0801]*| it/evals=1800/2837 eff=70.9499% N=300
Mono-modal Volume: ~exp(-10.51) * Expected Volume: exp(-6.03) 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(27.13%) | Like=-60.05..-58.75 [-60.0535..-60.0470]*| it/evals=1809/2848 eff=70.9969% N=300
Z=-66.2(29.17%) | Like=-59.99..-58.75 [-59.9887..-59.9872]*| it/evals=1827/2870 eff=71.0895% N=300
Z=-66.2(29.53%) | Like=-59.98..-58.75 [-59.9770..-59.9755]*| it/evals=1830/2873 eff=71.1232% N=300
Z=-66.2(30.99%) | Like=-59.90..-58.75 [-59.9043..-59.8966]*| it/evals=1844/2895 eff=71.0597% N=300
Z=-66.1(32.90%) | Like=-59.83..-58.75 [-59.8315..-59.8263]*| it/evals=1860/2916 eff=71.1009% N=300
Mono-modal Volume: ~exp(-10.52) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ***** +4.8e-11 | +1.0e-10
Z=-66.1(34.88%) | Like=-59.77..-58.75 [-59.7690..-59.7686]*| it/evals=1876/2938 eff=71.1145% N=300
Z=-66.0(36.57%) | Like=-59.71..-58.75 [-59.7138..-59.7028] | it/evals=1890/2954 eff=71.2133% N=300
Z=-66.0(38.54%) | Like=-59.64..-58.75 [-59.6410..-59.6406]*| it/evals=1906/2977 eff=71.1991% N=300
Z=-65.9(40.24%) | Like=-59.60..-58.75 [-59.6023..-59.6017]*| it/evals=1920/2993 eff=71.2960% N=300
Z=-65.9(42.30%) | Like=-59.55..-58.75 [-59.5537..-59.5480]*| it/evals=1937/3015 eff=71.3444% N=300
Mono-modal Volume: ~exp(-10.63) * Expected Volume: exp(-6.48) 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(43.13%) | Like=-59.54..-58.75 [-59.5413..-59.5364]*| it/evals=1943/3022 eff=71.3813% N=300
Z=-65.8(43.95%) | Like=-59.52..-58.75 [-59.5167..-59.5157]*| it/evals=1950/3032 eff=71.3763% N=300
Z=-65.8(46.06%) | Like=-59.47..-58.75 [-59.4734..-59.4692]*| it/evals=1967/3054 eff=71.4234% N=300
Z=-65.8(47.65%) | Like=-59.44..-58.75 [-59.4371..-59.4336]*| it/evals=1980/3070 eff=71.4801% N=300
Z=-65.7(49.23%) | Like=-59.40..-58.75 [-59.4029..-59.4011]*| it/evals=1994/3092 eff=71.4183% N=300
Mono-modal Volume: ~exp(-10.85) * 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=-65.7(50.96%) | Like=-59.36..-58.75 [-59.3616..-59.3603]*| it/evals=2010/3113 eff=71.4540% N=300
Z=-65.6(53.07%) | Like=-59.33..-58.75 [-59.3278..-59.3263]*| it/evals=2028/3136 eff=71.5092% N=300
Z=-65.6(54.43%) | Like=-59.30..-58.75 [-59.2991..-59.2972]*| it/evals=2040/3150 eff=71.5789% N=300
Z=-65.6(56.58%) | Like=-59.26..-58.75 [-59.2586..-59.2585]*| it/evals=2059/3172 eff=71.6922% N=300
Z=-65.6(57.81%) | Like=-59.24..-58.75 [-59.2355..-59.2353]*| it/evals=2070/3185 eff=71.7504% N=300
Mono-modal Volume: ~exp(-11.13) * Expected Volume: exp(-6.92) 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(58.64%) | Like=-59.22..-58.75 [-59.2194..-59.2190]*| it/evals=2077/3195 eff=71.7444% N=300
Z=-65.5(60.35%) | Like=-59.20..-58.75 [-59.1985..-59.1966]*| it/evals=2094/3217 eff=71.7861% N=300
Z=-65.5(60.96%) | Like=-59.19..-58.75 [-59.1873..-59.1838]*| it/evals=2100/3225 eff=71.7949% N=300
Z=-65.5(62.27%) | Like=-59.16..-58.75 [-59.1648..-59.1638]*| it/evals=2113/3249 eff=71.6514% N=300
Z=-65.5(63.95%) | Like=-59.15..-58.75 [-59.1455..-59.1436]*| it/evals=2130/3270 eff=71.7172% N=300
Mono-modal Volume: ~exp(-11.17) * 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=-65.4(65.29%) | Like=-59.12..-58.75 [-59.1195..-59.1192]*| it/evals=2144/3290 eff=71.7057% N=300
Z=-65.4(66.67%) | Like=-59.11..-58.75 [-59.1054..-59.1043]*| it/evals=2159/3312 eff=71.6799% N=300
Z=-65.4(66.75%) | Like=-59.10..-58.75 [-59.1043..-59.1030]*| it/evals=2160/3313 eff=71.6893% N=300
Z=-65.4(68.30%) | Like=-59.08..-58.75 [-59.0780..-59.0759]*| it/evals=2177/3335 eff=71.7298% N=300
Z=-65.4(69.46%) | Like=-59.06..-58.75 [-59.0585..-59.0581]*| it/evals=2190/3351 eff=71.7797% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3362
[ultranest] logZ = -65.02 +- 0.0871
[ultranest] Effective samples strategy satisfied (ESS = 995.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.09 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:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.012 +- 0.339
single instance: logZ = -65.012 +- 0.133
bootstrapped : logZ = -65.022 +- 0.215
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.381 │ ▁▁▁▁▁▁▁▂▃▃▃▃▅▅▆▇▇▇▇▆▆▅▆▄▄▂▂▂▁▁▁▁▁▁ ▁ │2.985 2.670 +- 0.083
amplitude : 0.0000000000340│ ▁▁▁▁▁▁▁▁▃▃▄▄▄▅▆▆▇▇▆▅▅▄▄▃▃▃▂▁▁▁▁▁▁▁▁▁▁ │0.0000000000561 0.0000000000443 +- 0.0000000000031
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.
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. 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.
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.
Results#
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 DatasetModels will be updated with the mean of
the posterior distributions.
print(result_joint.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.670 +/- 0.08
amplitude : 4.43e-11 +/- 3.1e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.67001100522129, 4.427769125942066e-11], 'stdev': [0.0828390870242246, 3.054253161726441e-12], 'median': [2.6714775324915045, 4.414647594894338e-11], 'errlo': [2.5865330325041707, 4.1259335571019636e-11], 'errup': [2.750409994667595, 4.742565653741921e-11], 'information_gain_bits': [2.6818047413837007, 3.11775830160564]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrunk 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.
The SamplerResult dictionary contains also other interesting
information :
print(result_joint.sampler_results.keys())
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'])
Of particular interest, the samples used in the process to approximate the posterior distribution can be accessed via :
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()
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 package. See the above link for optional keywords. Other packages exist for corner plots, like chainconsumer which is discussed later in this tutorial.
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()
Spectral model error band from samples#
To compute the spectral error band (“butterfly plots”), we will directly use the samples of the posterior distribution. This is more robust as compared to the traditional method of using the covariance matrix of the parameters which implicitly assumes Gaussian errors while for the posterior distribution there is no shape assumed. This difference can become significant when the parameter errors are non-Gaussian. For this we will need to convert the list of samples back to the spectral model parameters with the relevant units (e.g. normalisation units).
def get_samples_from_posterior(spectral_model, results):
"""
Create a list of spectral parameters with correct units
from the unitless parameters returned by the sampler.
"""
n_samples = results.samples.shape[0]
samples = []
for p in spectral_model.parameters:
try:
idx = spectral_model.parameters.free_unique_parameters.index(p)
samples.append(results.samples[:, idx] * p.unit)
except ValueError:
samples.append(np.ones(n_samples) * p.quantity)
return samples
samples = get_samples_from_posterior(datasets.models[0].spectral_model, result_joint)
Next we can provide these samples to the plot_error
method.
Individual run analysis#
Now we’ll analyse several Crab runs individually so that we can compare them.
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.99) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|********** *************************************| +5.0
amplitude: +1.0e-12|******************************* * * *** * ** **| +1.0e-10
Z=-inf(0.00%) | Like=-878.52..-20.47 [-878.5246..-111.9958] | it/evals=0/301 eff=0.0000% N=300
Z=-178.1(0.00%) | Like=-173.61..-20.47 [-878.5246..-111.9958] | it/evals=30/331 eff=96.7742% N=300
Z=-167.9(0.00%) | Like=-163.07..-20.47 [-878.5246..-111.9958] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-3.99) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************* * * *** ** * * | +1.0e-10
Z=-157.3(0.00%) | Like=-152.19..-20.47 [-878.5246..-111.9958] | it/evals=90/396 eff=93.7500% N=300
Z=-147.1(0.00%) | Like=-141.37..-20.47 [-878.5246..-111.9958] | it/evals=118/437 eff=86.1314% N=300
Z=-146.2(0.00%) | Like=-141.32..-20.47 [-878.5246..-111.9958] | it/evals=120/442 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.08) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************* **** *** **** * | +1.0e-10
Z=-141.3(0.00%) | Like=-135.32..-20.47 [-878.5246..-111.9958] | it/evals=134/456 eff=85.8974% N=300
Z=-137.4(0.00%) | Like=-132.37..-20.47 [-878.5246..-111.9958] | it/evals=150/479 eff=83.7989% N=300
Z=-125.8(0.00%) | Like=-119.85..-20.47 [-878.5246..-111.9958] | it/evals=180/513 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.42) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ***************************** ** ** *** ***** | +1.0e-10
Z=-118.9(0.00%) | Like=-112.91..-20.47 [-878.5246..-111.9958] | it/evals=201/538 eff=84.4538% N=300
Z=-115.8(0.00%) | Like=-110.65..-20.47 [-111.7361..-67.1879] | it/evals=210/552 eff=83.3333% N=300
Z=-107.9(0.00%) | Like=-102.22..-20.47 [-111.7361..-67.1879] | it/evals=240/590 eff=82.7586% N=300
Mono-modal Volume: ~exp(-4.42) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ***************************** ***** *** ***** | +1.0e-10
Z=-99.4(0.00%) | Like=-93.96..-20.47 [-111.7361..-67.1879] | it/evals=270/627 eff=82.5688% N=300
Z=-92.5(0.00%) | Like=-87.12..-20.47 [-111.7361..-67.1879] | it/evals=298/669 eff=80.7588% N=300
Z=-91.9(0.00%) | Like=-86.29..-20.47 [-111.7361..-67.1879] | it/evals=300/672 eff=80.6452% N=300
Z=-84.4(0.00%) | Like=-78.77..-20.47 [-111.7361..-67.1879] | it/evals=330/713 eff=79.9031% N=300
Mono-modal Volume: ~exp(-5.04) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| **************************** ***** *** * * | +1.0e-10
Z=-83.1(0.00%) | Like=-77.53..-20.47 [-111.7361..-67.1879] | it/evals=335/722 eff=79.3839% N=300
Z=-78.9(0.00%) | Like=-74.09..-20.47 [-111.7361..-67.1879] | it/evals=360/752 eff=79.6460% N=300
Z=-75.7(0.00%) | Like=-71.21..-20.47 [-111.7361..-67.1879] | it/evals=390/790 eff=79.5918% N=300
Mono-modal Volume: ~exp(-5.33) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| **************************** **** ** | +1.0e-10
Z=-74.8(0.00%) | Like=-70.06..-20.47 [-111.7361..-67.1879] | it/evals=402/805 eff=79.6040% N=300
Z=-72.3(0.00%) | Like=-67.58..-20.47 [-111.7361..-67.1879] | it/evals=420/832 eff=78.9474% N=300
Z=-69.8(0.00%) | Like=-65.04..-20.47 [-67.1841..-47.3822] | it/evals=447/874 eff=77.8746% N=300
Z=-69.5(0.00%) | Like=-64.52..-20.47 [-67.1841..-47.3822] | it/evals=450/879 eff=77.7202% N=300
Mono-modal Volume: ~exp(-5.64) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ****************************** * +7.5e-11 | +1.0e-10
Z=-67.6(0.00%) | Like=-61.89..-20.47 [-67.1841..-47.3822] | it/evals=469/906 eff=77.3927% N=300
Z=-66.2(0.00%) | Like=-61.01..-20.47 [-67.1841..-47.3822] | it/evals=480/921 eff=77.2947% N=300
Z=-63.2(0.00%) | Like=-58.14..-20.47 [-67.1841..-47.3822] | it/evals=510/959 eff=77.3900% N=300
Mono-modal Volume: ~exp(-5.85) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ***************************** +4.1 | +5.0
amplitude: +1.0e-12| ****************************** * +7.3e-11 | +1.0e-10
Z=-60.2(0.00%) | Like=-54.89..-20.47 [-67.1841..-47.3822] | it/evals=536/997 eff=76.9010% N=300
Z=-59.8(0.00%) | Like=-54.68..-20.47 [-67.1841..-47.3822] | it/evals=540/1001 eff=77.0328% N=300
Z=-57.0(0.00%) | Like=-51.60..-20.47 [-67.1841..-47.3822] | it/evals=569/1044 eff=76.4785% N=300
Z=-56.9(0.00%) | Like=-51.59..-20.47 [-67.1841..-47.3822] | it/evals=570/1047 eff=76.3052% N=300
Z=-54.9(0.00%) | Like=-49.74..-20.47 [-67.1841..-47.3822] | it/evals=594/1089 eff=75.2852% N=300
Z=-54.5(0.00%) | Like=-49.31..-20.47 [-67.1841..-47.3822] | it/evals=600/1103 eff=74.7198% N=300
Mono-modal Volume: ~exp(-6.05) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +3.9 | +5.0
amplitude: +1.0e-12| ***************************** +7.1e-11 | +1.0e-10
Z=-54.2(0.00%) | Like=-49.13..-20.47 [-67.1841..-47.3822] | it/evals=603/1111 eff=74.3527% N=300
Z=-52.5(0.00%) | Like=-47.66..-20.47 [-67.1841..-47.3822] | it/evals=630/1144 eff=74.6445% N=300
Z=-51.2(0.00%) | Like=-46.28..-20.47 [-47.3386..-34.9303] | it/evals=656/1186 eff=74.0406% N=300
Z=-51.0(0.00%) | Like=-45.52..-20.47 [-47.3386..-34.9303] | it/evals=660/1190 eff=74.1573% N=300
Mono-modal Volume: ~exp(-6.27) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.7 | +5.0
amplitude: +1.0e-12| ************************** +6.5e-11 | +1.0e-10
Z=-50.2(0.00%) | Like=-44.73..-20.47 [-47.3386..-34.9303] | it/evals=670/1205 eff=74.0331% N=300
Z=-48.7(0.00%) | Like=-43.28..-20.47 [-47.3386..-34.9303] | it/evals=690/1235 eff=73.7968% N=300
Z=-46.8(0.00%) | Like=-41.43..-20.47 [-47.3386..-34.9303] | it/evals=720/1276 eff=73.7705% N=300
Mono-modal Volume: ~exp(-6.39) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************* +3.6 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-45.7(0.00%) | Like=-40.48..-20.47 [-47.3386..-34.9303] | it/evals=737/1306 eff=73.2604% N=300
Z=-45.1(0.00%) | Like=-39.84..-20.47 [-47.3386..-34.9303] | it/evals=750/1327 eff=73.0282% N=300
Z=-43.5(0.00%) | Like=-38.09..-20.47 [-47.3386..-34.9303] | it/evals=775/1369 eff=72.4977% N=300
Z=-43.3(0.00%) | Like=-37.95..-20.47 [-47.3386..-34.9303] | it/evals=780/1377 eff=72.4234% N=300
Z=-42.2(0.00%) | Like=-36.98..-20.47 [-47.3386..-34.9303] | it/evals=802/1419 eff=71.6711% N=300
Mono-modal Volume: ~exp(-6.62) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +1.9 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-42.1(0.00%) | Like=-36.89..-20.47 [-47.3386..-34.9303] | it/evals=804/1421 eff=71.7217% N=300
Z=-41.9(0.00%) | Like=-36.41..-20.47 [-47.3386..-34.9303] | it/evals=810/1428 eff=71.8085% N=300
Z=-40.4(0.00%) | Like=-35.01..-20.47 [-47.3386..-34.9303] | it/evals=838/1471 eff=71.5628% N=300
Z=-40.3(0.00%) | Like=-35.00..-20.47 [-47.3386..-34.9303] | it/evals=840/1473 eff=71.6113% N=300
Z=-39.0(0.00%) | Like=-33.56..-20.47 [-34.9171..-27.8632] | it/evals=870/1511 eff=71.8415% N=300
Mono-modal Volume: ~exp(-6.62) 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=-38.2(0.00%) | Like=-32.82..-20.47 [-34.9171..-27.8632] | it/evals=892/1549 eff=71.4171% N=300
Z=-37.9(0.00%) | Like=-32.36..-20.47 [-34.9171..-27.8632] | it/evals=900/1560 eff=71.4286% N=300
Z=-36.7(0.00%) | Like=-31.14..-20.47 [-34.9171..-27.8632] | it/evals=929/1602 eff=71.3518% N=300
Z=-36.6(0.00%) | Like=-31.12..-20.47 [-34.9171..-27.8632] | it/evals=930/1603 eff=71.3738% N=300
Mono-modal Volume: ~exp(-6.91) * 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=-36.4(0.00%) | Like=-30.95..-20.47 [-34.9171..-27.8632] | it/evals=938/1612 eff=71.4939% N=300
Z=-35.7(0.01%) | Like=-30.12..-20.47 [-34.9171..-27.8632] | it/evals=960/1654 eff=70.9010% N=300
Z=-34.8(0.01%) | Like=-29.33..-20.47 [-34.9171..-27.8632] | it/evals=985/1697 eff=70.5082% N=300
Z=-34.7(0.01%) | Like=-29.16..-20.47 [-34.9171..-27.8632] | it/evals=990/1703 eff=70.5631% N=300
Mono-modal Volume: ~exp(-6.91) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| **************** +5.2e-11 | +1.0e-10
Z=-34.0(0.03%) | Like=-28.52..-20.47 [-34.9171..-27.8632] | it/evals=1012/1741 eff=70.2290% N=300
Z=-33.8(0.03%) | Like=-28.34..-20.47 [-34.9171..-27.8632] | it/evals=1020/1755 eff=70.1031% N=300
Z=-33.2(0.06%) | Like=-27.77..-20.47 [-27.8316..-26.8876] | it/evals=1045/1799 eff=69.7131% N=300
Z=-33.1(0.07%) | Like=-27.62..-20.47 [-27.8316..-26.8876] | it/evals=1050/1807 eff=69.6749% N=300
Mono-modal Volume: ~exp(-7.44) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************ +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-32.5(0.13%) | Like=-26.78..-20.47 [-26.7957..-26.7767] | it/evals=1072/1840 eff=69.6104% N=300
Z=-32.3(0.16%) | Like=-26.65..-20.47 [-26.6453..-26.5943] | it/evals=1080/1851 eff=69.6325% N=300
Z=-31.6(0.31%) | Like=-26.21..-20.47 [-26.2126..-26.1659] | it/evals=1109/1892 eff=69.6608% N=300
Z=-31.6(0.31%) | Like=-26.17..-20.47 [-26.2126..-26.1659] | it/evals=1110/1897 eff=69.5053% N=300
Z=-31.2(0.48%) | Like=-25.70..-20.46 [-25.7400..-25.7046] | it/evals=1134/1939 eff=69.1885% N=300
Mono-modal Volume: ~exp(-8.10) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.7e-11 | +1.0e-10
Z=-31.1(0.53%) | Like=-25.60..-20.46 [-25.6127..-25.6016] | it/evals=1139/1945 eff=69.2401% N=300
Z=-31.1(0.54%) | Like=-25.56..-20.46 [-25.5577..-25.5572]*| it/evals=1140/1946 eff=69.2588% N=300
Z=-30.6(0.87%) | Like=-25.16..-20.46 [-25.1760..-25.1624] | it/evals=1170/1982 eff=69.5600% N=300
Z=-30.2(1.27%) | Like=-24.87..-20.46 [-24.9026..-24.8656] | it/evals=1198/2022 eff=69.5703% N=300
Z=-30.2(1.30%) | Like=-24.85..-20.46 [-24.8547..-24.8413] | it/evals=1200/2025 eff=69.5652% N=300
Mono-modal Volume: ~exp(-8.10) 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=-29.8(1.88%) | Like=-24.34..-20.46 [-24.3584..-24.3428] | it/evals=1228/2062 eff=69.6935% N=300
Z=-29.8(1.94%) | Like=-24.29..-20.46 [-24.3420..-24.2948] | it/evals=1230/2064 eff=69.7279% N=300
Z=-29.5(2.71%) | Like=-23.90..-20.46 [-23.9127..-23.9025] | it/evals=1255/2105 eff=69.5291% N=300
Z=-29.4(2.93%) | Like=-23.80..-20.46 [-23.8013..-23.7849] | it/evals=1260/2114 eff=69.4598% N=300
Mono-modal Volume: ~exp(-8.45) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.4e-11 | +1.0e-10
Z=-29.2(3.52%) | Like=-23.71..-20.46 [-23.7073..-23.6974]*| it/evals=1273/2133 eff=69.4490% N=300
Z=-29.0(4.28%) | Like=-23.51..-20.46 [-23.5252..-23.5099] | it/evals=1290/2157 eff=69.4669% N=300
Z=-28.7(5.94%) | Like=-23.21..-20.46 [-23.2054..-23.1821] | it/evals=1320/2197 eff=69.5836% N=300
Mono-modal Volume: ~exp(-8.61) * 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=-28.5(7.30%) | Like=-23.03..-20.46 [-23.0250..-23.0224]*| it/evals=1340/2224 eff=69.6466% N=300
Z=-28.4(8.05%) | Like=-22.94..-20.46 [-22.9443..-22.9392]*| it/evals=1350/2236 eff=69.7314% N=300
Z=-28.2(10.36%) | Like=-22.81..-20.46 [-22.8106..-22.8042]*| it/evals=1380/2273 eff=69.9442% N=300
Z=-28.0(12.35%) | Like=-22.56..-20.46 [-22.5628..-22.5619]*| it/evals=1405/2312 eff=69.8310% N=300
Mono-modal Volume: ~exp(-8.61) 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=-28.0(12.76%) | Like=-22.54..-20.46 [-22.5447..-22.5219] | it/evals=1410/2317 eff=69.9058% N=300
Z=-27.8(15.47%) | Like=-22.38..-20.46 [-22.3754..-22.3747]*| it/evals=1439/2359 eff=69.8883% N=300
Z=-27.8(15.55%) | Like=-22.37..-20.46 [-22.3747..-22.3731]*| it/evals=1440/2360 eff=69.9029% N=300
Z=-27.7(17.64%) | Like=-22.21..-20.46 [-22.2054..-22.2038]*| it/evals=1461/2401 eff=69.5383% N=300
Z=-27.6(18.55%) | Like=-22.15..-20.46 [-22.1498..-22.1413]*| it/evals=1470/2413 eff=69.5693% N=300
Mono-modal Volume: ~exp(-8.61) 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=-27.4(21.36%) | Like=-21.99..-20.46 [-21.9950..-21.9814] | it/evals=1497/2452 eff=69.5632% N=300
Z=-27.4(21.71%) | Like=-21.98..-20.46 [-21.9794..-21.9592] | it/evals=1500/2455 eff=69.6056% N=300
Z=-27.3(24.48%) | Like=-21.87..-20.46 [-21.8658..-21.8655]*| it/evals=1525/2498 eff=69.3813% N=300
Z=-27.3(25.08%) | Like=-21.83..-20.46 [-21.8322..-21.8313]*| it/evals=1530/2504 eff=69.4192% N=300
Mono-modal Volume: ~exp(-9.45) * 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=-27.2(26.54%) | Like=-21.80..-20.46 [-21.7968..-21.7952]*| it/evals=1541/2518 eff=69.4770% N=300
Z=-27.2(28.81%) | Like=-21.72..-20.46 [-21.7361..-21.7239] | it/evals=1560/2539 eff=69.6740% N=300
Z=-27.0(32.32%) | Like=-21.61..-20.46 [-21.6082..-21.6077]*| it/evals=1588/2578 eff=69.7103% N=300
Z=-27.0(32.54%) | Like=-21.61..-20.46 [-21.6062..-21.5980]*| it/evals=1590/2582 eff=69.6757% N=300
Mono-modal Volume: ~exp(-9.45) 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=-26.9(35.94%) | Like=-21.50..-20.46 [-21.4983..-21.4930]*| it/evals=1618/2618 eff=69.8016% N=300
Z=-26.9(36.11%) | Like=-21.49..-20.46 [-21.4913..-21.4910]*| it/evals=1620/2621 eff=69.7975% N=300
Z=-26.8(39.01%) | Like=-21.44..-20.46 [-21.4389..-21.4346]*| it/evals=1644/2662 eff=69.6020% N=300
Z=-26.8(39.68%) | Like=-21.43..-20.46 [-21.4327..-21.4293]*| it/evals=1650/2672 eff=69.5616% N=300
Mono-modal Volume: ~exp(-9.49) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.8(42.58%) | Like=-21.37..-20.46 [-21.3667..-21.3577]*| it/evals=1675/2704 eff=69.6755% N=300
Z=-26.7(43.19%) | Like=-21.35..-20.46 [-21.3489..-21.3387] | it/evals=1680/2711 eff=69.6806% N=300
Z=-26.7(46.52%) | Like=-21.26..-20.46 [-21.2619..-21.2603]*| it/evals=1708/2751 eff=69.6858% N=300
Z=-26.7(46.72%) | Like=-21.26..-20.46 [-21.2598..-21.2537]*| it/evals=1710/2754 eff=69.6822% N=300
Z=-26.6(50.17%) | Like=-21.18..-20.46 [-21.1784..-21.1771]*| it/evals=1740/2792 eff=69.8234% N=300
Mono-modal Volume: ~exp(-10.03) * 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(50.41%) | Like=-21.17..-20.46 [-21.1725..-21.1692]*| it/evals=1742/2794 eff=69.8476% N=300
Z=-26.5(53.55%) | Like=-21.12..-20.46 [-21.1169..-21.1159]*| it/evals=1770/2828 eff=70.0158% N=300
Z=-26.5(56.80%) | Like=-21.05..-20.46 [-21.0547..-21.0505]*| it/evals=1800/2859 eff=70.3400% N=300
Mono-modal Volume: ~exp(-10.09) * 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=-26.5(57.75%) | Like=-21.03..-20.46 [-21.0303..-21.0300]*| it/evals=1809/2868 eff=70.4439% N=300
Z=-26.4(59.90%) | Like=-21.00..-20.46 [-21.0012..-21.0010]*| it/evals=1830/2896 eff=70.4931% N=300
Z=-26.4(62.75%) | Like=-20.97..-20.46 [-20.9664..-20.9641]*| it/evals=1859/2938 eff=70.4701% N=300
Z=-26.4(62.86%) | Like=-20.96..-20.46 [-20.9641..-20.9615]*| it/evals=1860/2939 eff=70.4812% N=300
Mono-modal Volume: ~exp(-10.11) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.3(64.29%) | Like=-20.93..-20.46 [-20.9326..-20.9302]*| it/evals=1876/2961 eff=70.4998% N=300
Z=-26.3(65.60%) | Like=-20.90..-20.46 [-20.9048..-20.9045]*| it/evals=1890/2976 eff=70.6278% N=300
Z=-26.3(67.85%) | Like=-20.87..-20.46 [-20.8661..-20.8659]*| it/evals=1915/3017 eff=70.4821% N=300
Z=-26.3(68.31%) | Like=-20.86..-20.46 [-20.8623..-20.8605]*| it/evals=1920/3023 eff=70.5105% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3053
[ultranest] logZ = -25.92 +- 0.09193
[ultranest] Effective samples strategy satisfied (ESS = 998.5, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.44+-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:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.905 +- 0.337
single instance: logZ = -25.905 +- 0.121
bootstrapped : logZ = -25.922 +- 0.212
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.17 │ ▁▁▁▁▁▁▂▂▃▃▄▄▅▆▇▇▆▇▆▆▅▅▄▄▄▃▁▂▁▁▁▁▁ ▁ ▁ │3.05 2.57 +- 0.13
amplitude : 0.0000000000204│ ▁▁▁▁ ▁▁▁▂▂▄▄▅▆▆▆▇▆▆▇▅▅▆▅▄▂▂▂▂▁▁▁▁▁▁ ▁ │0.0000000000486 0.0000000000341 +- 0.0000000000039
[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=-1292.59..-19.27 [-1292.5886..-137.7797] | it/evals=0/301 eff=0.0000% N=300
Z=-220.6(0.00%) | Like=-215.97..-19.27 [-1292.5886..-137.7797] | it/evals=30/333 eff=90.9091% N=300
Z=-203.6(0.00%) | Like=-198.38..-19.27 [-1292.5886..-137.7797] | it/evals=60/367 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.22) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** *********** * *** | +1.0e-10
Z=-201.0(0.00%) | Like=-196.03..-19.27 [-1292.5886..-137.7797] | it/evals=67/374 eff=90.5405% N=300
Z=-191.1(0.00%) | Like=-186.13..-19.27 [-1292.5886..-137.7797] | it/evals=90/399 eff=90.9091% N=300
Z=-180.8(0.00%) | Like=-172.61..-19.27 [-1292.5886..-137.7797] | it/evals=120/431 eff=91.6031% N=300
Mono-modal Volume: ~exp(-4.22) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ************* *** | +1.0e-10
Z=-165.4(0.00%) | Like=-159.34..-19.27 [-1292.5886..-137.7797] | it/evals=150/466 eff=90.3614% N=300
Z=-155.1(0.00%) | Like=-148.91..-19.27 [-1292.5886..-137.7797] | it/evals=180/501 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.22) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| **************************** ***************** | +1.0e-10
Z=-144.3(0.00%) | Like=-138.90..-19.27 [-1292.5886..-137.7797] | it/evals=207/538 eff=86.9748% N=300
Z=-142.8(0.00%) | Like=-136.38..-19.27 [-137.0381..-66.2814] | it/evals=210/541 eff=87.1369% N=300
Z=-131.7(0.00%) | Like=-126.88..-19.27 [-137.0381..-66.2814] | it/evals=238/582 eff=84.3972% N=300
Z=-131.1(0.00%) | Like=-125.17..-19.27 [-137.0381..-66.2814] | it/evals=240/584 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.29) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-121.4(0.00%) | Like=-116.05..-19.27 [-137.0381..-66.2814] | it/evals=268/620 eff=83.7500% N=300
Z=-120.9(0.00%) | Like=-115.69..-19.27 [-137.0381..-66.2814] | it/evals=270/624 eff=83.3333% N=300
Z=-107.6(0.00%) | Like=-101.55..-19.27 [-137.0381..-66.2814] | it/evals=300/663 eff=82.6446% N=300
Z=-97.6(0.00%) | Like=-91.77..-19.27 [-137.0381..-66.2814] | it/evals=327/705 eff=80.7407% N=300
Z=-96.8(0.00%) | Like=-91.12..-19.27 [-137.0381..-66.2814] | it/evals=330/711 eff=80.2920% N=300
Mono-modal Volume: ~exp(-4.71) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-95.1(0.00%) | Like=-89.78..-19.27 [-137.0381..-66.2814] | it/evals=335/718 eff=80.1435% N=300
Z=-88.4(0.00%) | Like=-82.78..-19.27 [-137.0381..-66.2814] | it/evals=360/752 eff=79.6460% N=300
Z=-78.5(0.00%) | Like=-72.75..-19.27 [-137.0381..-66.2814] | it/evals=390/789 eff=79.7546% N=300
Mono-modal Volume: ~exp(-5.35) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-75.4(0.00%) | Like=-69.86..-19.27 [-137.0381..-66.2814] | it/evals=402/806 eff=79.4466% N=300
Z=-70.8(0.00%) | Like=-65.09..-19.27 [-66.2773..-42.5754] | it/evals=420/828 eff=79.5455% N=300
Z=-65.0(0.00%) | Like=-59.45..-19.27 [-66.2773..-42.5754] | it/evals=450/867 eff=79.3651% N=300
Mono-modal Volume: ~exp(-5.35) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ***************************************| +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-60.4(0.00%) | Like=-55.20..-19.27 [-66.2773..-42.5754] | it/evals=478/904 eff=79.1391% N=300
Z=-60.1(0.00%) | Like=-54.76..-19.27 [-66.2773..-42.5754] | it/evals=480/908 eff=78.9474% N=300
Z=-56.7(0.00%) | Like=-51.43..-19.27 [-66.2773..-42.5754] | it/evals=504/952 eff=77.3006% N=300
Z=-56.0(0.00%) | Like=-50.90..-19.27 [-66.2773..-42.5754] | it/evals=510/964 eff=76.8072% N=300
Mono-modal Volume: ~exp(-5.95) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ************************************* *| +1.0e-10
Z=-54.1(0.00%) | Like=-49.44..-19.27 [-66.2773..-42.5754] | it/evals=536/1001 eff=76.4622% N=300
Z=-53.9(0.00%) | Like=-49.12..-19.27 [-66.2773..-42.5754] | it/evals=540/1007 eff=76.3791% N=300
Z=-51.4(0.00%) | Like=-46.36..-19.27 [-66.2773..-42.5754] | it/evals=570/1042 eff=76.8194% N=300
Z=-49.3(0.00%) | Like=-44.10..-19.27 [-66.2773..-42.5754] | it/evals=600/1082 eff=76.7263% N=300
Mono-modal Volume: ~exp(-5.95) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ****************************** | +5.0
amplitude: +1.0e-12| +2.4e-11 *************************************| +1.0e-10
Z=-46.9(0.00%) | Like=-41.57..-19.27 [-42.5198..-31.3837] | it/evals=625/1121 eff=76.1267% N=300
Z=-46.5(0.00%) | Like=-41.39..-19.27 [-42.5198..-31.3837] | it/evals=630/1132 eff=75.7212% N=300
Z=-44.9(0.00%) | Like=-40.00..-19.27 [-42.5198..-31.3837] | it/evals=657/1173 eff=75.2577% N=300
Z=-44.7(0.00%) | Like=-39.43..-19.27 [-42.5198..-31.3837] | it/evals=660/1177 eff=75.2566% N=300
Mono-modal Volume: ~exp(-6.21) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************* * | +5.0
amplitude: +1.0e-12| +2.5e-11 ************************************ | +1.0e-10
Z=-44.0(0.00%) | Like=-38.84..-19.27 [-42.5198..-31.3837] | it/evals=670/1193 eff=75.0280% N=300
Z=-42.9(0.00%) | Like=-37.78..-19.27 [-42.5198..-31.3837] | it/evals=690/1219 eff=75.0816% N=300
Z=-41.2(0.00%) | Like=-36.13..-19.27 [-42.5198..-31.3837] | it/evals=720/1260 eff=75.0000% N=300
Mono-modal Volume: ~exp(-6.28) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ************************ +4.0 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************************** | +1.0e-10
Z=-40.4(0.00%) | Like=-35.33..-19.27 [-42.5198..-31.3837] | it/evals=737/1290 eff=74.4444% N=300
Z=-39.9(0.00%) | Like=-34.67..-19.25 [-42.5198..-31.3837] | it/evals=750/1309 eff=74.3310% N=300
Z=-38.4(0.00%) | Like=-32.96..-19.25 [-42.5198..-31.3837] | it/evals=780/1350 eff=74.2857% N=300
Mono-modal Volume: ~exp(-6.52) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ********************* +3.8 | +5.0
amplitude: +1.0e-12| +3.1e-11 ****************************** | +1.0e-10
Z=-37.2(0.00%) | Like=-31.91..-19.25 [-42.5198..-31.3837] | it/evals=804/1386 eff=74.0331% N=300
Z=-37.0(0.00%) | Like=-31.85..-19.25 [-42.5198..-31.3837] | it/evals=810/1396 eff=73.9051% N=300
Z=-36.0(0.00%) | Like=-31.09..-19.25 [-31.3494..-26.2639] | it/evals=840/1436 eff=73.9437% N=300
Z=-35.3(0.00%) | Like=-30.24..-19.23 [-31.3494..-26.2639] | it/evals=868/1477 eff=73.7468% N=300
Z=-35.2(0.00%) | Like=-30.13..-19.23 [-31.3494..-26.2639] | it/evals=870/1479 eff=73.7913% N=300
Mono-modal Volume: ~exp(-6.76) * 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=-35.2(0.00%) | Like=-30.11..-19.23 [-31.3494..-26.2639] | it/evals=871/1482 eff=73.6887% N=300
Z=-34.4(0.01%) | Like=-29.27..-19.23 [-31.3494..-26.2639] | it/evals=900/1523 eff=73.5895% N=300
Z=-33.7(0.01%) | Like=-28.41..-19.23 [-31.3494..-26.2639] | it/evals=922/1567 eff=72.7703% N=300
Z=-33.5(0.01%) | Like=-28.14..-19.23 [-31.3494..-26.2639] | it/evals=930/1582 eff=72.5429% N=300
Mono-modal Volume: ~exp(-6.91) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 ***************** +3.6 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************************ | +1.0e-10
Z=-33.2(0.02%) | Like=-27.85..-19.23 [-31.3494..-26.2639] | it/evals=938/1592 eff=72.6006% N=300
Z=-32.6(0.03%) | Like=-27.20..-19.23 [-31.3494..-26.2639] | it/evals=960/1620 eff=72.7273% N=300
Z=-31.8(0.06%) | Like=-26.51..-19.23 [-31.3494..-26.2639] | it/evals=987/1661 eff=72.5202% N=300
Z=-31.8(0.07%) | Like=-26.51..-19.23 [-31.3494..-26.2639] | it/evals=990/1665 eff=72.5275% N=300
Mono-modal Volume: ~exp(-6.91) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 **************** +3.5 | +5.0
amplitude: +1.0e-12| +3.6e-11 *********************** | +1.0e-10
Z=-31.2(0.12%) | Like=-25.95..-19.22 [-26.2469..-25.6817] | it/evals=1012/1703 eff=72.1311% N=300
Z=-31.1(0.14%) | Like=-25.76..-19.22 [-26.2469..-25.6817] | it/evals=1020/1715 eff=72.0848% N=300
Z=-30.5(0.24%) | Like=-25.17..-19.21 [-25.1793..-25.1681] | it/evals=1043/1760 eff=71.4384% N=300
Z=-30.4(0.28%) | Like=-25.10..-19.21 [-25.0993..-25.0870] | it/evals=1050/1768 eff=71.5259% N=300
Mono-modal Volume: ~exp(-7.31) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************** +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.6e-11 | +1.0e-10
Z=-30.0(0.43%) | Like=-24.69..-19.21 [-24.6939..-24.6743] | it/evals=1072/1804 eff=71.2766% N=300
Z=-29.8(0.50%) | Like=-24.51..-19.21 [-24.5113..-24.4687] | it/evals=1080/1813 eff=71.3814% N=300
Z=-29.3(0.80%) | Like=-24.05..-19.21 [-24.0967..-24.0529] | it/evals=1108/1853 eff=71.3458% N=300
Z=-29.3(0.82%) | Like=-24.02..-19.21 [-24.0450..-24.0157] | it/evals=1110/1855 eff=71.3826% N=300
Mono-modal Volume: ~exp(-7.97) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************* +3.3 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****************** +7.4e-11 | +1.0e-10
Z=-28.9(1.26%) | Like=-23.61..-19.20 [-23.6122..-23.5480] | it/evals=1139/1894 eff=71.4555% N=300
Z=-28.8(1.27%) | Like=-23.55..-19.20 [-23.6122..-23.5480] | it/evals=1140/1895 eff=71.4734% N=300
Z=-28.4(1.85%) | Like=-23.20..-19.19 [-23.1986..-23.1637] | it/evals=1170/1929 eff=71.8232% N=300
Z=-28.1(2.67%) | Like=-22.77..-19.19 [-22.7800..-22.7698] | it/evals=1199/1969 eff=71.8394% N=300
Z=-28.1(2.71%) | Like=-22.77..-19.19 [-22.7675..-22.7377] | it/evals=1200/1970 eff=71.8563% N=300
Mono-modal Volume: ~exp(-7.97) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.2e-11 | +1.0e-10
Z=-27.7(3.75%) | Like=-22.44..-19.19 [-22.4356..-22.3998] | it/evals=1228/2006 eff=71.9812% N=300
Z=-27.7(3.82%) | Like=-22.39..-19.19 [-22.3865..-22.3833]*| it/evals=1230/2009 eff=71.9719% N=300
Z=-27.5(4.87%) | Like=-22.24..-19.19 [-22.2589..-22.2447] | it/evals=1252/2050 eff=71.5429% N=300
Z=-27.4(5.28%) | Like=-22.13..-19.19 [-22.1896..-22.1340] | it/evals=1260/2061 eff=71.5503% N=300
Mono-modal Volume: ~exp(-7.97) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.2e-11 *************** +7.0e-11 | +1.0e-10
Z=-27.2(6.58%) | Like=-21.95..-19.19 [-21.9538..-21.9353] | it/evals=1281/2098 eff=71.2458% N=300
Z=-27.1(7.16%) | Like=-21.82..-19.19 [-21.8507..-21.8167] | it/evals=1290/2110 eff=71.2707% N=300
Z=-26.9(9.25%) | Like=-21.60..-19.18 [-21.6048..-21.5870] | it/evals=1319/2150 eff=71.2973% N=300
Z=-26.9(9.30%) | Like=-21.59..-19.18 [-21.6048..-21.5870] | it/evals=1320/2151 eff=71.3128% N=300
Mono-modal Volume: ~exp(-8.18) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.7(10.72%) | Like=-21.39..-19.18 [-21.3936..-21.3880]*| it/evals=1340/2183 eff=71.1630% N=300
Z=-26.6(11.56%) | Like=-21.35..-19.18 [-21.3653..-21.3542] | it/evals=1350/2195 eff=71.2401% N=300
Z=-26.4(13.96%) | Like=-21.20..-19.18 [-21.2007..-21.1981]*| it/evals=1380/2234 eff=71.3547% N=300
Mono-modal Volume: ~exp(-8.50) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.7e-11 | +1.0e-10
Z=-26.3(16.31%) | Like=-21.02..-19.18 [-21.0231..-21.0208]*| it/evals=1407/2270 eff=71.4213% N=300
Z=-26.3(16.61%) | Like=-21.01..-19.18 [-21.0110..-21.0027]*| it/evals=1410/2273 eff=71.4648% N=300
Z=-26.1(19.73%) | Like=-20.82..-19.18 [-20.8192..-20.8115]*| it/evals=1440/2311 eff=71.6062% N=300
Z=-25.9(23.06%) | Like=-20.63..-19.17 [-20.6264..-20.6261]*| it/evals=1468/2351 eff=71.5748% N=300
Z=-25.9(23.23%) | Like=-20.62..-19.17 [-20.6243..-20.6228]*| it/evals=1470/2353 eff=71.6025% N=300
Mono-modal Volume: ~exp(-8.83) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.6e-11 | +1.0e-10
Z=-25.9(23.66%) | Like=-20.61..-19.17 [-20.6187..-20.6070] | it/evals=1474/2360 eff=71.5534% N=300
Z=-25.8(26.78%) | Like=-20.50..-19.17 [-20.5007..-20.4912]*| it/evals=1500/2387 eff=71.8735% N=300
Z=-25.7(30.35%) | Like=-20.37..-19.17 [-20.3705..-20.3699]*| it/evals=1529/2428 eff=71.8515% N=300
Z=-25.7(30.50%) | Like=-20.37..-19.17 [-20.3699..-20.3598] | it/evals=1530/2429 eff=71.8647% N=300
Mono-modal Volume: ~exp(-9.25) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.6(31.79%) | Like=-20.33..-19.17 [-20.3310..-20.3133] | it/evals=1541/2445 eff=71.8415% N=300
Z=-25.5(34.10%) | Like=-20.25..-19.17 [-20.2470..-20.2459]*| it/evals=1560/2470 eff=71.8894% N=300
Z=-25.4(37.99%) | Like=-20.15..-19.17 [-20.1545..-20.1539]*| it/evals=1590/2505 eff=72.1088% N=300
Mono-modal Volume: ~exp(-9.40) * 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=-25.4(40.21%) | Like=-20.10..-19.17 [-20.0962..-20.0903]*| it/evals=1608/2532 eff=72.0430% N=300
Z=-25.4(41.66%) | Like=-20.05..-19.17 [-20.0508..-20.0502]*| it/evals=1620/2547 eff=72.0961% N=300
Z=-25.3(45.41%) | Like=-19.99..-19.17 [-19.9880..-19.9860]*| it/evals=1650/2583 eff=72.2733% N=300
Mono-modal Volume: ~exp(-9.40) Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ******** +6.2e-11 | +1.0e-10
Z=-25.2(48.65%) | Like=-19.94..-19.16 [-19.9424..-19.9403]*| it/evals=1679/2619 eff=72.4019% N=300
Z=-25.2(48.74%) | Like=-19.94..-19.16 [-19.9403..-19.9372]*| it/evals=1680/2620 eff=72.4138% N=300
Z=-25.1(52.18%) | Like=-19.86..-19.16 [-19.8615..-19.8591]*| it/evals=1710/2659 eff=72.4883% N=300
Z=-25.1(54.87%) | Like=-19.82..-19.16 [-19.8168..-19.8126]*| it/evals=1735/2700 eff=72.2917% N=300
Z=-25.1(55.41%) | Like=-19.80..-19.16 [-19.7978..-19.7961]*| it/evals=1740/2708 eff=72.2591% N=300
Mono-modal Volume: ~exp(-9.76) * Expected Volume: exp(-5.81) 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=-25.1(55.63%) | Like=-19.80..-19.16 [-19.7961..-19.7956]*| it/evals=1742/2711 eff=72.2522% N=300
Z=-25.0(58.57%) | Like=-19.74..-19.16 [-19.7395..-19.7387]*| it/evals=1770/2748 eff=72.3039% N=300
Z=-25.0(61.66%) | Like=-19.69..-19.16 [-19.6868..-19.6862]*| it/evals=1800/2785 eff=72.4346% N=300
Mono-modal Volume: ~exp(-9.82) * Expected Volume: exp(-6.03) 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(62.50%) | Like=-19.67..-19.16 [-19.6699..-19.6694]*| it/evals=1809/2798 eff=72.4179% N=300
Z=-24.9(64.37%) | Like=-19.63..-19.16 [-19.6347..-19.6326]*| it/evals=1830/2826 eff=72.4466% N=300
Z=-24.9(67.02%) | Like=-19.60..-19.16 [-19.6014..-19.6000]*| it/evals=1859/2866 eff=72.4474% N=300
Z=-24.9(67.11%) | Like=-19.60..-19.16 [-19.6000..-19.5968]*| it/evals=1860/2867 eff=72.4581% N=300
Mono-modal Volume: ~exp(-10.02) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.0e-11 | +1.0e-10
Z=-24.8(68.47%) | Like=-19.57..-19.16 [-19.5729..-19.5726]*| it/evals=1876/2891 eff=72.4045% N=300
Z=-24.8(69.64%) | Like=-19.55..-19.16 [-19.5513..-19.5512]*| it/evals=1890/2908 eff=72.4693% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2913
[ultranest] logZ = -24.46 +- 0.09106
[ultranest] Effective samples strategy satisfied (ESS = 997.6, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.06 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:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -24.469 +- 0.363
single instance: logZ = -24.469 +- 0.119
bootstrapped : logZ = -24.458 +- 0.251
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.28 │ ▁▁▁▁▁▁▂▃▄▄▆▅▅▇▇▆▆▅▅▆▄▄▃▂▂▁▁▁▁▁▁ ▁▁▁ ▁ │3.59 2.82 +- 0.17
amplitude : 0.0000000000326│ ▁ ▁▁▁▁▁▂▂▂▃▄▅▆▇▇▆▇▆▅▅▄▄▃▃▁▁▂▁▁▁▁▁▁ ▁ │0.0000000000817 0.0000000000552 +- 0.0000000000061
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.95) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ******* ** **** *| +1.0e-10
Z=-inf(0.00%) | Like=-913.97..-13.32 [-913.9669..-93.1436] | it/evals=0/301 eff=0.0000% N=300
Z=-151.7(0.00%) | Like=-147.30..-13.32 [-913.9669..-93.1436] | it/evals=30/331 eff=96.7742% N=300
Z=-142.6(0.00%) | Like=-138.04..-13.32 [-913.9669..-93.1436] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.24) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ******* ** * ** *| +1.0e-10
Z=-141.0(0.00%) | Like=-136.69..-13.32 [-913.9669..-93.1436] | it/evals=67/372 eff=93.0556% N=300
Z=-136.0(0.00%) | Like=-131.45..-13.32 [-913.9669..-93.1436] | it/evals=90/396 eff=93.7500% N=300
Z=-124.1(0.00%) | Like=-119.45..-13.32 [-913.9669..-93.1436] | it/evals=120/426 eff=95.2381% 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=-119.5(0.00%) | Like=-114.56..-13.32 [-913.9669..-93.1436] | it/evals=134/442 eff=94.3662% N=300
Z=-114.8(0.00%) | Like=-109.59..-13.32 [-913.9669..-93.1436] | it/evals=150/459 eff=94.3396% N=300
Z=-106.8(0.00%) | Like=-101.76..-13.32 [-913.9669..-93.1436] | it/evals=179/500 eff=89.5000% N=300
Z=-106.5(0.00%) | Like=-101.60..-13.32 [-913.9669..-93.1436] | it/evals=180/501 eff=89.5522% N=300
Mono-modal Volume: ~exp(-5.00) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ***********************************************| +1.0e-10
Z=-100.3(0.00%) | Like=-94.01..-13.32 [-913.9669..-93.1436] | it/evals=201/525 eff=89.3333% N=300
Z=-97.1(0.00%) | Like=-90.97..-13.32 [-93.0750..-49.3892] | it/evals=210/535 eff=89.3617% N=300
Z=-90.4(0.00%) | Like=-85.37..-13.32 [-93.0750..-49.3892] | it/evals=240/569 eff=89.2193% N=300
Mono-modal Volume: ~exp(-5.00) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-83.9(0.00%) | Like=-78.47..-13.32 [-93.0750..-49.3892] | it/evals=270/604 eff=88.8158% N=300
Z=-75.3(0.00%) | Like=-70.15..-13.32 [-93.0750..-49.3892] | it/evals=300/644 eff=87.2093% N=300
Z=-68.8(0.00%) | Like=-62.78..-13.32 [-93.0750..-49.3892] | it/evals=330/679 eff=87.0712% N=300
Mono-modal Volume: ~exp(-5.22) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-67.3(0.00%) | Like=-61.71..-13.32 [-93.0750..-49.3892] | it/evals=335/687 eff=86.5633% N=300
Z=-63.2(0.00%) | Like=-57.65..-13.32 [-93.0750..-49.3892] | it/evals=360/719 eff=85.9189% N=300
Z=-57.1(0.00%) | Like=-52.09..-13.32 [-93.0750..-49.3892] | it/evals=390/759 eff=84.9673% N=300
Mono-modal Volume: ~exp(-5.36) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-55.7(0.00%) | Like=-50.81..-13.32 [-93.0750..-49.3892] | it/evals=402/777 eff=84.2767% N=300
Z=-53.8(0.00%) | Like=-48.87..-13.32 [-49.3345..-31.3311] | it/evals=420/801 eff=83.8323% N=300
Z=-51.2(0.00%) | Like=-46.50..-13.32 [-49.3345..-31.3311] | it/evals=448/842 eff=82.6568% N=300
Z=-51.1(0.00%) | Like=-46.43..-13.32 [-49.3345..-31.3311] | it/evals=450/844 eff=82.7206% N=300
Mono-modal Volume: ~exp(-5.36) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-48.9(0.00%) | Like=-44.29..-13.32 [-49.3345..-31.3311] | it/evals=476/881 eff=81.9277% N=300
Z=-48.6(0.00%) | Like=-44.05..-13.32 [-49.3345..-31.3311] | it/evals=480/887 eff=81.7717% N=300
Z=-46.2(0.00%) | Like=-41.32..-13.32 [-49.3345..-31.3311] | it/evals=510/928 eff=81.2102% N=300
Mono-modal Volume: ~exp(-5.69) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-43.5(0.00%) | Like=-38.26..-13.32 [-49.3345..-31.3311] | it/evals=536/960 eff=81.2121% N=300
Z=-43.2(0.00%) | Like=-38.00..-13.32 [-49.3345..-31.3311] | it/evals=540/965 eff=81.2030% N=300
Z=-40.5(0.00%) | Like=-35.23..-13.32 [-49.3345..-31.3311] | it/evals=570/1007 eff=80.6223% N=300
Z=-38.0(0.00%) | Like=-32.99..-13.32 [-49.3345..-31.3311] | it/evals=599/1048 eff=80.0802% N=300
Z=-38.0(0.00%) | Like=-32.86..-13.32 [-49.3345..-31.3311] | it/evals=600/1049 eff=80.1068% N=300
Mono-modal Volume: ~exp(-6.04) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| ******************************** *******| +1.0e-10
Z=-37.7(0.00%) | Like=-32.35..-13.32 [-49.3345..-31.3311] | it/evals=603/1052 eff=80.1862% N=300
Z=-35.7(0.00%) | Like=-30.24..-13.31 [-31.3218..-21.1233] | it/evals=629/1094 eff=79.2191% N=300
Z=-35.6(0.00%) | Like=-30.21..-13.31 [-31.3218..-21.1233] | it/evals=630/1096 eff=79.1457% N=300
Z=-33.6(0.00%) | Like=-28.51..-13.31 [-31.3218..-21.1233] | it/evals=657/1137 eff=78.4946% N=300
Z=-33.4(0.00%) | Like=-28.42..-13.31 [-31.3218..-21.1233] | it/evals=660/1141 eff=78.4780% N=300
Mono-modal Volume: ~exp(-6.04) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 *********************** +3.9 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-32.1(0.00%) | Like=-27.04..-13.31 [-31.3218..-21.1233] | it/evals=686/1178 eff=78.1321% N=300
Z=-31.9(0.00%) | Like=-26.81..-13.31 [-31.3218..-21.1233] | it/evals=690/1186 eff=77.8781% N=300
Z=-30.5(0.00%) | Like=-25.64..-13.31 [-31.3218..-21.1233] | it/evals=718/1226 eff=77.5378% N=300
Z=-30.4(0.00%) | Like=-25.49..-13.31 [-31.3218..-21.1233] | it/evals=720/1228 eff=77.5862% N=300
Mono-modal Volume: ~exp(-6.60) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************* +3.7 | +5.0
amplitude: +1.0e-12| +2.5e-11 *************************************| +1.0e-10
Z=-29.7(0.00%) | Like=-24.73..-13.31 [-31.3218..-21.1233] | it/evals=737/1253 eff=77.3347% N=300
Z=-29.2(0.00%) | Like=-24.17..-13.31 [-31.3218..-21.1233] | it/evals=750/1271 eff=77.2400% N=300
Z=-28.2(0.00%) | Like=-23.18..-13.31 [-31.3218..-21.1233] | it/evals=778/1311 eff=76.9535% N=300
Z=-28.1(0.00%) | Like=-23.02..-13.31 [-31.3218..-21.1233] | it/evals=780/1315 eff=76.8473% N=300
Mono-modal Volume: ~exp(-6.60) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******************************** | +1.0e-10
Z=-27.1(0.01%) | Like=-22.23..-13.31 [-31.3218..-21.1233] | it/evals=810/1351 eff=77.0695% N=300
Z=-26.3(0.03%) | Like=-21.18..-13.31 [-31.3218..-21.1233] | it/evals=838/1391 eff=76.8103% N=300
Z=-26.2(0.03%) | Like=-21.07..-13.31 [-21.0696..-19.8569] | it/evals=840/1393 eff=76.8527% N=300
Z=-25.5(0.07%) | Like=-20.37..-13.31 [-21.0696..-19.8569] | it/evals=863/1434 eff=76.1023% N=300
Z=-25.3(0.08%) | Like=-20.21..-13.31 [-21.0696..-19.8569] | it/evals=870/1445 eff=75.9825% N=300
Mono-modal Volume: ~exp(-6.92) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 **************************** | +1.0e-10
Z=-25.2(0.08%) | Like=-20.20..-13.31 [-21.0696..-19.8569] | it/evals=871/1447 eff=75.9372% N=300
Z=-24.5(0.18%) | Like=-19.53..-13.31 [-19.5675..-19.5272] | it/evals=900/1482 eff=76.1421% N=300
Z=-23.9(0.31%) | Like=-18.95..-13.31 [-18.9549..-18.9515]*| it/evals=927/1522 eff=75.8592% N=300
Z=-23.8(0.33%) | Like=-18.90..-13.31 [-18.8955..-18.8780] | it/evals=930/1525 eff=75.9184% N=300
Mono-modal Volume: ~exp(-7.09) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.3 | +5.0
amplitude: +1.0e-12| +3.1e-11 ************************** | +1.0e-10
Z=-23.7(0.40%) | Like=-18.69..-13.31 [-18.7360..-18.6937] | it/evals=938/1537 eff=75.8286% N=300
Z=-23.3(0.59%) | Like=-18.40..-13.31 [-18.4460..-18.3990] | it/evals=960/1566 eff=75.8294% N=300
Z=-22.8(0.92%) | Like=-18.12..-13.31 [-18.1175..-18.0265] | it/evals=987/1609 eff=75.4011% N=300
Z=-22.8(0.97%) | Like=-17.95..-13.31 [-17.9841..-17.9505] | it/evals=990/1612 eff=75.4573% N=300
Mono-modal Volume: ~exp(-7.29) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 *********************** +7.9e-11| +1.0e-10
Z=-22.6(1.25%) | Like=-17.66..-13.31 [-17.6771..-17.6622] | it/evals=1005/1636 eff=75.2246% N=300
Z=-22.4(1.52%) | Like=-17.54..-13.31 [-17.5682..-17.5395] | it/evals=1020/1659 eff=75.0552% N=300
Z=-22.0(2.16%) | Like=-17.19..-13.31 [-17.1920..-17.1896]*| it/evals=1046/1700 eff=74.7143% N=300
Z=-22.0(2.29%) | Like=-17.14..-13.31 [-17.1380..-17.1357]*| it/evals=1050/1707 eff=74.6269% N=300
Mono-modal Volume: ~exp(-7.58) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************* +7.6e-11 | +1.0e-10
Z=-21.7(2.89%) | Like=-16.90..-13.31 [-16.9015..-16.8916]*| it/evals=1072/1738 eff=74.5480% N=300
Z=-21.6(3.13%) | Like=-16.78..-13.31 [-16.7774..-16.7645] | it/evals=1080/1748 eff=74.5856% N=300
Z=-21.3(4.25%) | Like=-16.44..-13.31 [-16.4415..-16.4407]*| it/evals=1108/1789 eff=74.4124% N=300
Z=-21.3(4.35%) | Like=-16.43..-13.31 [-16.4407..-16.4269] | it/evals=1110/1791 eff=74.4467% N=300
Mono-modal Volume: ~exp(-8.25) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.3e-11 | +1.0e-10
Z=-21.0(5.77%) | Like=-16.13..-13.31 [-16.1442..-16.1322] | it/evals=1139/1831 eff=74.3958% N=300
Z=-21.0(5.81%) | Like=-16.13..-13.31 [-16.1273..-16.1221]*| it/evals=1140/1833 eff=74.3640% N=300
Z=-20.7(7.79%) | Like=-15.86..-13.31 [-15.8642..-15.8608]*| it/evals=1170/1869 eff=74.5698% N=300
Z=-20.5(9.72%) | Like=-15.63..-13.31 [-15.6270..-15.6268]*| it/evals=1200/1902 eff=74.9064% N=300
Mono-modal Volume: ~exp(-8.25) Expected Volume: exp(-4.02) 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=-20.3(11.54%) | Like=-15.46..-13.31 [-15.4614..-15.4600]*| it/evals=1223/1939 eff=74.6187% N=300
Z=-20.2(12.20%) | Like=-15.41..-13.31 [-15.4105..-15.4072]*| it/evals=1230/1949 eff=74.5907% N=300
Z=-20.1(14.80%) | Like=-15.27..-13.31 [-15.2688..-15.2658]*| it/evals=1260/1989 eff=74.6004% N=300
Mono-modal Volume: ~exp(-8.57) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 *************** +6.9e-11 | +1.0e-10
Z=-20.0(16.04%) | Like=-15.18..-13.31 [-15.1802..-15.1755]*| it/evals=1273/2008 eff=74.5316% N=300
Z=-19.9(17.59%) | Like=-15.08..-13.31 [-15.0817..-15.0702] | it/evals=1290/2030 eff=74.5665% N=300
Z=-19.7(20.72%) | Like=-14.89..-13.30 [-14.8893..-14.8882]*| it/evals=1320/2069 eff=74.6184% N=300
Mono-modal Volume: ~exp(-8.71) * Expected Volume: exp(-4.47) 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=-19.6(22.97%) | Like=-14.78..-13.30 [-14.7835..-14.7777]*| it/evals=1340/2096 eff=74.6102% N=300
Z=-19.6(24.03%) | Like=-14.74..-13.30 [-14.7442..-14.7395]*| it/evals=1350/2106 eff=74.7508% N=300
Z=-19.4(27.36%) | Like=-14.60..-13.30 [-14.5995..-14.5899]*| it/evals=1380/2142 eff=74.9186% N=300
Mono-modal Volume: ~exp(-8.92) * Expected Volume: exp(-4.69) 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=-19.3(30.50%) | Like=-14.52..-13.30 [-14.5220..-14.5172]*| it/evals=1407/2179 eff=74.8803% N=300
Z=-19.3(30.83%) | Like=-14.51..-13.30 [-14.5070..-14.5048]*| it/evals=1410/2182 eff=74.9203% N=300
Z=-19.2(34.35%) | Like=-14.41..-13.30 [-14.4089..-14.4050]*| it/evals=1440/2218 eff=75.0782% N=300
Z=-19.1(36.83%) | Like=-14.34..-13.30 [-14.3443..-14.3371]*| it/evals=1462/2259 eff=74.6299% N=300
Z=-19.1(37.75%) | Like=-14.31..-13.30 [-14.3135..-14.3135]*| it/evals=1470/2272 eff=74.5436% N=300
Mono-modal Volume: ~exp(-8.92) Expected Volume: exp(-4.91) 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=-19.1(40.33%) | Like=-14.26..-13.30 [-14.2578..-14.2576]*| it/evals=1492/2309 eff=74.2658% N=300
Z=-19.0(41.29%) | Like=-14.23..-13.30 [-14.2324..-14.2179] | it/evals=1500/2318 eff=74.3310% N=300
Z=-19.0(44.33%) | Like=-14.13..-13.30 [-14.1333..-14.1260]*| it/evals=1525/2359 eff=74.0651% N=300
Z=-19.0(44.85%) | Like=-14.11..-13.30 [-14.1085..-14.1043]*| it/evals=1530/2365 eff=74.0920% N=300
Mono-modal Volume: ~exp(-9.07) * Expected Volume: exp(-5.14) 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=-18.9(46.29%) | Like=-14.08..-13.30 [-14.0812..-14.0790]*| it/evals=1541/2380 eff=74.0865% N=300
Z=-18.9(48.65%) | Like=-14.03..-13.30 [-14.0274..-14.0186]*| it/evals=1560/2404 eff=74.1445% N=300
Z=-18.8(52.01%) | Like=-13.94..-13.30 [-13.9400..-13.9400]*| it/evals=1590/2437 eff=74.4034% N=300
Mono-modal Volume: ~exp(-9.64) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.2e-11 | +1.0e-10
Z=-18.8(54.00%) | Like=-13.91..-13.30 [-13.9083..-13.9061]*| it/evals=1608/2458 eff=74.5134% N=300
Z=-18.7(55.38%) | Like=-13.88..-13.30 [-13.8819..-13.8812]*| it/evals=1620/2472 eff=74.5856% N=300
Z=-18.7(58.60%) | Like=-13.83..-13.30 [-13.8255..-13.8214]*| it/evals=1650/2508 eff=74.7283% N=300
Mono-modal Volume: ~exp(-9.64) Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ********* +6.1e-11 | +1.0e-10
Z=-18.6(61.65%) | Like=-13.78..-13.30 [-13.7787..-13.7762]*| it/evals=1680/2543 eff=74.8997% N=300
Z=-18.6(64.46%) | Like=-13.73..-13.30 [-13.7323..-13.7290]*| it/evals=1709/2583 eff=74.8576% N=300
Z=-18.6(64.54%) | Like=-13.73..-13.30 [-13.7290..-13.7282]*| it/evals=1710/2586 eff=74.8031% N=300
Z=-18.6(67.29%) | Like=-13.67..-13.30 [-13.6726..-13.6710]*| it/evals=1740/2624 eff=74.8709% N=300
Mono-modal Volume: ~exp(-9.91) * Expected Volume: exp(-5.81) 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=-18.6(67.46%) | Like=-13.67..-13.30 [-13.6705..-13.6685]*| it/evals=1742/2626 eff=74.8925% N=300
Z=-18.5(69.91%) | Like=-13.64..-13.30 [-13.6361..-13.6343]*| it/evals=1770/2654 eff=75.1912% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2654
[ultranest] logZ = -18.18 +- 0.08325
[ultranest] Effective samples strategy satisfied (ESS = 1011.5, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.10 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.11 bs:0.08 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.159 +- 0.295
single instance: logZ = -18.159 +- 0.113
bootstrapped : logZ = -18.176 +- 0.135
tail : logZ = +- 0.263
insert order U test : converged: True correlation: inf iterations
index : 2.16 │ ▁ ▁▁▁▂▂▃▄▄▄▆▇▆▇▇▇▆▆▅▄▃▃▂▂▂▁▁▁▁▁▁▁▁ ▁ │3.49 2.75 +- 0.17
amplitude : 0.0000000000272│ ▁▁▁▁▁▁▁▂▂▃▃▄▇▆▅▆▇▇▇▇▆▄▅▄▃▂▃▂▁▁▁▁▁▁ ▁▁ │0.0000000000814 0.0000000000531 +- 0.0000000000077
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/
# 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()
Corner plot comparison#
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).
Total running time of the script: (0 minutes 39.900 seconds)
