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(-4.19) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************* ** ******************** *| +1.0e-10
Z=-inf(0.00%) | Like=-2550.27..-67.38 [-2550.2737..-313.0113] | it/evals=0/301 eff=0.0000% N=300
Z=-555.0(0.00%) | Like=-548.86..-67.38 [-2550.2737..-313.0113] | it/evals=19/323 eff=82.6087% N=300
Z=-541.9(0.00%) | Like=-531.03..-67.38 [-2550.2737..-313.0113] | it/evals=30/334 eff=88.2353% N=300
Z=-508.1(0.00%) | Like=-501.83..-63.95 [-2550.2737..-313.0113] | it/evals=49/356 eff=87.5000% N=300
Z=-493.7(0.00%) | Like=-487.67..-62.72 [-2550.2737..-313.0113] | it/evals=60/367 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.32) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************ ***** ************** | +1.0e-10
Z=-486.0(0.00%) | Like=-478.89..-62.72 [-2550.2737..-313.0113] | it/evals=67/376 eff=88.1579% N=300
Z=-466.1(0.00%) | Like=-460.56..-62.72 [-2550.2737..-313.0113] | it/evals=88/398 eff=89.7959% N=300
Z=-463.3(0.00%) | Like=-456.81..-62.72 [-2550.2737..-313.0113] | it/evals=90/400 eff=90.0000% N=300
Z=-439.7(0.00%) | Like=-433.90..-62.72 [-2550.2737..-313.0113] | it/evals=108/422 eff=88.5246% N=300
Z=-426.8(0.00%) | Like=-418.14..-62.72 [-2550.2737..-313.0113] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.57) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|******* ****************************************| +5.0
amplitude: +1.0e-12|****************************** ************** * | +1.0e-10
Z=-408.4(0.00%) | Like=-402.76..-62.72 [-2550.2737..-313.0113] | it/evals=134/454 eff=87.0130% N=300
Z=-392.4(0.00%) | Like=-386.31..-59.84 [-2550.2737..-313.0113] | it/evals=150/473 eff=86.7052% N=300
Z=-363.6(0.00%) | Like=-355.80..-58.80 [-2550.2737..-313.0113] | it/evals=169/495 eff=86.6667% N=300
Z=-351.4(0.00%) | Like=-343.57..-58.80 [-2550.2737..-313.0113] | it/evals=180/506 eff=87.3786% N=300
Z=-326.2(0.00%) | Like=-318.65..-58.80 [-2550.2737..-313.0113] | it/evals=199/530 eff=86.5217% N=300
Mono-modal Volume: ~exp(-4.71) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ***************************** ************** * | +1.0e-10
Z=-323.7(0.00%) | Like=-314.12..-58.80 [-2550.2737..-313.0113] | it/evals=201/533 eff=86.2661% N=300
Z=-311.3(0.00%) | Like=-304.75..-58.80 [-312.2472..-168.7495] | it/evals=210/542 eff=86.7769% N=300
Z=-298.3(0.00%) | Like=-291.20..-58.80 [-312.2472..-168.7495] | it/evals=227/564 eff=85.9848% N=300
Z=-286.9(0.00%) | Like=-280.35..-58.80 [-312.2472..-168.7495] | it/evals=240/584 eff=84.5070% N=300
Z=-273.8(0.00%) | Like=-266.14..-58.80 [-312.2472..-168.7495] | it/evals=256/607 eff=83.3876% N=300
Mono-modal Volume: ~exp(-4.90) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *************************** ************** * | +1.0e-10
Z=-261.5(0.00%) | Like=-254.05..-58.80 [-312.2472..-168.7495] | it/evals=268/621 eff=83.4891% N=300
Z=-259.3(0.00%) | Like=-252.85..-58.80 [-312.2472..-168.7495] | it/evals=270/623 eff=83.5913% N=300
Z=-240.4(0.00%) | Like=-231.86..-58.80 [-312.2472..-168.7495] | it/evals=290/645 eff=84.0580% N=300
Z=-231.6(0.00%) | Like=-225.81..-58.80 [-312.2472..-168.7495] | it/evals=300/656 eff=84.2697% N=300
Z=-222.3(0.00%) | Like=-216.41..-58.80 [-312.2472..-168.7495] | it/evals=316/678 eff=83.5979% N=300
Z=-211.7(0.00%) | Like=-206.01..-58.80 [-312.2472..-168.7495] | it/evals=330/695 eff=83.5443% N=300
Mono-modal Volume: ~exp(-5.15) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ***************************************** **| +1.0e-10
Z=-209.9(0.00%) | Like=-204.26..-58.80 [-312.2472..-168.7495] | it/evals=335/703 eff=83.1266% N=300
Z=-202.7(0.00%) | Like=-196.71..-58.80 [-312.2472..-168.7495] | it/evals=349/726 eff=81.9249% N=300
Z=-199.6(0.00%) | Like=-193.89..-58.80 [-312.2472..-168.7495] | it/evals=360/741 eff=81.6327% N=300
Z=-190.3(0.00%) | Like=-183.73..-58.80 [-312.2472..-168.7495] | it/evals=378/763 eff=81.6415% N=300
Z=-184.8(0.00%) | Like=-178.70..-58.80 [-312.2472..-168.7495] | it/evals=390/781 eff=81.0811% N=300
Mono-modal Volume: ~exp(-5.60) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| ************************************** * **| +1.0e-10
Z=-180.6(0.00%) | Like=-174.56..-58.80 [-312.2472..-168.7495] | it/evals=402/800 eff=80.4000% N=300
Z=-175.6(0.00%) | Like=-170.01..-58.80 [-312.2472..-168.7495] | it/evals=420/821 eff=80.6142% N=300
Z=-171.1(0.00%) | Like=-165.39..-58.80 [-168.7058..-121.9110] | it/evals=439/843 eff=80.8471% N=300
Z=-169.2(0.00%) | Like=-163.15..-58.80 [-168.7058..-121.9110] | it/evals=450/863 eff=79.9290% N=300
Z=-165.4(0.00%) | Like=-159.55..-58.80 [-168.7058..-121.9110] | it/evals=465/885 eff=79.4872% N=300
Mono-modal Volume: ~exp(-5.77) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ************************************* ***| +1.0e-10
Z=-164.4(0.00%) | Like=-158.35..-58.80 [-168.7058..-121.9110] | it/evals=469/891 eff=79.3570% N=300
Z=-161.6(0.00%) | Like=-155.39..-58.80 [-168.7058..-121.9110] | it/evals=480/903 eff=79.6020% N=300
Z=-157.7(0.00%) | Like=-151.85..-58.80 [-168.7058..-121.9110] | it/evals=497/925 eff=79.5200% N=300
Z=-154.9(0.00%) | Like=-149.23..-58.80 [-168.7058..-121.9110] | it/evals=510/945 eff=79.0698% N=300
Z=-151.4(0.00%) | Like=-145.33..-58.80 [-168.7058..-121.9110] | it/evals=527/967 eff=79.0105% N=300
Mono-modal Volume: ~exp(-6.10) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ***************************** +4.2 | +5.0
amplitude: +1.0e-12| ********************************* ** | +1.0e-10
Z=-148.7(0.00%) | Like=-142.50..-58.80 [-168.7058..-121.9110] | it/evals=536/979 eff=78.9396% N=300
Z=-147.9(0.00%) | Like=-142.15..-58.80 [-168.7058..-121.9110] | it/evals=540/984 eff=78.9474% N=300
Z=-143.8(0.00%) | Like=-137.51..-58.80 [-168.7058..-121.9110] | it/evals=555/1010 eff=78.1690% N=300
Z=-141.2(0.00%) | Like=-135.27..-58.80 [-168.7058..-121.9110] | it/evals=570/1031 eff=77.9754% N=300
Z=-136.8(0.00%) | Like=-130.55..-58.80 [-168.7058..-121.9110] | it/evals=587/1053 eff=77.9548% N=300
Z=-134.0(0.00%) | Like=-127.76..-58.80 [-168.7058..-121.9110] | it/evals=600/1074 eff=77.5194% N=300
Mono-modal Volume: ~exp(-6.15) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +4.0 | +5.0
amplitude: +1.0e-12| ******************************** ** | +1.0e-10
Z=-133.4(0.00%) | Like=-127.61..-58.80 [-168.7058..-121.9110] | it/evals=603/1079 eff=77.4069% N=300
Z=-130.6(0.00%) | Like=-124.24..-58.80 [-168.7058..-121.9110] | it/evals=621/1101 eff=77.5281% N=300
Z=-128.8(0.00%) | Like=-122.71..-58.80 [-168.7058..-121.9110] | it/evals=630/1114 eff=77.3956% N=300
Z=-126.5(0.00%) | Like=-120.13..-58.80 [-121.8454..-92.5187] | it/evals=645/1137 eff=77.0609% N=300
Z=-124.4(0.00%) | Like=-118.67..-58.80 [-121.8454..-92.5187] | it/evals=659/1160 eff=76.6279% N=300
Z=-124.3(0.00%) | Like=-118.38..-58.80 [-121.8454..-92.5187] | it/evals=660/1162 eff=76.5661% N=300
Mono-modal Volume: ~exp(-6.27) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 ************************ +3.8 | +5.0
amplitude: +1.0e-12| ******************************** | +1.0e-10
Z=-123.2(0.00%) | Like=-117.47..-58.80 [-121.8454..-92.5187] | it/evals=670/1177 eff=76.3968% N=300
Z=-121.6(0.00%) | Like=-115.58..-58.80 [-121.8454..-92.5187] | it/evals=687/1199 eff=76.4182% N=300
Z=-121.2(0.00%) | Like=-115.44..-58.80 [-121.8454..-92.5187] | it/evals=690/1202 eff=76.4967% N=300
Z=-119.0(0.00%) | Like=-112.06..-58.80 [-121.8454..-92.5187] | it/evals=707/1224 eff=76.5152% N=300
Z=-116.6(0.00%) | Like=-110.22..-58.80 [-121.8454..-92.5187] | it/evals=720/1239 eff=76.6773% N=300
Z=-115.0(0.00%) | Like=-109.16..-58.80 [-121.8454..-92.5187] | it/evals=732/1262 eff=76.0915% N=300
Mono-modal Volume: ~exp(-6.32) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.7 | +5.0
amplitude: +1.0e-12| ****************************** | +1.0e-10
Z=-114.4(0.00%) | Like=-108.21..-58.80 [-121.8454..-92.5187] | it/evals=737/1268 eff=76.1364% N=300
Z=-112.6(0.00%) | Like=-106.01..-58.80 [-121.8454..-92.5187] | it/evals=750/1286 eff=76.0649% N=300
Z=-110.7(0.00%) | Like=-103.89..-58.80 [-121.8454..-92.5187] | it/evals=761/1309 eff=75.4212% N=300
Z=-107.3(0.00%) | Like=-100.98..-58.80 [-121.8454..-92.5187] | it/evals=776/1332 eff=75.1938% N=300
Z=-106.7(0.00%) | Like=-100.05..-58.80 [-121.8454..-92.5187] | it/evals=780/1338 eff=75.1445% N=300
Z=-104.1(0.00%) | Like=-97.87..-58.80 [-121.8454..-92.5187] | it/evals=797/1361 eff=75.1178% N=300
Mono-modal Volume: ~exp(-6.48) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.5 | +5.0
amplitude: +1.0e-12| *************************** +7.6e-11 | +1.0e-10
Z=-103.4(0.00%) | Like=-97.34..-58.80 [-121.8454..-92.5187] | it/evals=804/1375 eff=74.7907% N=300
Z=-102.8(0.00%) | Like=-96.80..-58.80 [-121.8454..-92.5187] | it/evals=810/1387 eff=74.5170% N=300
Z=-101.6(0.00%) | Like=-95.37..-58.80 [-121.8454..-92.5187] | it/evals=823/1410 eff=74.1441% N=300
Z=-100.1(0.00%) | Like=-94.19..-58.80 [-121.8454..-92.5187] | it/evals=840/1429 eff=74.4021% N=300
Z=-98.5(0.00%) | Like=-92.35..-58.80 [-92.5161..-76.3095] | it/evals=857/1451 eff=74.4570% N=300
Z=-97.3(0.00%) | Like=-91.29..-58.80 [-92.5161..-76.3095] | it/evals=870/1468 eff=74.4863% N=300
Mono-modal Volume: ~exp(-7.14) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.4 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********************** +7.0e-11 | +1.0e-10
Z=-97.2(0.00%) | Like=-91.12..-58.80 [-92.5161..-76.3095] | it/evals=871/1469 eff=74.5081% N=300
Z=-95.4(0.00%) | Like=-88.79..-58.80 [-92.5161..-76.3095] | it/evals=889/1492 eff=74.5805% N=300
Z=-94.3(0.00%) | Like=-88.07..-58.80 [-92.5161..-76.3095] | it/evals=900/1504 eff=74.7508% N=300
Z=-92.8(0.00%) | Like=-86.50..-58.80 [-92.5161..-76.3095] | it/evals=918/1526 eff=74.8777% N=300
Z=-91.9(0.00%) | Like=-85.45..-58.80 [-92.5161..-76.3095] | it/evals=930/1545 eff=74.6988% N=300
Mono-modal Volume: ~exp(-7.29) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.8e-11 | +1.0e-10
Z=-91.0(0.00%) | Like=-84.25..-58.80 [-92.5161..-76.3095] | it/evals=938/1554 eff=74.8006% N=300
Z=-89.4(0.00%) | Like=-82.86..-58.80 [-92.5161..-76.3095] | it/evals=954/1576 eff=74.7649% N=300
Z=-88.9(0.00%) | Like=-82.57..-58.80 [-92.5161..-76.3095] | it/evals=960/1583 eff=74.8246% N=300
Z=-87.7(0.00%) | Like=-81.61..-58.80 [-92.5161..-76.3095] | it/evals=980/1606 eff=75.0383% N=300
Z=-87.2(0.00%) | Like=-80.93..-58.80 [-92.5161..-76.3095] | it/evals=990/1617 eff=75.1708% N=300
Mono-modal Volume: ~exp(-7.29) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************** +6.6e-11 | +1.0e-10
Z=-86.1(0.00%) | Like=-79.72..-58.80 [-92.5161..-76.3095] | it/evals=1007/1637 eff=75.3179% N=300
Z=-85.1(0.00%) | Like=-78.49..-58.80 [-92.5161..-76.3095] | it/evals=1020/1656 eff=75.2212% N=300
Z=-84.0(0.00%) | Like=-77.70..-58.80 [-92.5161..-76.3095] | it/evals=1036/1678 eff=75.1814% N=300
Z=-83.3(0.00%) | Like=-77.18..-58.80 [-92.5161..-76.3095] | it/evals=1050/1699 eff=75.0536% N=300
Z=-82.4(0.00%) | Like=-76.23..-58.80 [-76.2962..-68.1480] | it/evals=1070/1721 eff=75.2991% N=300
Mono-modal Volume: ~exp(-7.75) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.3e-11 | +1.0e-10
Z=-82.3(0.00%) | Like=-76.15..-58.80 [-76.2962..-68.1480] | it/evals=1072/1723 eff=75.3338% N=300
Z=-81.9(0.00%) | Like=-75.74..-58.80 [-76.2962..-68.1480] | it/evals=1080/1734 eff=75.3138% N=300
Z=-81.1(0.00%) | Like=-74.76..-58.80 [-76.2962..-68.1480] | it/evals=1100/1756 eff=75.5495% N=300
Z=-80.6(0.00%) | Like=-74.15..-58.80 [-76.2962..-68.1480] | it/evals=1110/1775 eff=75.2542% N=300
Z=-79.8(0.00%) | Like=-73.33..-58.80 [-76.2962..-68.1480] | it/evals=1126/1797 eff=75.2171% N=300
Mono-modal Volume: ~exp(-7.75) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.1e-11 **************** +6.2e-11 | +1.0e-10
Z=-79.1(0.00%) | Like=-72.68..-58.80 [-76.2962..-68.1480] | it/evals=1140/1814 eff=75.2972% N=300
Z=-78.4(0.00%) | Like=-72.09..-58.80 [-76.2962..-68.1480] | it/evals=1157/1838 eff=75.2276% N=300
Z=-78.0(0.00%) | Like=-71.67..-58.80 [-76.2962..-68.1480] | it/evals=1170/1855 eff=75.2412% N=300
Z=-77.5(0.00%) | Like=-71.23..-58.80 [-76.2962..-68.1480] | it/evals=1184/1877 eff=75.0793% N=300
Z=-77.0(0.00%) | Like=-70.89..-58.80 [-76.2962..-68.1480] | it/evals=1200/1899 eff=75.0469% N=300
Mono-modal Volume: ~exp(-7.75) 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=-76.6(0.00%) | Like=-70.44..-58.80 [-76.2962..-68.1480] | it/evals=1212/1920 eff=74.8148% N=300
Z=-76.2(0.00%) | Like=-69.96..-58.80 [-76.2962..-68.1480] | it/evals=1228/1942 eff=74.7868% N=300
Z=-76.1(0.00%) | Like=-69.86..-58.80 [-76.2962..-68.1480] | it/evals=1230/1944 eff=74.8175% N=300
Z=-75.6(0.00%) | Like=-69.35..-58.80 [-76.2962..-68.1480] | it/evals=1247/1966 eff=74.8499% N=300
Z=-75.3(0.01%) | Like=-68.93..-58.80 [-76.2962..-68.1480] | it/evals=1259/1989 eff=74.5411% N=300
Z=-75.3(0.01%) | Like=-68.91..-58.80 [-76.2962..-68.1480] | it/evals=1260/1990 eff=74.5562% N=300
Mono-modal Volume: ~exp(-8.22) * Expected Volume: exp(-4.24) 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=-74.9(0.01%) | Like=-68.45..-58.80 [-76.2962..-68.1480] | it/evals=1273/2009 eff=74.4880% N=300
Z=-74.4(0.01%) | Like=-68.03..-58.80 [-68.1378..-65.5980] | it/evals=1289/2032 eff=74.4226% N=300
Z=-74.4(0.01%) | Like=-68.03..-58.80 [-68.1378..-65.5980] | it/evals=1290/2033 eff=74.4374% N=300
Z=-73.9(0.02%) | Like=-67.48..-58.80 [-68.1378..-65.5980] | it/evals=1309/2055 eff=74.5869% N=300
Z=-73.6(0.03%) | Like=-67.17..-58.80 [-68.1378..-65.5980] | it/evals=1320/2069 eff=74.6184% N=300
Z=-73.2(0.04%) | Like=-66.89..-58.80 [-68.1378..-65.5980] | it/evals=1337/2091 eff=74.6510% N=300
Mono-modal Volume: ~exp(-8.60) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.7e-11 | +1.0e-10
Z=-73.2(0.04%) | Like=-66.74..-58.80 [-68.1378..-65.5980] | it/evals=1340/2094 eff=74.6934% N=300
Z=-73.0(0.05%) | Like=-66.62..-58.77 [-68.1378..-65.5980] | it/evals=1350/2107 eff=74.7095% N=300
Z=-72.6(0.07%) | Like=-66.13..-58.77 [-68.1378..-65.5980] | it/evals=1367/2129 eff=74.7403% N=300
Z=-72.3(0.09%) | Like=-65.79..-58.77 [-68.1378..-65.5980] | it/evals=1380/2151 eff=74.5543% N=300
Z=-72.0(0.13%) | Like=-65.51..-58.77 [-65.5501..-65.1577] | it/evals=1395/2175 eff=74.4000% N=300
Mono-modal Volume: ~exp(-8.60) Expected Volume: exp(-4.69) 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=-71.7(0.17%) | Like=-65.20..-58.77 [-65.5501..-65.1577] | it/evals=1410/2191 eff=74.5637% N=300
Z=-71.4(0.22%) | Like=-64.85..-58.77 [-64.8510..-64.8329] | it/evals=1423/2213 eff=74.3858% N=300
Z=-71.1(0.30%) | Like=-64.50..-58.77 [-64.5013..-64.4868] | it/evals=1438/2235 eff=74.3152% N=300
Z=-71.1(0.31%) | Like=-64.46..-58.77 [-64.4615..-64.4571]*| it/evals=1440/2237 eff=74.3418% N=300
Z=-70.7(0.45%) | Like=-64.27..-58.77 [-64.2729..-64.2662]*| it/evals=1460/2259 eff=74.5278% N=300
Z=-70.6(0.52%) | Like=-64.16..-58.77 [-64.1560..-64.1478]*| it/evals=1470/2272 eff=74.5436% N=300
Mono-modal Volume: ~exp(-8.81) * Expected Volume: exp(-4.91) 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=-70.5(0.56%) | Like=-64.13..-58.77 [-64.1268..-64.1051] | it/evals=1474/2284 eff=74.2944% N=300
Z=-70.3(0.70%) | Like=-63.74..-58.77 [-63.7404..-63.7401]*| it/evals=1490/2306 eff=74.2772% N=300
Z=-70.1(0.82%) | Like=-63.58..-58.77 [-63.6316..-63.5807] | it/evals=1500/2320 eff=74.2574% N=300
Z=-69.9(1.02%) | Like=-63.35..-58.77 [-63.3481..-63.3212] | it/evals=1514/2343 eff=74.1067% N=300
Z=-69.6(1.32%) | Like=-62.96..-58.77 [-62.9882..-62.9578] | it/evals=1530/2365 eff=74.0920% N=300
Mono-modal Volume: ~exp(-9.11) * 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=-69.5(1.58%) | Like=-62.74..-58.77 [-62.7409..-62.7345]*| it/evals=1541/2386 eff=73.8734% N=300
Z=-69.2(2.00%) | Like=-62.48..-58.77 [-62.4780..-62.4693]*| it/evals=1557/2411 eff=73.7565% N=300
Z=-69.2(2.09%) | Like=-62.46..-58.77 [-62.4665..-62.4564] | it/evals=1560/2416 eff=73.7240% N=300
Z=-69.0(2.56%) | Like=-62.32..-58.77 [-62.3577..-62.3195] | it/evals=1574/2438 eff=73.6202% N=300
Z=-68.8(3.07%) | Like=-62.17..-58.76 [-62.1853..-62.1719] | it/evals=1590/2458 eff=73.6793% N=300
Z=-68.6(3.70%) | Like=-61.93..-58.76 [-61.9305..-61.9268]*| it/evals=1604/2481 eff=73.5442% N=300
Mono-modal Volume: ~exp(-9.11) Expected Volume: exp(-5.36) 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=-68.4(4.38%) | Like=-61.78..-58.76 [-61.7759..-61.7741]*| it/evals=1619/2501 eff=73.5575% N=300
Z=-68.4(4.43%) | Like=-61.77..-58.76 [-61.7741..-61.7581] | it/evals=1620/2502 eff=73.5695% N=300
Z=-68.3(5.09%) | Like=-61.67..-58.76 [-61.6687..-61.6670]*| it/evals=1633/2525 eff=73.3933% N=300
Z=-68.1(5.95%) | Like=-61.51..-58.76 [-61.5209..-61.5093] | it/evals=1648/2547 eff=73.3422% N=300
Z=-68.1(6.04%) | Like=-61.50..-58.76 [-61.4981..-61.4893]*| it/evals=1650/2550 eff=73.3333% N=300
Z=-67.9(7.06%) | Like=-61.31..-58.76 [-61.3091..-61.3088]*| it/evals=1667/2573 eff=73.3392% N=300
Mono-modal Volume: ~exp(-9.38) * Expected Volume: exp(-5.58) 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=-67.8(7.54%) | Like=-61.25..-58.76 [-61.2488..-61.2487]*| it/evals=1675/2596 eff=72.9530% N=300
Z=-67.8(7.88%) | Like=-61.22..-58.76 [-61.2245..-61.2083] | it/evals=1680/2602 eff=72.9800% N=300
Z=-67.7(9.10%) | Like=-61.15..-58.76 [-61.1465..-61.1450]*| it/evals=1696/2625 eff=72.9462% N=300
Z=-67.6(10.14%) | Like=-61.05..-58.76 [-61.0514..-61.0482]*| it/evals=1710/2643 eff=72.9834% N=300
Z=-67.4(11.59%) | Like=-60.92..-58.76 [-60.9238..-60.9195]*| it/evals=1729/2665 eff=73.1078% N=300
Z=-67.4(12.37%) | Like=-60.87..-58.76 [-60.8679..-60.8483] | it/evals=1740/2680 eff=73.1092% N=300
Mono-modal Volume: ~exp(-9.67) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 ***** +2.8 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.3(12.49%) | Like=-60.84..-58.76 [-60.8404..-60.8372]*| it/evals=1742/2682 eff=73.1318% N=300
Z=-67.2(13.73%) | Like=-60.71..-58.75 [-60.7221..-60.7103] | it/evals=1757/2704 eff=73.0865% N=300
Z=-67.2(14.69%) | Like=-60.63..-58.75 [-60.6268..-60.6156] | it/evals=1770/2722 eff=73.0801% N=300
Z=-67.1(15.81%) | Like=-60.57..-58.75 [-60.5653..-60.5626]*| it/evals=1783/2744 eff=72.9542% N=300
Z=-67.0(17.46%) | Like=-60.45..-58.75 [-60.4699..-60.4500] | it/evals=1800/2764 eff=73.0519% N=300
Mono-modal Volume: ~exp(-9.79) * 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=-66.9(18.43%) | Like=-60.38..-58.75 [-60.3969..-60.3808] | it/evals=1809/2778 eff=73.0024% N=300
Z=-66.8(20.38%) | Like=-60.25..-58.75 [-60.2507..-60.2493]*| it/evals=1827/2801 eff=73.0508% N=300
Z=-66.8(20.77%) | Like=-60.21..-58.75 [-60.2483..-60.2142] | it/evals=1830/2805 eff=73.0539% N=300
Z=-66.7(22.62%) | Like=-60.17..-58.75 [-60.1695..-60.1666]*| it/evals=1846/2826 eff=73.0800% N=300
Z=-66.7(24.27%) | Like=-60.08..-58.75 [-60.0844..-60.0753]*| it/evals=1860/2846 eff=73.0558% N=300
Mono-modal Volume: ~exp(-9.95) * 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=-66.6(26.20%) | Like=-60.02..-58.75 [-60.0376..-60.0174] | it/evals=1876/2865 eff=73.1384% N=300
Z=-66.5(28.00%) | Like=-59.96..-58.75 [-59.9641..-59.9542]*| it/evals=1890/2885 eff=73.1141% N=300
Z=-66.5(30.14%) | Like=-59.88..-58.75 [-59.8850..-59.8755]*| it/evals=1907/2907 eff=73.1492% N=300
Z=-66.4(31.75%) | Like=-59.84..-58.75 [-59.8380..-59.8296]*| it/evals=1920/2929 eff=73.0316% N=300
Z=-66.4(33.42%) | Like=-59.80..-58.75 [-59.8036..-59.7991]*| it/evals=1933/2952 eff=72.8884% N=300
Mono-modal Volume: ~exp(-10.30) * 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=-66.3(34.67%) | Like=-59.77..-58.75 [-59.7674..-59.7658]*| it/evals=1943/2964 eff=72.9354% N=300
Z=-66.3(35.60%) | Like=-59.73..-58.75 [-59.7284..-59.7268]*| it/evals=1950/2973 eff=72.9517% N=300
Z=-66.2(37.68%) | Like=-59.67..-58.75 [-59.6700..-59.6667]*| it/evals=1968/2995 eff=73.0241% N=300
Z=-66.2(39.19%) | Like=-59.63..-58.75 [-59.6259..-59.6213]*| it/evals=1980/3012 eff=73.0088% N=300
Z=-66.1(41.15%) | Like=-59.60..-58.75 [-59.6004..-59.5921]*| it/evals=1996/3034 eff=73.0066% N=300
Mono-modal Volume: ~exp(-10.39) * 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=-66.1(43.06%) | Like=-59.57..-58.75 [-59.5718..-59.5716]*| it/evals=2010/3057 eff=72.9053% N=300
Z=-66.0(45.37%) | Like=-59.53..-58.75 [-59.5348..-59.5343]*| it/evals=2029/3079 eff=73.0119% N=300
Z=-66.0(46.61%) | Like=-59.49..-58.75 [-59.5049..-59.4918] | it/evals=2040/3091 eff=73.0921% N=300
Z=-66.0(48.57%) | Like=-59.44..-58.75 [-59.4412..-59.4400]*| it/evals=2056/3112 eff=73.1152% N=300
Z=-65.9(50.22%) | Like=-59.40..-58.75 [-59.3998..-59.3990]*| it/evals=2070/3131 eff=73.1190% N=300
Mono-modal Volume: ~exp(-10.81) * 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=-65.9(51.08%) | Like=-59.39..-58.75 [-59.3873..-59.3843]*| it/evals=2077/3142 eff=73.0823% N=300
Z=-65.9(53.30%) | Like=-59.35..-58.75 [-59.3538..-59.3536]*| it/evals=2096/3164 eff=73.1844% N=300
Z=-65.9(53.72%) | Like=-59.33..-58.75 [-59.3322..-59.3320]*| it/evals=2100/3170 eff=73.1707% N=300
Z=-65.8(55.34%) | Like=-59.30..-58.75 [-59.3011..-59.2994]*| it/evals=2115/3193 eff=73.1075% N=300
Z=-65.8(56.93%) | Like=-59.28..-58.75 [-59.2797..-59.2796]*| it/evals=2129/3215 eff=73.0360% N=300
Z=-65.8(57.04%) | Like=-59.28..-58.75 [-59.2796..-59.2775]*| it/evals=2130/3217 eff=73.0202% N=300
Mono-modal Volume: ~exp(-11.00) * Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.7e-11 | +1.0e-10
Z=-65.8(58.59%) | Like=-59.26..-58.75 [-59.2582..-59.2576]*| it/evals=2144/3233 eff=73.0992% N=300
Z=-65.8(60.31%) | Like=-59.24..-58.75 [-59.2375..-59.2370]*| it/evals=2160/3251 eff=73.1955% N=300
Z=-65.7(61.94%) | Like=-59.21..-58.75 [-59.2129..-59.2123]*| it/evals=2176/3273 eff=73.1921% N=300
Z=-65.7(63.24%) | Like=-59.19..-58.75 [-59.1908..-59.1891]*| it/evals=2190/3294 eff=73.1463% N=300
Z=-65.7(64.81%) | Like=-59.17..-58.75 [-59.1702..-59.1700]*| it/evals=2206/3316 eff=73.1432% N=300
Mono-modal Volume: ~exp(-11.00) 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=-65.7(66.10%) | Like=-59.15..-58.75 [-59.1518..-59.1501]*| it/evals=2220/3332 eff=73.2190% N=300
Z=-65.7(67.47%) | Like=-59.12..-58.75 [-59.1200..-59.1183]*| it/evals=2236/3354 eff=73.2155% N=300
Z=-65.6(68.69%) | Like=-59.10..-58.75 [-59.1012..-59.1003]*| it/evals=2250/3373 eff=73.2184% N=300
Z=-65.6(69.58%) | Like=-59.09..-58.75 [-59.0898..-59.0888]*| it/evals=2260/3395 eff=73.0210% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3399
[ultranest] logZ = -65.25 +- 0.1154
[ultranest] Effective samples strategy satisfied (ESS = 987.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.47+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.29, need <0.5)
[ultranest] logZ error budget: single: 0.14 bs:0.12 tail:0.26 total:0.29 required:<0.50
[ultranest] done iterating.
logZ = -65.259 +- 0.328
single instance: logZ = -65.259 +- 0.136
bootstrapped : logZ = -65.246 +- 0.198
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.365 │ ▁▁▁▁▁▁▁▂▂▂▄▆▅▅▆▇▆▇▇▆▅▄▄▄▂▂▁▁▁▁▁▁▁▁ ▁ │3.037 2.673 +- 0.085
amplitude : 0.0000000000328│ ▁ ▁▁▁▁▁▁▂▂▃▃▃▅▅▆▆▇▇▇▆▅▅▄▄▂▂▂▁▁▁▁▁▁ ▁▁ │0.0000000000563 0.0000000000444 +- 0.0000000000030
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 : 2.673 +/- 0.08
amplitude : 4.44e-11 +/- 3.0e-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': [2.6726094983964317, 4.440415534753019e-11], 'stdev': [0.08473335312102624, 2.970920713269077e-12], 'median': [2.6723634443996076, 4.440891996238534e-11], 'errlo': [2.584882147050954, 4.1424222787964657e-11], 'errup': [2.7609619992889054, 4.733548790389904e-11], 'information_gain_bits': [2.6883100942698723, 3.107579342305461]}
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(-4.11) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************** **** **** | +1.0e-10
Z=-inf(0.00%) | Like=-1944.90..-20.55 [-1944.8954..-105.0153] | it/evals=0/301 eff=0.0000% N=300
Z=-171.7(0.00%) | Like=-166.81..-20.55 [-1944.8954..-105.0153] | it/evals=30/331 eff=96.7742% N=300
Z=-157.8(0.00%) | Like=-152.36..-20.55 [-1944.8954..-105.0153] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.11) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************ **** **** | +1.0e-10
Z=-144.2(0.00%) | Like=-137.45..-20.55 [-1944.8954..-105.0153] | it/evals=89/404 eff=85.5769% N=300
Z=-143.1(0.00%) | Like=-137.43..-20.55 [-1944.8954..-105.0153] | it/evals=90/406 eff=84.9057% N=300
Z=-133.9(0.00%) | Like=-129.18..-20.55 [-1944.8954..-105.0153] | it/evals=117/449 eff=78.5235% N=300
Z=-133.3(0.00%) | Like=-128.91..-20.55 [-1944.8954..-105.0153] | it/evals=120/454 eff=77.9221% N=300
Mono-modal Volume: ~exp(-4.53) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|************************************ ***** **** | +1.0e-10
Z=-128.4(0.00%) | Like=-123.29..-20.55 [-1944.8954..-105.0153] | it/evals=134/472 eff=77.9070% N=300
Z=-123.0(0.00%) | Like=-116.59..-20.55 [-1944.8954..-105.0153] | it/evals=150/491 eff=78.5340% N=300
Z=-114.8(0.00%) | Like=-109.94..-20.55 [-1944.8954..-105.0153] | it/evals=180/529 eff=78.6026% 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=-110.5(0.00%) | Like=-105.49..-20.55 [-1944.8954..-105.0153] | it/evals=201/551 eff=80.0797% N=300
Z=-107.7(0.00%) | Like=-101.89..-20.55 [-104.7850..-66.4496] | it/evals=210/564 eff=79.5455% N=300
Z=-100.6(0.00%) | Like=-95.57..-20.55 [-104.7850..-66.4496] | it/evals=236/605 eff=77.3770% N=300
Z=-99.8(0.00%) | Like=-94.90..-20.55 [-104.7850..-66.4496] | it/evals=240/612 eff=76.9231% N=300
Mono-modal Volume: ~exp(-4.86) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************** *** * | +1.0e-10
Z=-94.4(0.00%) | Like=-89.10..-20.55 [-104.7850..-66.4496] | it/evals=268/648 eff=77.0115% N=300
Z=-94.0(0.00%) | Like=-88.61..-20.55 [-104.7850..-66.4496] | it/evals=270/650 eff=77.1429% N=300
Z=-88.2(0.00%) | Like=-82.85..-20.55 [-104.7850..-66.4496] | it/evals=298/691 eff=76.2148% N=300
Z=-87.7(0.00%) | Like=-82.12..-20.55 [-104.7850..-66.4496] | it/evals=300/693 eff=76.3359% N=300
Z=-81.6(0.00%) | Like=-76.73..-20.55 [-104.7850..-66.4496] | it/evals=328/734 eff=75.5760% N=300
Z=-81.3(0.00%) | Like=-76.69..-20.55 [-104.7850..-66.4496] | it/evals=330/736 eff=75.6881% N=300
Mono-modal Volume: ~exp(-5.26) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-80.9(0.00%) | Like=-76.31..-20.55 [-104.7850..-66.4496] | it/evals=335/744 eff=75.4505% N=300
Z=-77.9(0.00%) | Like=-73.29..-20.55 [-104.7850..-66.4496] | it/evals=360/782 eff=74.6888% N=300
Z=-74.6(0.00%) | Like=-69.96..-20.55 [-104.7850..-66.4496] | it/evals=388/822 eff=74.3295% N=300
Z=-74.5(0.00%) | Like=-69.67..-20.55 [-104.7850..-66.4496] | it/evals=390/827 eff=74.0038% N=300
Mono-modal Volume: ~exp(-5.26) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ******************************** ****** | +5.0
amplitude: +1.0e-12| ************************************** | +1.0e-10
Z=-72.6(0.00%) | Like=-68.00..-20.55 [-104.7850..-66.4496] | it/evals=411/865 eff=72.7434% N=300
Z=-71.9(0.00%) | Like=-67.18..-20.55 [-104.7850..-66.4496] | it/evals=420/879 eff=72.5389% N=300
Z=-69.1(0.00%) | Like=-64.30..-20.55 [-66.4333..-45.8209] | it/evals=446/919 eff=72.0517% N=300
Z=-68.7(0.00%) | Like=-63.75..-20.55 [-66.4333..-45.8209] | it/evals=450/924 eff=72.1154% N=300
Mono-modal Volume: ~exp(-5.63) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ********************************* +7.7e-11 | +1.0e-10
Z=-66.4(0.00%) | Like=-61.09..-20.55 [-66.4333..-45.8209] | it/evals=469/955 eff=71.6031% N=300
Z=-64.5(0.00%) | Like=-58.55..-20.55 [-66.4333..-45.8209] | it/evals=480/969 eff=71.7489% N=300
Z=-60.5(0.00%) | Like=-55.62..-20.55 [-66.4333..-45.8209] | it/evals=509/1009 eff=71.7913% N=300
Z=-60.5(0.00%) | Like=-55.49..-20.55 [-66.4333..-45.8209] | it/evals=510/1010 eff=71.8310% N=300
Z=-58.0(0.00%) | Like=-52.91..-20.55 [-66.4333..-45.8209] | it/evals=535/1054 eff=70.9549% N=300
Mono-modal Volume: ~exp(-6.16) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| **************************** +4.0 | +5.0
amplitude: +1.0e-12| ****************************** +7.2e-11 | +1.0e-10
Z=-57.9(0.00%) | Like=-52.83..-20.55 [-66.4333..-45.8209] | it/evals=536/1057 eff=70.8058% N=300
Z=-57.6(0.00%) | Like=-52.56..-20.55 [-66.4333..-45.8209] | it/evals=540/1063 eff=70.7733% N=300
Z=-55.4(0.00%) | Like=-50.44..-20.55 [-66.4333..-45.8209] | it/evals=570/1098 eff=71.4286% N=300
Z=-53.9(0.00%) | Like=-48.96..-20.55 [-66.4333..-45.8209] | it/evals=594/1140 eff=70.7143% N=300
Z=-53.5(0.00%) | Like=-48.43..-20.55 [-66.4333..-45.8209] | it/evals=600/1147 eff=70.8383% N=300
Mono-modal Volume: ~exp(-6.16) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************* +3.8 | +5.0
amplitude: +1.0e-12| **************************** +6.7e-11 | +1.0e-10
Z=-51.4(0.00%) | Like=-46.35..-20.55 [-66.4333..-45.8209] | it/evals=628/1184 eff=71.0407% N=300
Z=-51.2(0.00%) | Like=-46.27..-20.55 [-66.4333..-45.8209] | it/evals=630/1187 eff=71.0259% N=300
Z=-49.7(0.00%) | Like=-44.89..-20.55 [-45.8024..-32.0340] | it/evals=660/1229 eff=71.0441% N=300
Mono-modal Volume: ~exp(-6.22) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| *********************** +3.7 | +5.0
amplitude: +1.0e-12| ************************ +6.3e-11 | +1.0e-10
Z=-49.3(0.00%) | Like=-44.41..-20.55 [-45.8024..-32.0340] | it/evals=670/1241 eff=71.2009% N=300
Z=-48.0(0.00%) | Like=-42.71..-20.55 [-45.8024..-32.0340] | it/evals=690/1273 eff=70.9147% N=300
Z=-46.0(0.00%) | Like=-40.19..-20.55 [-45.8024..-32.0340] | it/evals=716/1314 eff=70.6114% N=300
Z=-45.5(0.00%) | Like=-39.56..-20.55 [-45.8024..-32.0340] | it/evals=720/1318 eff=70.7269% N=300
Mono-modal Volume: ~exp(-6.83) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ******************** +3.5 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-44.0(0.00%) | Like=-38.49..-20.55 [-45.8024..-32.0340] | it/evals=737/1344 eff=70.5939% N=300
Z=-43.2(0.00%) | Like=-37.68..-20.55 [-45.8024..-32.0340] | it/evals=750/1359 eff=70.8215% N=300
Z=-41.6(0.00%) | Like=-36.29..-20.55 [-45.8024..-32.0340] | it/evals=779/1399 eff=70.8826% N=300
Z=-41.5(0.00%) | Like=-36.28..-20.55 [-45.8024..-32.0340] | it/evals=780/1401 eff=70.8447% N=300
Mono-modal Volume: ~exp(-6.83) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ****************** +3.4 | +5.0
amplitude: +1.0e-12| ********************* +5.7e-11 | +1.0e-10
Z=-40.3(0.00%) | Like=-34.91..-20.55 [-45.8024..-32.0340] | it/evals=805/1438 eff=70.7381% N=300
Z=-40.1(0.00%) | Like=-34.82..-20.55 [-45.8024..-32.0340] | it/evals=810/1447 eff=70.6190% N=300
Z=-38.9(0.00%) | Like=-33.54..-20.55 [-45.8024..-32.0340] | it/evals=837/1488 eff=70.4545% N=300
Z=-38.8(0.00%) | Like=-33.33..-20.55 [-45.8024..-32.0340] | it/evals=840/1492 eff=70.4698% N=300
Z=-37.6(0.00%) | Like=-31.95..-20.55 [-31.9928..-27.4384] | it/evals=868/1532 eff=70.4545% N=300
Z=-37.5(0.00%) | Like=-31.74..-20.55 [-31.9928..-27.4384] | it/evals=870/1535 eff=70.4453% N=300
Mono-modal Volume: ~exp(-6.98) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 *************** +3.2 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-37.4(0.00%) | Like=-31.72..-20.55 [-31.9928..-27.4384] | it/evals=871/1536 eff=70.4693% N=300
Z=-36.2(0.00%) | Like=-30.87..-20.55 [-31.9928..-27.4384] | it/evals=898/1575 eff=70.4314% N=300
Z=-36.1(0.00%) | Like=-30.82..-20.55 [-31.9928..-27.4384] | it/evals=900/1578 eff=70.4225% N=300
Z=-35.3(0.01%) | Like=-30.26..-20.55 [-31.9928..-27.4384] | it/evals=929/1618 eff=70.4856% N=300
Z=-35.3(0.01%) | Like=-30.18..-20.48 [-31.9928..-27.4384] | it/evals=930/1619 eff=70.5080% N=300
Mono-modal Volume: ~exp(-7.52) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 *************** +3.2 | +5.0
amplitude: +1.0e-12| **************** +5.2e-11 | +1.0e-10
Z=-35.1(0.01%) | Like=-30.10..-20.48 [-31.9928..-27.4384] | it/evals=938/1629 eff=70.5794% N=300
Z=-34.6(0.02%) | Like=-29.49..-20.48 [-31.9928..-27.4384] | it/evals=960/1661 eff=70.5364% N=300
Z=-34.0(0.03%) | Like=-28.65..-20.48 [-31.9928..-27.4384] | it/evals=987/1702 eff=70.3994% N=300
Z=-33.9(0.04%) | Like=-28.59..-20.48 [-31.9928..-27.4384] | it/evals=990/1706 eff=70.4125% N=300
Mono-modal Volume: ~exp(-7.66) * 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=-33.5(0.05%) | Like=-28.36..-20.48 [-31.9928..-27.4384] | it/evals=1005/1726 eff=70.4769% N=300
Z=-33.2(0.07%) | Like=-28.08..-20.48 [-31.9928..-27.4384] | it/evals=1020/1745 eff=70.5882% N=300
Z=-32.7(0.12%) | Like=-27.35..-20.48 [-27.3821..-26.9438] | it/evals=1050/1786 eff=70.6595% N=300
Mono-modal Volume: ~exp(-7.96) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************ +3.1 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-32.2(0.18%) | Like=-26.91..-20.46 [-26.9370..-26.8300] | it/evals=1072/1814 eff=70.8058% N=300
Z=-32.1(0.21%) | Like=-26.73..-20.46 [-26.7304..-26.7254]*| it/evals=1080/1823 eff=70.9127% N=300
Z=-31.5(0.36%) | Like=-26.03..-20.46 [-26.0263..-25.9632] | it/evals=1110/1862 eff=71.0627% N=300
Z=-31.1(0.57%) | Like=-25.67..-20.46 [-25.6892..-25.6705] | it/evals=1133/1904 eff=70.6359% N=300
Mono-modal Volume: ~exp(-7.96) 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=-30.9(0.64%) | Like=-25.53..-20.46 [-25.5293..-25.5160] | it/evals=1140/1912 eff=70.7196% N=300
Z=-30.5(1.05%) | Like=-25.00..-20.46 [-24.9991..-24.9883] | it/evals=1169/1952 eff=70.7627% N=300
Z=-30.5(1.07%) | Like=-24.99..-20.46 [-24.9991..-24.9883] | it/evals=1170/1954 eff=70.7376% N=300
Z=-30.0(1.68%) | Like=-24.61..-20.46 [-24.6069..-24.5851] | it/evals=1200/1994 eff=70.8383% N=300
Mono-modal Volume: ~exp(-8.12) * 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=-29.9(1.84%) | Like=-24.52..-20.46 [-24.5407..-24.5207] | it/evals=1206/2003 eff=70.8162% N=300
Z=-29.6(2.45%) | Like=-24.27..-20.46 [-24.2669..-24.2620]*| it/evals=1230/2032 eff=71.0162% N=300
Z=-29.3(3.49%) | Like=-23.96..-20.46 [-23.9719..-23.9600] | it/evals=1260/2067 eff=71.3073% N=300
Mono-modal Volume: ~exp(-8.56) * 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(3.98%) | Like=-23.82..-20.46 [-23.8246..-23.8194]*| it/evals=1273/2084 eff=71.3565% N=300
Z=-29.0(4.67%) | Like=-23.72..-20.46 [-23.7237..-23.7184]*| it/evals=1290/2111 eff=71.2314% N=300
Z=-28.8(6.11%) | Like=-23.48..-20.46 [-23.4752..-23.4715]*| it/evals=1320/2150 eff=71.3514% N=300
Mono-modal Volume: ~exp(-8.56) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********* +4.4e-11 | +1.0e-10
Z=-28.5(7.60%) | Like=-23.25..-20.46 [-23.2473..-23.2349] | it/evals=1347/2188 eff=71.3453% N=300
Z=-28.5(7.75%) | Like=-23.22..-20.46 [-23.2237..-23.2153]*| it/evals=1350/2195 eff=71.2401% N=300
Z=-28.3(9.54%) | Like=-22.96..-20.46 [-22.9550..-22.9491]*| it/evals=1378/2235 eff=71.2145% N=300
Z=-28.3(9.67%) | Like=-22.95..-20.46 [-22.9454..-22.9400]*| it/evals=1380/2238 eff=71.2074% N=300
Z=-28.1(11.29%) | Like=-22.77..-20.46 [-22.7735..-22.7345] | it/evals=1406/2278 eff=71.0819% N=300
Mono-modal Volume: ~exp(-8.94) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.1(11.37%) | Like=-22.73..-20.46 [-22.7735..-22.7345] | it/evals=1407/2279 eff=71.0965% N=300
Z=-28.1(11.64%) | Like=-22.71..-20.46 [-22.7097..-22.7078]*| it/evals=1410/2285 eff=71.0327% N=300
Z=-27.9(13.57%) | Like=-22.59..-20.46 [-22.5864..-22.5848]*| it/evals=1436/2326 eff=70.8786% N=300
Z=-27.9(13.76%) | Like=-22.58..-20.46 [-22.5784..-22.5702]*| it/evals=1440/2332 eff=70.8661% N=300
Z=-27.7(16.11%) | Like=-22.40..-20.46 [-22.4015..-22.4013]*| it/evals=1470/2372 eff=70.9459% N=300
Mono-modal Volume: ~exp(-9.05) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ******* +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.2e-11 | +1.0e-10
Z=-27.7(16.47%) | Like=-22.35..-20.46 [-22.3647..-22.3546] | it/evals=1474/2376 eff=71.0019% N=300
Z=-27.6(18.86%) | Like=-22.13..-20.46 [-22.1451..-22.1326] | it/evals=1500/2406 eff=71.2251% N=300
Z=-27.5(21.72%) | Like=-21.96..-20.46 [-21.9644..-21.9628]*| it/evals=1527/2446 eff=71.1556% N=300
Z=-27.4(22.08%) | Like=-21.96..-20.46 [-21.9577..-21.9347] | it/evals=1530/2452 eff=71.0967% N=300
Mono-modal Volume: ~exp(-9.38) * 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=-27.4(23.27%) | Like=-21.89..-20.46 [-21.8912..-21.8660] | it/evals=1541/2470 eff=71.0138% N=300
Z=-27.3(25.44%) | Like=-21.77..-20.46 [-21.7721..-21.7605] | it/evals=1560/2496 eff=71.0383% N=300
Z=-27.2(28.92%) | Like=-21.68..-20.46 [-21.6754..-21.6729]*| it/evals=1590/2535 eff=71.1409% N=300
Mono-modal Volume: ~exp(-9.46) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.1(31.08%) | Like=-21.57..-20.46 [-21.5650..-21.5631]*| it/evals=1608/2563 eff=71.0561% N=300
Z=-27.1(32.64%) | Like=-21.53..-20.46 [-21.5288..-21.5266]*| it/evals=1620/2577 eff=71.1462% N=300
Z=-26.9(36.21%) | Like=-21.41..-20.46 [-21.4114..-21.4058]*| it/evals=1650/2614 eff=71.3051% N=300
Mono-modal Volume: ~exp(-9.51) * 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=-26.9(39.29%) | Like=-21.34..-20.46 [-21.3416..-21.3371]*| it/evals=1675/2651 eff=71.2463% N=300
Z=-26.8(39.92%) | Like=-21.33..-20.46 [-21.3307..-21.3264]*| it/evals=1680/2659 eff=71.2166% N=300
Z=-26.8(43.16%) | Like=-21.25..-20.46 [-21.2513..-21.2506]*| it/evals=1705/2699 eff=71.0713% N=300
Z=-26.8(43.81%) | Like=-21.24..-20.46 [-21.2370..-21.2342]*| it/evals=1710/2706 eff=71.0723% N=300
Z=-26.7(47.02%) | Like=-21.17..-20.46 [-21.1743..-21.1723]*| it/evals=1737/2746 eff=71.0139% N=300
Z=-26.7(47.33%) | Like=-21.17..-20.46 [-21.1705..-21.1635]*| it/evals=1740/2750 eff=71.0204% N=300
Mono-modal Volume: ~exp(-9.88) * 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=-26.7(47.59%) | Like=-21.16..-20.46 [-21.1596..-21.1574]*| it/evals=1742/2752 eff=71.0440% N=300
Z=-26.6(50.75%) | Like=-21.09..-20.46 [-21.0895..-21.0858]*| it/evals=1770/2787 eff=71.1701% N=300
Z=-26.5(54.04%) | Like=-21.01..-20.46 [-21.0130..-21.0082]*| it/evals=1798/2827 eff=71.1516% N=300
Z=-26.5(54.28%) | Like=-21.01..-20.46 [-21.0075..-21.0074]*| it/evals=1800/2831 eff=71.1181% N=300
Mono-modal Volume: ~exp(-10.25) * 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(55.28%) | Like=-20.99..-20.46 [-20.9890..-20.9874]*| it/evals=1809/2848 eff=70.9969% N=300
Z=-26.5(57.60%) | Like=-20.96..-20.46 [-20.9578..-20.9524]*| it/evals=1830/2871 eff=71.1785% N=300
Z=-26.4(60.89%) | Like=-20.90..-20.46 [-20.9006..-20.9006]*| it/evals=1860/2908 eff=71.3190% N=300
Mono-modal Volume: ~exp(-10.25) 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=-26.4(63.09%) | Like=-20.87..-20.46 [-20.8700..-20.8676]*| it/evals=1882/2949 eff=71.0457% N=300
Z=-26.4(63.89%) | Like=-20.86..-20.46 [-20.8608..-20.8600]*| it/evals=1890/2958 eff=71.1061% N=300
Z=-26.3(66.77%) | Like=-20.82..-20.46 [-20.8247..-20.8221]*| it/evals=1920/2996 eff=71.2166% N=300
Mono-modal Volume: ~exp(-10.67) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.7e-11 | +1.0e-10
Z=-26.3(68.80%) | Like=-20.80..-20.46 [-20.7976..-20.7974]*| it/evals=1943/3036 eff=71.0161% N=300
Z=-26.3(69.38%) | Like=-20.79..-20.46 [-20.7887..-20.7875]*| it/evals=1950/3045 eff=71.0383% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3055
[ultranest] logZ = -25.95 +- 0.1039
[ultranest] Effective samples strategy satisfied (ESS = 1008.0, 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:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.932 +- 0.329
single instance: logZ = -25.932 +- 0.122
bootstrapped : logZ = -25.950 +- 0.199
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.11 │ ▁ ▁▁▁▁▁▁▃▃▄▄▆▇▆▇▇▇▅▅▃▄▃▂▂▁▁▁▁▁▁▁ ▁ │3.12 2.58 +- 0.13
amplitude : 0.0000000000198│ ▁ ▁▁▁▁▁▂▂▃▃▅▆▇▇▇▇▆▆▄▄▃▃▂▁▁▁▁▁▁▁ ▁ │0.0000000000535 0.0000000000341 +- 0.0000000000038
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.23) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** **** **** ** *** *| +1.0e-10
Z=-inf(0.00%) | Like=-547.01..-19.55 [-547.0103..-128.7928] | it/evals=0/301 eff=0.0000% N=300
Z=-217.9(0.00%) | Like=-211.76..-19.55 [-547.0103..-128.7928] | it/evals=30/333 eff=90.9091% N=300
Z=-202.0(0.00%) | Like=-197.30..-19.55 [-547.0103..-128.7928] | it/evals=60/365 eff=92.3077% N=300
Mono-modal Volume: ~exp(-4.23) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ***** ** ** *| +1.0e-10
Z=-185.9(0.00%) | Like=-180.49..-19.55 [-547.0103..-128.7928] | it/evals=90/396 eff=93.7500% N=300
Z=-171.6(0.00%) | Like=-166.40..-19.55 [-547.0103..-128.7928] | it/evals=120/434 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.47) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ************************************** ** ** *| +1.0e-10
Z=-165.9(0.00%) | Like=-160.98..-19.55 [-547.0103..-128.7928] | it/evals=134/454 eff=87.0130% N=300
Z=-161.2(0.00%) | Like=-156.10..-19.55 [-547.0103..-128.7928] | it/evals=150/474 eff=86.2069% N=300
Z=-145.6(0.00%) | Like=-138.45..-19.55 [-547.0103..-128.7928] | it/evals=180/508 eff=86.5385% N=300
Mono-modal Volume: ~exp(-4.86) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ************************************* *** ****| +1.0e-10
Z=-135.5(0.00%) | Like=-129.97..-19.55 [-547.0103..-128.7928] | it/evals=201/536 eff=85.1695% N=300
Z=-131.5(0.00%) | Like=-126.16..-19.55 [-128.5255..-66.0900] | it/evals=210/552 eff=83.3333% N=300
Z=-119.7(0.00%) | Like=-114.75..-19.55 [-128.5255..-66.0900] | it/evals=240/592 eff=82.1918% N=300
Mono-modal Volume: ~exp(-4.86) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************ *** ****| +1.0e-10
Z=-110.8(0.00%) | Like=-104.92..-19.55 [-128.5255..-66.0900] | it/evals=268/632 eff=80.7229% N=300
Z=-110.0(0.00%) | Like=-104.16..-19.55 [-128.5255..-66.0900] | it/evals=270/634 eff=80.8383% N=300
Z=-101.5(0.00%) | Like=-95.62..-19.55 [-128.5255..-66.0900] | it/evals=300/670 eff=81.0811% N=300
Z=-89.0(0.00%) | Like=-82.46..-19.55 [-128.5255..-66.0900] | it/evals=330/708 eff=80.8824% N=300
Mono-modal Volume: ~exp(-4.86) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************** *** ****| +1.0e-10
Z=-81.3(0.00%) | Like=-76.06..-19.28 [-128.5255..-66.0900] | it/evals=357/744 eff=80.4054% N=300
Z=-80.8(0.00%) | Like=-75.74..-19.28 [-128.5255..-66.0900] | it/evals=360/749 eff=80.1782% N=300
Z=-75.5(0.00%) | Like=-68.86..-19.28 [-128.5255..-66.0900] | it/evals=388/789 eff=79.3456% N=300
Z=-74.6(0.00%) | Like=-68.77..-19.28 [-128.5255..-66.0900] | it/evals=390/791 eff=79.4297% N=300
Mono-modal Volume: ~exp(-4.86) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-68.5(0.00%) | Like=-63.00..-19.28 [-65.9863..-41.9020] | it/evals=414/828 eff=78.4091% N=300
Z=-67.2(0.00%) | Like=-61.71..-19.28 [-65.9863..-41.9020] | it/evals=420/837 eff=78.2123% N=300
Z=-62.8(0.00%) | Like=-57.69..-19.28 [-65.9863..-41.9020] | it/evals=450/876 eff=78.1250% N=300
Mono-modal Volume: ~exp(-5.63) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| **************************************| +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-60.1(0.00%) | Like=-54.62..-19.28 [-65.9863..-41.9020] | it/evals=469/918 eff=75.8900% N=300
Z=-58.4(0.00%) | Like=-52.75..-19.28 [-65.9863..-41.9020] | it/evals=480/930 eff=76.1905% N=300
Z=-54.5(0.00%) | Like=-49.34..-19.28 [-65.9863..-41.9020] | it/evals=510/964 eff=76.8072% N=300
Mono-modal Volume: ~exp(-5.65) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-51.7(0.00%) | Like=-46.37..-19.28 [-65.9863..-41.9020] | it/evals=536/1000 eff=76.5714% N=300
Z=-51.2(0.00%) | Like=-45.75..-19.28 [-65.9863..-41.9020] | it/evals=540/1005 eff=76.5957% N=300
Z=-48.7(0.00%) | Like=-43.89..-19.21 [-65.9863..-41.9020] | it/evals=569/1046 eff=76.2735% N=300
Z=-48.7(0.00%) | Like=-43.86..-19.21 [-65.9863..-41.9020] | it/evals=570/1048 eff=76.2032% N=300
Z=-47.0(0.00%) | Like=-41.78..-19.21 [-41.7787..-29.7672] | it/evals=596/1089 eff=75.5387% N=300
Z=-46.7(0.00%) | Like=-41.62..-19.21 [-41.7787..-29.7672] | it/evals=600/1094 eff=75.5668% N=300
Mono-modal Volume: ~exp(-6.01) * 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=-46.5(0.00%) | Like=-41.58..-19.21 [-41.7787..-29.7672] | it/evals=603/1098 eff=75.5639% N=300
Z=-44.7(0.00%) | Like=-39.66..-19.21 [-41.7787..-29.7672] | it/evals=630/1130 eff=75.9036% N=300
Z=-42.8(0.00%) | Like=-37.52..-19.21 [-41.7787..-29.7672] | it/evals=659/1170 eff=75.7471% N=300
Z=-42.7(0.00%) | Like=-37.49..-19.21 [-41.7787..-29.7672] | it/evals=660/1171 eff=75.7750% N=300
Mono-modal Volume: ~exp(-6.30) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************** +4.1 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******************************** ** | +1.0e-10
Z=-42.0(0.00%) | Like=-36.77..-19.21 [-41.7787..-29.7672] | it/evals=670/1184 eff=75.7919% N=300
Z=-40.8(0.00%) | Like=-35.34..-19.21 [-41.7787..-29.7672] | it/evals=690/1217 eff=75.2454% N=300
Z=-39.1(0.00%) | Like=-33.81..-19.21 [-41.7787..-29.7672] | it/evals=716/1259 eff=74.6611% N=300
Z=-38.8(0.00%) | Like=-33.31..-19.21 [-41.7787..-29.7672] | it/evals=720/1269 eff=74.3034% N=300
Mono-modal Volume: ~exp(-6.55) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************** +3.9 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************************** | +1.0e-10
Z=-37.8(0.00%) | Like=-32.59..-19.21 [-41.7787..-29.7672] | it/evals=737/1293 eff=74.2195% N=300
Z=-37.1(0.00%) | Like=-31.96..-19.21 [-41.7787..-29.7672] | it/evals=750/1313 eff=74.0375% N=300
Z=-35.9(0.00%) | Like=-30.80..-19.21 [-41.7787..-29.7672] | it/evals=780/1352 eff=74.1445% N=300
Z=-35.1(0.00%) | Like=-30.19..-19.21 [-41.7787..-29.7672] | it/evals=803/1398 eff=73.1330% N=300
Mono-modal Volume: ~exp(-6.94) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ******************* +3.7 | +5.0
amplitude: +1.0e-12| +3.2e-11 *************************** | +1.0e-10
Z=-35.1(0.00%) | Like=-30.19..-19.21 [-41.7787..-29.7672] | it/evals=804/1400 eff=73.0909% N=300
Z=-34.9(0.00%) | Like=-29.91..-19.21 [-41.7787..-29.7672] | it/evals=810/1408 eff=73.1047% N=300
Z=-34.1(0.01%) | Like=-29.22..-19.21 [-29.7055..-26.0177] | it/evals=840/1444 eff=73.4266% N=300
Z=-33.3(0.01%) | Like=-28.29..-19.21 [-29.7055..-26.0177] | it/evals=870/1482 eff=73.6041% N=300
Mono-modal Volume: ~exp(-7.25) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.3 ***************** +3.6 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************************ | +1.0e-10
Z=-33.2(0.01%) | Like=-28.26..-19.21 [-29.7055..-26.0177] | it/evals=871/1483 eff=73.6264% N=300
Z=-32.6(0.03%) | Like=-27.44..-19.21 [-29.7055..-26.0177] | it/evals=898/1523 eff=73.4260% N=300
Z=-32.5(0.03%) | Like=-27.42..-19.21 [-29.7055..-26.0177] | it/evals=900/1527 eff=73.3496% N=300
Z=-31.8(0.05%) | Like=-26.65..-19.21 [-29.7055..-26.0177] | it/evals=930/1566 eff=73.4597% N=300
Mono-modal Volume: ~exp(-7.25) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 **************** +3.5 | +5.0
amplitude: +1.0e-12| +3.6e-11 *********************** | +1.0e-10
Z=-31.2(0.09%) | Like=-26.19..-19.21 [-29.7055..-26.0177] | it/evals=954/1603 eff=73.2157% N=300
Z=-31.1(0.11%) | Like=-25.89..-19.21 [-26.0077..-25.5528] | it/evals=960/1615 eff=73.0038% N=300
Z=-30.5(0.20%) | Like=-25.28..-19.21 [-25.2939..-25.2793] | it/evals=988/1656 eff=72.8614% N=300
Z=-30.4(0.21%) | Like=-25.18..-19.21 [-25.1985..-25.1774] | it/evals=990/1661 eff=72.7406% N=300
Mono-modal Volume: ~exp(-7.51) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************** +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.7e-11 | +1.0e-10
Z=-30.1(0.28%) | Like=-24.93..-19.21 [-24.9256..-24.9210]*| it/evals=1005/1681 eff=72.7734% N=300
Z=-29.8(0.38%) | Like=-24.58..-19.21 [-24.5777..-24.5636] | it/evals=1020/1701 eff=72.8051% N=300
Z=-29.3(0.59%) | Like=-24.07..-19.16 [-24.0831..-24.0702] | it/evals=1045/1741 eff=72.5191% N=300
Z=-29.2(0.62%) | Like=-23.94..-19.16 [-23.9695..-23.9385] | it/evals=1050/1749 eff=72.4638% N=300
Mono-modal Volume: ~exp(-7.99) * Expected Volume: exp(-3.57) 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=-28.8(0.96%) | Like=-23.54..-19.16 [-23.5393..-23.5375]*| it/evals=1072/1782 eff=72.3347% N=300
Z=-28.7(1.11%) | Like=-23.49..-19.16 [-23.4939..-23.4938]*| it/evals=1080/1790 eff=72.4832% N=300
Z=-28.2(1.71%) | Like=-23.10..-19.16 [-23.1365..-23.1042] | it/evals=1110/1828 eff=72.6440% N=300
Mono-modal Volume: ~exp(-7.99) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.1e-11 | +1.0e-10
Z=-27.8(2.59%) | Like=-22.66..-19.16 [-22.6632..-22.6222] | it/evals=1140/1863 eff=72.9367% N=300
Z=-27.5(3.64%) | Like=-22.36..-19.16 [-22.3649..-22.3495] | it/evals=1166/1904 eff=72.6933% N=300
Z=-27.5(3.81%) | Like=-22.32..-19.16 [-22.3180..-22.2961] | it/evals=1170/1911 eff=72.6257% N=300
Z=-27.1(5.27%) | Like=-22.01..-19.16 [-22.0101..-22.0086]*| it/evals=1198/1951 eff=72.5621% N=300
Z=-27.1(5.40%) | Like=-21.99..-19.16 [-22.0086..-21.9943] | it/evals=1200/1953 eff=72.5953% N=300
Mono-modal Volume: ~exp(-8.26) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +7.0e-11 | +1.0e-10
Z=-27.1(5.72%) | Like=-21.89..-19.16 [-21.8858..-21.8817]*| it/evals=1206/1966 eff=72.3890% N=300
Z=-26.8(7.26%) | Like=-21.68..-19.16 [-21.6778..-21.6760]*| it/evals=1230/2001 eff=72.3104% N=300
Z=-26.6(9.20%) | Like=-21.47..-19.16 [-21.4976..-21.4739] | it/evals=1257/2036 eff=72.4078% N=300
Z=-26.6(9.48%) | Like=-21.46..-19.16 [-21.4575..-21.4527]*| it/evals=1260/2040 eff=72.4138% N=300
Mono-modal Volume: ~exp(-8.42) * Expected Volume: exp(-4.24) 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(10.56%) | Like=-21.38..-19.16 [-21.3772..-21.3745]*| it/evals=1273/2056 eff=72.4943% N=300
Z=-26.3(12.02%) | Like=-21.27..-19.16 [-21.2749..-21.2456] | it/evals=1290/2076 eff=72.6351% N=300
Z=-26.1(14.88%) | Like=-21.07..-19.16 [-21.0708..-21.0631]*| it/evals=1320/2112 eff=72.8477% N=300
Mono-modal Volume: ~exp(-8.42) Expected Volume: exp(-4.47) 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=-26.0(17.39%) | Like=-20.94..-19.16 [-20.9360..-20.9313]*| it/evals=1344/2147 eff=72.7666% N=300
Z=-25.9(18.02%) | Like=-20.90..-19.16 [-20.9033..-20.9025]*| it/evals=1350/2157 eff=72.6979% N=300
Z=-25.8(20.86%) | Like=-20.74..-19.16 [-20.7377..-20.7318]*| it/evals=1376/2197 eff=72.5356% N=300
Z=-25.8(21.32%) | Like=-20.72..-19.16 [-20.7293..-20.7184] | it/evals=1380/2203 eff=72.5171% N=300
Mono-modal Volume: ~exp(-8.82) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.5e-11 | +1.0e-10
Z=-25.6(24.27%) | Like=-20.62..-19.16 [-20.6223..-20.6113] | it/evals=1407/2242 eff=72.4511% N=300
Z=-25.6(24.57%) | Like=-20.60..-19.16 [-20.6035..-20.6007]*| it/evals=1410/2246 eff=72.4563% N=300
Z=-25.5(28.02%) | Like=-20.47..-19.16 [-20.4743..-20.4731]*| it/evals=1440/2281 eff=72.6906% N=300
Z=-25.4(31.45%) | Like=-20.35..-19.16 [-20.3536..-20.3520]*| it/evals=1470/2320 eff=72.7723% 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| +4.7e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.4(31.95%) | Like=-20.34..-19.16 [-20.3405..-20.3340]*| it/evals=1474/2325 eff=72.7901% N=300
Z=-25.3(35.09%) | Like=-20.22..-19.16 [-20.2339..-20.2204] | it/evals=1500/2360 eff=72.8155% N=300
Z=-25.2(38.95%) | Like=-20.13..-19.16 [-20.1283..-20.1249]*| it/evals=1530/2397 eff=72.9614% N=300
Mono-modal Volume: ~exp(-9.15) Expected Volume: exp(-5.14) 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=-25.1(42.20%) | Like=-20.07..-19.16 [-20.0694..-20.0648]*| it/evals=1557/2434 eff=72.9616% N=300
Z=-25.1(42.53%) | Like=-20.06..-19.16 [-20.0603..-20.0550]*| it/evals=1560/2440 eff=72.8972% N=300
Z=-25.0(45.34%) | Like=-19.97..-19.16 [-19.9720..-19.9714]*| it/evals=1586/2480 eff=72.7523% N=300
Z=-25.0(45.78%) | Like=-19.97..-19.16 [-19.9674..-19.9648]*| it/evals=1590/2485 eff=72.7689% N=300
Mono-modal Volume: ~exp(-9.25) * Expected Volume: exp(-5.36) 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=-25.0(47.82%) | Like=-19.91..-19.16 [-19.9081..-19.9077]*| it/evals=1608/2510 eff=72.7602% N=300
Z=-24.9(49.25%) | Like=-19.88..-19.16 [-19.8782..-19.8759]*| it/evals=1620/2527 eff=72.7436% N=300
Z=-24.9(52.80%) | Like=-19.82..-19.16 [-19.8173..-19.8168]*| it/evals=1650/2565 eff=72.8477% N=300
Mono-modal Volume: ~exp(-9.94) * 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=-24.8(55.50%) | Like=-19.77..-19.16 [-19.7727..-19.7722]*| it/evals=1675/2595 eff=72.9847% N=300
Z=-24.8(56.02%) | Like=-19.77..-19.16 [-19.7672..-19.7657]*| it/evals=1680/2604 eff=72.9167% N=300
Z=-24.7(59.16%) | Like=-19.72..-19.16 [-19.7151..-19.7108]*| it/evals=1710/2637 eff=73.1707% N=300
Z=-24.7(61.97%) | Like=-19.66..-19.16 [-19.6629..-19.6616]*| it/evals=1739/2676 eff=73.1902% N=300
Z=-24.7(62.05%) | Like=-19.66..-19.16 [-19.6616..-19.6608]*| it/evals=1740/2677 eff=73.2015% 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| +5.0e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.7(64.93%) | Like=-19.61..-19.16 [-19.6080..-19.6069]*| it/evals=1770/2712 eff=73.3831% N=300
Z=-24.6(67.65%) | Like=-19.56..-19.16 [-19.5582..-19.5567]*| it/evals=1800/2748 eff=73.5294% N=300
Mono-modal Volume: ~exp(-10.06) * 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=-24.6(68.44%) | Like=-19.55..-19.16 [-19.5518..-19.5505]*| it/evals=1809/2761 eff=73.5067% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2782
[ultranest] logZ = -24.21 +- 0.07214
[ultranest] Effective samples strategy satisfied (ESS = 999.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-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:0.07 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -24.223 +- 0.295
single instance: logZ = -24.223 +- 0.116
bootstrapped : logZ = -24.210 +- 0.136
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.25 │ ▁ ▁▁▁▁▂▃▃▄▅▆▅▇▆▆▄▄▄▄▃▃▂▂▁▁▁▁▁▁▁▁▁ ▁ │3.58 2.82 +- 0.17
amplitude : 0.0000000000357│ ▁▁▁▁▁▁▂▂▃▂▄▅▄▅▅▅▇▆▅▆▅▅▄▃▃▂▁▁▁▁▁▁▁▁▁▁▁ │0.0000000000773 0.0000000000548 +- 0.0000000000059
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.93) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************** ****** * ** * *******| +1.0e-10
Z=-inf(0.00%) | Like=-855.64..-13.54 [-855.6360..-83.2366] | it/evals=0/301 eff=0.0000% N=300
Z=-150.1(0.00%) | Like=-144.51..-13.54 [-855.6360..-83.2366] | it/evals=30/332 eff=93.7500% N=300
Z=-138.0(0.00%) | Like=-133.04..-13.54 [-855.6360..-83.2366] | it/evals=60/369 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.15) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* * ** * *******| +1.0e-10
Z=-135.7(0.00%) | Like=-130.61..-13.54 [-855.6360..-83.2366] | it/evals=67/376 eff=88.1579% N=300
Z=-127.8(0.00%) | Like=-122.81..-13.54 [-855.6360..-83.2366] | it/evals=90/406 eff=84.9057% N=300
Z=-120.4(0.00%) | Like=-115.44..-13.54 [-855.6360..-83.2366] | it/evals=120/445 eff=82.7586% N=300
Mono-modal Volume: ~exp(-4.15) Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************* * ** *********| +1.0e-10
Z=-111.5(0.00%) | Like=-105.26..-13.54 [-855.6360..-83.2366] | it/evals=148/481 eff=81.7680% N=300
Z=-110.3(0.00%) | Like=-105.05..-13.54 [-855.6360..-83.2366] | it/evals=150/483 eff=81.9672% N=300
Z=-99.1(0.00%) | Like=-93.58..-13.54 [-855.6360..-83.2366] | it/evals=180/522 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.24) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ******************************** * ** *********| +1.0e-10
Z=-92.7(0.00%) | Like=-86.95..-13.54 [-855.6360..-83.2366] | it/evals=201/551 eff=80.0797% N=300
Z=-89.0(0.00%) | Like=-83.55..-13.54 [-855.6360..-83.2366] | it/evals=210/565 eff=79.2453% N=300
Z=-83.1(0.00%) | Like=-78.29..-13.54 [-83.2060..-46.2682] | it/evals=240/605 eff=78.6885% N=300
Mono-modal Volume: ~exp(-5.02) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************* **** *********| +1.0e-10
Z=-77.5(0.00%) | Like=-72.86..-13.54 [-83.2060..-46.2682] | it/evals=268/639 eff=79.0560% N=300
Z=-77.2(0.00%) | Like=-72.09..-13.54 [-83.2060..-46.2682] | it/evals=270/642 eff=78.9474% N=300
Z=-72.1(0.00%) | Like=-67.19..-13.54 [-83.2060..-46.2682] | it/evals=300/676 eff=79.7872% N=300
Z=-66.9(0.00%) | Like=-61.56..-13.54 [-83.2060..-46.2682] | it/evals=327/718 eff=78.2297% N=300
Z=-66.3(0.00%) | Like=-60.96..-13.54 [-83.2060..-46.2682] | it/evals=330/722 eff=78.1991% N=300
Mono-modal Volume: ~exp(-5.19) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-65.3(0.00%) | Like=-60.29..-13.54 [-83.2060..-46.2682] | it/evals=335/731 eff=77.7262% N=300
Z=-60.8(0.00%) | Like=-56.13..-13.54 [-83.2060..-46.2682] | it/evals=360/765 eff=77.4194% N=300
Z=-54.9(0.00%) | Like=-49.53..-13.54 [-83.2060..-46.2682] | it/evals=390/801 eff=77.8443% N=300
Mono-modal Volume: ~exp(-5.19) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************* *********| +1.0e-10
Z=-51.5(0.00%) | Like=-46.67..-13.54 [-83.2060..-46.2682] | it/evals=417/838 eff=77.5093% N=300
Z=-51.2(0.00%) | Like=-46.39..-13.54 [-83.2060..-46.2682] | it/evals=420/841 eff=77.6340% N=300
Z=-49.2(0.00%) | Like=-44.56..-13.54 [-46.2566..-29.7936] | it/evals=448/881 eff=77.1084% N=300
Z=-49.1(0.00%) | Like=-44.53..-13.54 [-46.2566..-29.7936] | it/evals=450/883 eff=77.1870% N=300
Mono-modal Volume: ~exp(-5.21) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-47.8(0.00%) | Like=-43.27..-13.54 [-46.2566..-29.7936] | it/evals=469/906 eff=77.3927% N=300
Z=-46.9(0.00%) | Like=-42.11..-13.54 [-46.2566..-29.7936] | it/evals=480/919 eff=77.5444% N=300
Z=-44.7(0.00%) | Like=-39.75..-13.54 [-46.2566..-29.7936] | it/evals=510/958 eff=77.5076% N=300
Mono-modal Volume: ~exp(-5.74) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-42.4(0.00%) | Like=-37.58..-13.54 [-46.2566..-29.7936] | it/evals=536/993 eff=77.3449% N=300
Z=-42.1(0.00%) | Like=-37.34..-13.54 [-46.2566..-29.7936] | it/evals=540/999 eff=77.2532% N=300
Z=-39.7(0.00%) | Like=-34.39..-13.36 [-46.2566..-29.7936] | it/evals=570/1041 eff=76.9231% N=300
Z=-37.4(0.00%) | Like=-32.29..-13.36 [-46.2566..-29.7936] | it/evals=599/1081 eff=76.6965% N=300
Z=-37.3(0.00%) | Like=-32.27..-13.36 [-46.2566..-29.7936] | it/evals=600/1082 eff=76.7263% N=300
Mono-modal Volume: ~exp(-6.38) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 *************************** +4.1 | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-37.1(0.00%) | Like=-32.13..-13.36 [-46.2566..-29.7936] | it/evals=603/1086 eff=76.7176% N=300
Z=-35.5(0.00%) | Like=-30.26..-13.31 [-46.2566..-29.7936] | it/evals=630/1119 eff=76.9231% N=300
Z=-33.6(0.00%) | Like=-28.49..-13.31 [-29.7806..-21.4599] | it/evals=660/1156 eff=77.1028% N=300
Mono-modal Volume: ~exp(-6.43) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************ +3.9 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-32.9(0.00%) | Like=-27.78..-13.31 [-29.7806..-21.4599] | it/evals=670/1168 eff=77.1889% N=300
Z=-31.9(0.00%) | Like=-27.06..-13.31 [-29.7806..-21.4599] | it/evals=690/1195 eff=77.0950% N=300
Z=-30.7(0.00%) | Like=-25.99..-13.31 [-29.7806..-21.4599] | it/evals=720/1235 eff=77.0053% N=300
Mono-modal Volume: ~exp(-6.46) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.8 | +5.0
amplitude: +1.0e-12| ************************************ *| +1.0e-10
Z=-30.2(0.00%) | Like=-25.60..-13.31 [-29.7806..-21.4599] | it/evals=737/1257 eff=77.0115% N=300
Z=-29.9(0.00%) | Like=-25.14..-13.31 [-29.7806..-21.4599] | it/evals=750/1275 eff=76.9231% N=300
Z=-28.9(0.00%) | Like=-23.83..-13.31 [-29.7806..-21.4599] | it/evals=779/1315 eff=76.7488% N=300
Z=-28.8(0.00%) | Like=-23.82..-13.31 [-29.7806..-21.4599] | it/evals=780/1316 eff=76.7717% N=300
Mono-modal Volume: ~exp(-6.68) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************* | +1.0e-10
Z=-28.0(0.01%) | Like=-22.95..-13.31 [-29.7806..-21.4599] | it/evals=804/1349 eff=76.6444% N=300
Z=-27.8(0.01%) | Like=-22.74..-13.31 [-29.7806..-21.4599] | it/evals=810/1356 eff=76.7045% N=300
Z=-26.8(0.02%) | Like=-21.68..-13.31 [-29.7806..-21.4599] | it/evals=840/1394 eff=76.7824% N=300
Z=-25.9(0.05%) | Like=-20.86..-13.31 [-21.3948..-20.0070] | it/evals=868/1434 eff=76.5432% N=300
Z=-25.9(0.05%) | Like=-20.83..-13.31 [-21.3948..-20.0070] | it/evals=870/1439 eff=76.3828% N=300
Mono-modal Volume: ~exp(-6.87) * 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=-25.9(0.05%) | Like=-20.79..-13.31 [-21.3948..-20.0070] | it/evals=871/1440 eff=76.4035% N=300
Z=-25.0(0.13%) | Like=-19.99..-13.31 [-19.9860..-19.8600] | it/evals=900/1478 eff=76.4007% N=300
Z=-24.3(0.24%) | Like=-19.40..-13.30 [-19.3955..-19.3938]*| it/evals=929/1518 eff=76.2726% N=300
Z=-24.3(0.24%) | Like=-19.39..-13.30 [-19.3938..-19.3867]*| it/evals=930/1521 eff=76.1671% N=300
Mono-modal Volume: ~exp(-7.43) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************************** | +1.0e-10
Z=-24.2(0.29%) | Like=-19.23..-13.30 [-19.2465..-19.2317] | it/evals=938/1530 eff=76.2602% N=300
Z=-23.7(0.41%) | Like=-18.93..-13.30 [-18.9453..-18.9311] | it/evals=960/1562 eff=76.0697% N=300
Z=-23.3(0.67%) | Like=-18.39..-13.30 [-18.3895..-18.3816]*| it/evals=989/1602 eff=75.9601% N=300
Z=-23.3(0.68%) | Like=-18.38..-13.30 [-18.3816..-18.3733]*| it/evals=990/1603 eff=75.9785% N=300
Mono-modal Volume: ~exp(-7.43) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.2e-11 *********************** +7.9e-11| +1.0e-10
Z=-22.8(1.05%) | Like=-17.84..-13.30 [-17.8702..-17.8398] | it/evals=1020/1639 eff=76.1763% N=300
Z=-22.4(1.62%) | Like=-17.45..-13.30 [-17.4985..-17.4547] | it/evals=1048/1679 eff=75.9971% N=300
Z=-22.4(1.67%) | Like=-17.44..-13.30 [-17.4518..-17.4370] | it/evals=1050/1681 eff=76.0319% N=300
Mono-modal Volume: ~exp(-7.43) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************* +7.7e-11 | +1.0e-10
Z=-22.0(2.33%) | Like=-17.07..-13.30 [-17.0914..-17.0672] | it/evals=1076/1717 eff=75.9351% N=300
Z=-22.0(2.44%) | Like=-17.04..-13.30 [-17.0398..-17.0215] | it/evals=1080/1723 eff=75.8960% N=300
Z=-21.6(3.37%) | Like=-16.68..-13.30 [-16.6824..-16.6822]*| it/evals=1108/1763 eff=75.7348% N=300
Z=-21.6(3.46%) | Like=-16.67..-13.30 [-16.6822..-16.6692] | it/evals=1110/1765 eff=75.7679% N=300
Mono-modal Volume: ~exp(-7.58) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.4e-11 | +1.0e-10
Z=-21.3(4.77%) | Like=-16.38..-13.30 [-16.3833..-16.3751]*| it/evals=1139/1804 eff=75.7314% N=300
Z=-21.3(4.83%) | Like=-16.38..-13.30 [-16.3751..-16.3664]*| it/evals=1140/1806 eff=75.6972% N=300
Z=-21.1(5.99%) | Like=-16.24..-13.30 [-16.2437..-16.2229] | it/evals=1164/1846 eff=75.2911% N=300
Z=-21.0(6.28%) | Like=-16.15..-13.30 [-16.1535..-16.1483]*| it/evals=1170/1855 eff=75.2412% N=300
Z=-20.8(8.05%) | Like=-15.93..-13.30 [-15.9325..-15.9125] | it/evals=1200/1893 eff=75.3296% N=300
Mono-modal Volume: ~exp(-7.80) * Expected Volume: exp(-4.02) 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=-20.7(8.38%) | Like=-15.86..-13.30 [-15.8764..-15.8574] | it/evals=1206/1902 eff=75.2809% N=300
Z=-20.5(9.81%) | Like=-15.65..-13.30 [-15.6520..-15.6486]*| it/evals=1230/1932 eff=75.3676% N=300
Z=-20.3(12.13%) | Like=-15.44..-13.30 [-15.4358..-15.4346]*| it/evals=1259/1972 eff=75.2990% N=300
Z=-20.3(12.22%) | Like=-15.43..-13.30 [-15.4346..-15.4296]*| it/evals=1260/1974 eff=75.2688% N=300
Mono-modal Volume: ~exp(-8.60) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********* +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 **************** +7.0e-11 | +1.0e-10
Z=-20.2(13.36%) | Like=-15.32..-13.30 [-15.3197..-15.3140]*| it/evals=1273/1990 eff=75.3254% N=300
Z=-20.1(14.83%) | Like=-15.20..-13.30 [-15.1993..-15.1815] | it/evals=1290/2015 eff=75.2187% N=300
Z=-20.0(17.64%) | Like=-14.97..-13.30 [-14.9932..-14.9733] | it/evals=1319/2054 eff=75.1995% N=300
Z=-20.0(17.74%) | Like=-14.97..-13.30 [-14.9732..-14.9662]*| it/evals=1320/2055 eff=75.2137% N=300
Mono-modal Volume: ~exp(-8.60) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.0e-11 ************** +6.8e-11 | +1.0e-10
Z=-19.8(20.57%) | Like=-14.80..-13.30 [-14.8034..-14.8008]*| it/evals=1347/2091 eff=75.2094% N=300
Z=-19.8(20.92%) | Like=-14.79..-13.30 [-14.8005..-14.7903] | it/evals=1350/2096 eff=75.1670% N=300
Z=-19.6(24.36%) | Like=-14.62..-13.30 [-14.6166..-14.6141]*| it/evals=1380/2134 eff=75.2454% N=300
Mono-modal Volume: ~exp(-8.93) * 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=-19.5(27.89%) | Like=-14.52..-13.30 [-14.5224..-14.5198]*| it/evals=1407/2174 eff=75.0800% N=300
Z=-19.5(28.22%) | Like=-14.50..-13.30 [-14.4996..-14.4899]*| it/evals=1410/2177 eff=75.1199% N=300
Z=-19.4(31.21%) | Like=-14.40..-13.30 [-14.3976..-14.3965]*| it/evals=1434/2216 eff=74.8434% N=300
Z=-19.4(32.04%) | Like=-14.37..-13.30 [-14.3680..-14.3661]*| it/evals=1440/2224 eff=74.8441% N=300
Z=-19.2(35.70%) | Like=-14.25..-13.30 [-14.2468..-14.2454]*| it/evals=1470/2263 eff=74.8854% N=300
Mono-modal Volume: ~exp(-9.13) * 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=-19.2(36.16%) | Like=-14.24..-13.30 [-14.2401..-14.2372]*| it/evals=1474/2268 eff=74.8984% N=300
Z=-19.1(39.74%) | Like=-14.16..-13.30 [-14.1650..-14.1648]*| it/evals=1500/2307 eff=74.7384% N=300
Z=-19.1(43.00%) | Like=-14.10..-13.30 [-14.0972..-14.0954]*| it/evals=1526/2347 eff=74.5481% N=300
Z=-19.0(43.47%) | Like=-14.08..-13.30 [-14.0812..-14.0797]*| it/evals=1530/2354 eff=74.4888% 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| +4.4e-11 ********** +6.3e-11 | +1.0e-10
Z=-19.0(44.77%) | Like=-14.05..-13.30 [-14.0512..-14.0476]*| it/evals=1541/2368 eff=74.5164% N=300
Z=-19.0(47.22%) | Like=-14.01..-13.30 [-14.0059..-14.0057]*| it/evals=1560/2396 eff=74.4275% N=300
Z=-18.9(50.75%) | Like=-13.97..-13.30 [-13.9665..-13.9654]*| it/evals=1590/2434 eff=74.5080% N=300
Mono-modal Volume: ~exp(-9.72) * 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=-18.9(52.86%) | Like=-13.94..-13.30 [-13.9386..-13.9357]*| it/evals=1608/2462 eff=74.3756% N=300
Z=-18.8(54.23%) | Like=-13.93..-13.30 [-13.9255..-13.9231]*| it/evals=1620/2478 eff=74.3802% N=300
Z=-18.8(57.41%) | Like=-13.85..-13.30 [-13.8528..-13.8506]*| it/evals=1650/2512 eff=74.5931% N=300
Mono-modal Volume: ~exp(-9.90) * 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=-18.7(59.99%) | Like=-13.81..-13.30 [-13.8147..-13.8133]*| it/evals=1675/2549 eff=74.4775% N=300
Z=-18.7(60.54%) | Like=-13.81..-13.30 [-13.8073..-13.8069]*| it/evals=1680/2554 eff=74.5342% N=300
Z=-18.7(63.35%) | Like=-13.76..-13.30 [-13.7646..-13.7646]*| it/evals=1708/2594 eff=74.4551% N=300
Z=-18.7(63.54%) | Like=-13.76..-13.30 [-13.7641..-13.7624]*| it/evals=1710/2598 eff=74.4125% N=300
Z=-18.6(66.35%) | Like=-13.73..-13.30 [-13.7270..-13.7259]*| it/evals=1740/2635 eff=74.5182% N=300
Mono-modal Volume: ~exp(-9.90) 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(68.71%) | Like=-13.69..-13.30 [-13.6891..-13.6825]*| it/evals=1767/2674 eff=74.4313% N=300
Z=-18.6(68.98%) | Like=-13.68..-13.30 [-13.6797..-13.6763]*| it/evals=1770/2677 eff=74.4636% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2692
[ultranest] logZ = -18.2 +- 0.08723
[ultranest] Effective samples strategy satisfied (ESS = 982.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.05 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:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.217 +- 0.342
single instance: logZ = -18.217 +- 0.115
bootstrapped : logZ = -18.200 +- 0.220
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.16 │ ▁▁▁▁▁▁▁▁▃▃▄▅▅▇▇▆▇▇▇▅▅▄▃▂▂▁▂▁▁▁▁▁▁ ▁ │3.49 2.75 +- 0.17
amplitude : 0.0000000000232│ ▁ ▁▁▁▁▁▁▂▂▅▅▆▇▇▇▆▆▄▅▄▃▃▂▁▂▁▁▁▁▁▁ ▁ ▁ │0.0000000000881 0.0000000000530 +- 0.0000000000078
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 40.508 seconds)
